# Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905–1910

###### Abstract

The law of gravitational attraction is a window on three formal approaches to laws of nature based on Lorentz-invariance: Poincaré’s four-dimensional vector space (1906), Minkowski’s matrix calculus and spacetime geometry (1908), and Sommerfeld’s 4-vector algebra (1910). In virtue of a common appeal to 4-vectors for the characterization of gravitational attraction, these three contributions track the emergence and early development of four-dimensional physics.

###### Contents

## Introduction

In July, 1905, Henri Poincaré (1854–1912) proposed two laws of gravitational attraction compatible with the principle of relativity and all astronomical observations explained by Newton’s law. Two years later, in the fall of 1907, Albert Einstein (1879–1955) began to investigate the consequences of the principle of equivalence for the behavior of light rays in a gravitational field (Einstein 1907). The following year, Hermann Minkowski (1864–1909), Einstein’s former mathematics instructor, borrowed Poincaré’s notion of a four-dimensional vector space for his new matrix calculus, in which he expressed a novel theory of the electrodynamics of moving media, a spacetime mechanics, and two laws of gravitational attraction. Following another two-year hiatus, Arnold Sommerfeld (1868–1951) characterized the relationship between the laws proposed by Poincaré and Minkowski, calling for this purpose both on spacetime diagrams and a new 4-vector formalism.

Of these four efforts to capture gravitation in a relativistic
framework, Einstein’s has attracted the lion’s share of attention, and
understandably so in hindsight, but at the expense of a full
understanding of what is arguably the most significant innovation in
contemporary mathematical physics: the four-dimensional approach to
laws of physics. In virtue of the common appeal made by Poincaré,
Minkowski, and Sommerfeld to four-dimensional vectors in their studies
of gravitational attraction, their respective contributions track the
evolving form of four-dimensional physics in the early days of
relativity theory.^{1}^{1}
1
In limiting the scope of this paper to the
methods applied by their authors to the problem of gravitation, four
contributions to four-dimensional physics are neglected: that of
Richard Hargreaves, based on integral invariants (1908),
two 4-vector systems due to Max Abraham (1910) and Gilbert Newton
Lewis (1910a), and Vladimir Varičak’s hyperbolic-function
based approach (1910). The objective of this paper
is to describe in terms of theorists’ intentions and peer readings the
emergence of a four-dimensional language for physics, as applied to
the geometric and symbolic expression of gravitational action.

The subject of gravitational action at the turn of the twentieth
century is well-suited for an investigation of this sort. This is not
to say that the reform of Newton’s law was a burning issue for
theorists. While several theories of gravitation claimed corroboration
on a par with that of classical Newtonian theory, contemporary
theoretical interest in gravitation as a research topic–including the
Lorentz-invariant variety–was sharply curtailed by the absence of
fresh empirical challenges to the inverse-square law. Rather, in
virtue of the stability of the empirical knowledge base, and two
centuries of research in celestial mechanics, the physics of
gravitation was a well-worked, stable terrain, familiar to physicists,
mathematicians and astronomers alike.^{2}^{2}
2
For an overview of
research on gravitation from 1850 to 1915, see
Roseveare (1982). On
early 20th-century investigations of gravitational
absorption, see
Andrade Martins (1999).
While only Lorentz-covariant
theories are considered in this paper, the relative acceptance of
the principle of relativity among theorists is understood as one
parameter among several influencing the development of
four-dimensional physics.

The leading theory of gravitation in 1905 was the one discovered by Isaac Newton over two centuries earlier, based on instantaneous action at a distance. When Poincaré sought to bring gravitational attraction within the purview of the principle of relativity, he saw it had to propagate with a velocity no greater than that of light in empty space, such that a reformulation of Newton’s law as a retarded action afforded a simple solution.

Newton’s law was the principal model for Poincaré, but it was not
the only one. With the success of Maxwell’s theory in explaining
electromagnetic phenomena (including the behavior of light) during the
latter third of the nineteenth century, theories of contiguous action
gained greater favor with physicists. In 1892, the Dutch theorist
H.-A. Lorentz produced a theory of mobile charged particles
interacting in an immobile ether, that was an habile synthesis of
Maxwell’s field theory and Wilhelm Weber’s particle theory of
electrodynamics. After the discovery of the electron in 1897, and
Lorentz’s elegant explanation of the Zeeman effect, certain charged
microscopic particles were understood to be electrons, and electrons
the building-blocks of matter.^{3}^{3}
3
Buchwald (1985, 242),
Darrigol (2000, 325),
Buchwald (2001).

In this new theoretical context of ether and electrons, Lorentz
derived the force on an electron moving in microscopic versions of
Maxwell’s electric and magnetic fields. To determine the
electromagnetic field of an electron in motion, Alfred Liénard and
Emil Wiechert derived a formula for a potential propagating with
finite velocity. In virtue of these two laws, both of which fell out
of a Lagrangian from Karl Schwarzschild, the theory of electrons
provided a means of calculating the force on a charged particle in
motion due to the fields of a second charged particle in
motion.^{4}^{4}
4
Lorentz took the force per unit charge on a volume
element of charged matter moving with velocity $\U0001d533$ in the
electric and magnetic fields $\U0001d521$ and $\U0001d525$ to be
$\U0001d523=\U0001d521+\frac{1}{c}[\U0001d533\cdot \U0001d525]$, where the brackets indicate a vector product
(Lorentz 1904c, 156–157). For a comparison of electrodynamic
Lagrangians from Maxwell to Schwarzschild, see
Darrigol (2000, App. 9).

An electron-based analogy to gravitational attraction of neutral mass
points was then close at hand. Lorentz’s electron theory was held in
high esteem by early twentieth-century theorists, including both
Poincaré and Minkowski, who naturally catered to the most promising
research program of the moment. They each proposed two force laws: one
based on retarded action at a distance, the other appealing directly
to contiguous action propagated in a medium. All four particle laws
were taken up in turn by Sommerfeld.^{5}^{5}
5
On the Maxwellian
approach to gravitation, see North (1965, chap. 3),
Roseveare (1982, 129–31), and
Norton (1992, 32). The
distinction drawn here between retarded action at a distance and
field representations reflects that of Lorentz
(1904b), for whom this was largely a matter of
convenience. On nineteenth-century conceptions of the
electromagnetic field, see
Cantor & Hodge (1981).

Several other writers have discussed Poincaré’s
and Minkowski’s work on gravitation. Of the first four substantial
synoptic reviews of the two theories, none
employed the notation of the original works, although this fact itself
reflects the rapid evolution of formal approaches in physics. Early
comparisons were carried out with either Sommerfeld’s 4-vector
formalism (Sommerfeld 1910b;
Kretschmann 1914), a relative coordinate notation
(De Sitter 1911),
or a mix of ordinary vector algebra and tensor calculus
(Kottler 1922). No further comparison studies were published after
1922, excepting one summary (North 1965, 49–50),
although since the 1960s, the work of Poincaré and Minkowski has
continued to incite historical interest.^{6}^{6}
6
On Poincaré’s theory
see
Cunningham (1914, 173),
Whitrow (1965, 20),
Harvey (1965, 452),
Cuvaj (1970, App. 5),
Schwartz (1972),
Zaher (1989, 192),
Torretti (1996, 132).
On Minkowski’s theory see
Weinstein (1914, 61),
Pyenson (1985a, 88),
Corry (1997, 287).
Sommerfeld’s contribution, while it inflected
theoretical practice in general, and contemporary reception
of Lorentz-covariant gravitation theory in particular, has been
neglected by historians.

The present study has three sections, beginning with Poincaré’s contribution, moving on in the second section to Minkowski’s initial response to Poincaré’s theory, and a review of his formalism and laws of gravitation. A third section is taken up by Sommerfeld’s interpretation of the laws proposed by Poincaré and Minkowski. The period of study is thus bracketed on one end by the discovery of special relativity in 1905, and on the other end by Sommerfeld’s paper. While the latter work did not spell the end of either 4-vector formalisms or Lorentz-covariant theories of gravitation, it was the first four-dimensional vector algebra, and represents a point of closure for a study of the emergence of a conceptual framework for four-dimensional physics.

## 1 Henri Poincaré’s Lorentz-invariant laws of gravitation

Poincaré’s memoir on the dynamics of the electron
(1906), like Einstein’s relativity paper (Einstein
1905),
contains the fundamental insight of the physical
significance of the group of Lorentz transformations, not only for
electrodynamics, but for all natural phenomena. The law of
gravitation, to no lesser extent than the laws of electrodynamics,
fell presumably within the purview of Einstein’s theory, but this is
not a point that Einstein, then working full time as a patent examiner
in Bern, chose to elaborate upon immediately. Poincaré, on the other
hand, as Professor of Mathematical Astronomy and Celestial Mechanics
at the Sorbonne, could hardly finesse the question of gravitation. In
particular, his address to the scientific congress at the St. Louis
World’s Fair, on 24 September, 1904, had pinpointed Laplace’s
calculation of the propagation velocity of gravitation as a potential
spoiler for the principle of relativity.^{7}^{7}
7
Laplace estimated
the propagation velocity of gravitation to be ${10}^{6}$ times that of
light, and Poincaré noted that such a signal velocity would allow
inertial observers to detect their motion with respect to the ether
(Poincaré 1904, 312).

There may have been another reason for Poincaré to investigate a
relativistic theory of gravitation. In the course of his study of
Lorentz’s contractile electron, Poincaré noted that the required
relations between electromagnetic energy and momentum were not
satisfied in general. Raised earlier by Max Abraham, the problem was
considered by Lorentz to be a fundamental one for his electron
theory.^{8}^{8}
8
Poincaré (1906, 153–154);
Miller (1973, 230–233). Following Abraham’s account
(Abraham 1905, 205), the problem may be presented in outline
as follows (using modified notation and units). Consider a
deformable massless sphere of radius $a$ and uniformly distributed
surface charge, and assume that this is a good model of the
electron. The longitudinal mass ${m}_{\parallel}$ of this sphere
may be defined as the quotient of external force and acceleration,
${m}_{\parallel}=d|\mathbf{G}|/d|\mathbf{v}|$, where $\mathbf{G}$ is
the electromagnetic momentum resulting from the electron’s
self-fields, and $\mathbf{v}$ is electron velocity. Defining the
electromagnetic momentum to be $\mathbf{G}=\int \mathbf{E}\times \mathbf{B}\mathit{d}V$, where $\mathbf{E}$ and $\mathbf{B}$ denote the electric
and magnetic self-fields, and $V$ is for volume, we let $c=1$, and
find the longitudinal mass for small velocities to be
${m}_{\parallel}=\frac{{e}^{2}}{6\pi a}{\left(1-{v}^{2}\right)}^{-3/2}$.
Longitudinal electron mass may also be defined in terms of the
electromagnetic energy W of the electron’s self-fields, assuming
quasistationary motion: ${m}_{\parallel}=\frac{1}{|\mathbf{v}|}\frac{dW}{d|\mathbf{v}|}$, where $W=\frac{{e}^{2}}{6\pi a}{\left(1-{v}^{2}\right)}^{-1/2}+\frac{{e}^{2}}{24\pi a}{\left(1-{v}^{2}\right)}^{1/2}$.
This leads, however, to an expression for longitudinal mass
different from the previous one: ${m}_{\parallel}=\frac{{e}^{2}}{6\pi a}\left[{\left(1-{v}^{2}\right)}^{-3/2}+\frac{1}{4}{\left(1-{v}^{2}\right)}^{-1/2}\right]$. From the difference in these two
expressions for longitudinal mass, Abraham concluded that the
Lorentz electron required the postulation of a non-electromagnetic
force and was thereby not compatible with a purely electromagnetic
foundation of physics.

Solving the stability problem of Lorentz’s contractile electron was a
trivial matter for Poincaré, as it meant transposing to electron
theory a special solution to a general problem he had treated earlier
at some length: to find the equilibrium form of a rotating fluid
mass.^{9}^{9}
9
See
Poincaré (1885,
1902a,
1902b).
In the limit of null angular velocity, gravitational attraction can be replaced by electrostatic
repulsion, with a sign reversal in the pressure gradient. He
postulated a non-electromagnetic, Lorentz-invariant “supplementary”
potential that exerts a binding (negative) pressure inside the
electron, and reduces the total energy of the electron in an amount
proportional to the volume decrease resulting from Lorentz
contraction. When combined with the electromagnetic field Lagrangian,
this binding potential yields a total Lagrangian invariant with
respect to the Lorentz group, as Poincaré required.

In accordance with the electromagnetic world-picture and the results of Kaufmann’s experiments, Poincaré supposed the inertia of matter to be exclusively of electromagnetic origin, and he set out, as he wrote in §6 of his paper,

to determine the total energy due to electron motion, the corresponding action, and the quantity of electromagnetic momentum, in order to calculate the electromagnetic masses of the electron.

Non-electromagnetic mass does not figure in this analysis, and
consequently, one would not expect the non-electromagnetic binding
potential to contribute to the tensorial electromagnetic mass of the electron,
although Poincaré did not state this in so many words. Instead,
immediately after obtaining an expression for the binding potential,
he derived the small-velocity, “experimental” mass from the
electromagnetic field Lagrangian alone, neglecting a contribution from
the binding potential. The mass of the slowly-moving Lorentz electron
was then equal to the electrostatic mass, just as one would want for
an electromagnetic foundation of mechanics. This fortuitous result,
which revised Lorentz’s electron mass value downward by a quarter,
was obtained independently by Einstein, using a method that did not
constrain electron structure.^{10}^{10}
10
Einstein (1905, 917).
Poincaré also neglected the mass contribution of the binding
potential in his 1906–1907 Sorbonne lectures, according to student
notes
(Poincaré 1953, 233). For reviews of Poincaré’s derivation of
the binding potential, see
Cuvaj (1970, App. 11) and
Miller (1973).
On post-Minkowskian interpretations of the binding
potential (also known as Poincaré pressure), see
Cuvaj (1970, 203),
Miller (1981, 382, n. 29), and
Yaghjian (1992).
Although the question of electron mass was far
from resolved, Poincaré had shown that the stability problem represented no
fundamental obstacle to the pursuit of a new mechanics based on the
concept of a contractile electron.

With this obstacle out of the way, Poincaré proceeded as if the laws
of mechanics were applicable to the experimental mass of the
electron.^{11}^{11}
11
In this paper Poincaré made no distinction between
inertial and gravitational mass. Noting that the negative pressure
deriving from his binding potential is proportional to the fourth
power of mass, and furthermore, that Newtonian attraction is itself
proportional to mass, Poincaré conjectured that

there is some relation between the cause giving rise to gravitation and that giving rise to the supplementary potential.

On the basis of a formal relation between experimental mass and the
binding potential, in other words, Poincaré predicted the unification
of his negative internal electron pressure with the gravitational
force, in a future theory encompassing all three forces.^{12}^{12}
12
As
Cuvaj (1968, 1112) points out, Poincaré may have
found inspiration for this conjecture in Paul Langevin’s remark that
gravitation stabilized the electron against Coulomb repulsion.
Unlike Langevin, Poincaré anticipated a unified theory of
gravitation and electrons, in the spirit of theories pursued later
by Gustav Mie, Gunnar Nordström, David Hilbert, Hans Reissner,
Hermann Weyl and Einstein; for an overview see
Vizgin (1994).

On this hopeful note, Poincaré began his memoir’s ninth and final section, entitled “Hypotheses concerning gravitation.” Lorentz’s theory, Poincaré explained, promised to account for the observed relativity of motion:

In this way Lorentz’s theory would fully explain the impossibility of detecting absolute motion, if all forces were of electromagnetic origin.

^{13}^{13}13 “Ainsi la théorie de Lorentz expliquerait complètement l’impossibilité de mettre en évidence le mouvement absolu, si toutes les forces étaient d’origine électromagnétique” (Poincaré 1906, 166).

The hypothesis of an electromagnetic origin of gravitational force had
been advanced by Lorentz at the turn of the
century. On the assumption that the force between
“ions” (later “electrons”) of unlike sign was of greater magnitude
at a given separation than that between ions of like sign (following
Mossotti’s conjecture), Lorentz represented gravitational attraction
as a field-theoretical phenomenon analogous to electromagnetism,
reducing to the Newtonian law for bodies at rest with respect to the
ether. Lorentz’s theory tacitly assumed negative energy density for
the “gravitational” field, and a gravitational ether of huge
intrinsic positive energy density, two well-known
sticking-points for
Maxwell.
Another difficulty stemmed from the
dependence of gravitational force on absolute velocities.^{14}^{14}
14
See
Lorentz (1900),
Havas (1979, 83),
Torretti (1996, 131). On
Lorentz’s precursors see
Whittaker (1951, 2:149) and
Zenneck (1903).
Lorentz’s theory of gravitation failed to convince
Oliver Heaviside, who had carefully weighed the analogy from
electromagnetism to gravitation (1893). In a letter to
Lorentz, Heaviside called into question the theory’s electromagnetic
nature, by characterizing Lorentz’s gravitational force as “action
at a distance of a double kind” (18 July, 1901, Lorentz Papers,
Rijksarchief in Noord-Holland te Haarlem). Aware of these
difficulties, Lorentz eventually discarded his theory, citing its
incompatibility with the principle of relativity
(Lorentz 1914, 32).

Neither Lorentz’s gravitation theory nor Maxwell’s sticking-points
were mentioned by Poincaré in the ninth section of his memoir.
Instead, he recalled a well-known empirical fact: two bodies that
generate identical electromagnetic fields need not exert the same
attraction on electrically neutral masses. Although Lorentz’s theory
clearly accounts for this fact, Poincaré concluded that the
gravitational field was distinct from the electromagnetic field. What
this tells us is that Poincaré’s attention was not focused on
Lorentz’s theory of gravitation.^{15}^{15}
15
In his
1906–1907 Sorbonne lectures (1953), Poincaré discussed a
different theory (based on an idea due to Le Sage) that Lorentz had
proposed in the same paper, without mentioning the Mossotti-style
theory. His first discussion of the latter theory was in 1908, when
he considered it to be an authentic relativistic theory, and one in
which the force of gravitation was of electromagnetic origin
(Poincaré 1908, 399).

To Poincaré’s way of thinking, it was the impossibility of an electromagnetic reduction of gravitation that had driven Lorentz to suppose that all forces transform like electromagnetic ones:

The gravitational field is therefore distinct from the electromagnetic field. Lorentz was obliged thereby to extend his hypothesis with the assumption that forces of any origin whatsoever, and gravitation in particular, are affected by a translation (or, if one prefers, by the Lorentz transformation) in the same manner as electromagnetic forces.

^{16}^{16}16 Poincaré (1906, 166). Poincaré’s account of Lorentz’s reasoning should be taken with a grain of salt, as Lorentz made no mention of his theory of gravitation in the 1904 publication referred to by Poincaré, “Electromagnetic phenomena in a system moving with any velocity less than that of light.” While the electron theory developed in the latter paper did not address the question of the origin of the gravitational force, it admitted the possibility of a reduction to electromagnetism (such as that of his own theory) by means of the additional hypothesis referred to in the quotation: all forces of interaction transformed in the same way as electric forces in an electrostatic system (Lorentz 1904a, § 8). The contraction hypothesis formerly invoked to account for the null result of the Michelson-Morley experiments, Lorentz added, was subsumed by the new hypothesis.

It was the cogency of the latter hypothesis that Poincaré set out to examine in detail, with respect to gravitational attraction. The situation was analogous to the one Poincaré had encountered in the case of electron energy and momentum mentioned above, where he had considered constraining internal forces of the electron to be Lorentz-invariant. Such a constraint solved the problem immediately, but Poincaré recognized that it was inadmissible nonetheless, because it violated Maxwell’s theory (p. 136). A similar violation in the realm of mechanics could not be ruled out in the case of gravitation, such that a careful analysis of the admissibility of the formal requirement of Lorentz-invariance was called for.

Poincaré set out to determine a general expression for the law of gravitation in accordance with the principle of relativity. A relativistic law of gravitation, he reasoned, must obey two constraints distinguishing it from the Newtonian law. First of all, the new force law could no longer depend solely on the masses of the two gravitating bodies and the distance between them. The force had to depend on their velocities, as well. Furthermore, gravitational action could no longer be considered instantaneous, but had to propagate with some finite velocity, so that the force acting on the passive mass would depend on the position and velocity of the active mass at some earlier instant in time. A gravitational propagation velocity greater than the speed of light, Poincaré observed, would be “difficult to understand,” because attraction would then be a function of a position in space not yet occupied by the active mass (p. 167).

These were not the only conditions Poincaré wanted to satisfy. The new law of gravitation had also (1) to behave in the same way as electromagnetic forces under a Lorentz transformation, (2) to reduce to Newton’s law in the case of relative rest of the two bodies, and (3) to come as close as possible to Newton’s law in the case of small velocities. Posed in this way, Poincaré noted, the problem remains indeterminate, save in the case of null relative velocity, where the propagation velocity of gravitation does not enter into consideration. Poincaré reasoned that if two bodies have a common rectilinear velocity, then the force on the passive mass is orthogonal to an ellipsoid, at the center of which lies the active mass.

Undeterred by the indeterminacy of the question in general, Poincaré set about identifying quantities invariant with respect to the Lorentz group, from which he wanted to construct a law of gravitation satisfying the constraints just mentioned. To assist in the identification and interpretation of these invariants, Poincaré referred to a space of four dimensions. “Let us regard,” he wrote,

$$x,y,z,t\sqrt{-1}$$ | ||

$$\delta x,\delta y,\delta z,\delta t\sqrt{-1}$$ | ||

$${\delta}_{1}x,{\delta}_{1}y,{\delta}_{1}z,{\delta}_{1}t\sqrt{-1},$$ |

as the coordinates of 3 points $P$, ${P}^{\prime}$, ${P}^{\prime \prime}$, in space of 4 dimensions. We see that the Lorentz transformation is merely a rotation in this space about the origin, regarded as fixed. Consequently, we will have no distinct invariants apart from the 6 distances between the 3 points $P$, ${P}^{\prime}$, ${P}^{\prime \prime}$, considered separately and with the origin, or, if one prefers, apart from the 2 expressions:

$${x}^{2}+{y}^{2}+{z}^{2}-{t}^{2},x\delta x+y\delta y+z\delta z-t\delta t,$$ or the 4 expressions of like form deduced by arbitrary permutation of the 3 points $P$, ${P}^{\prime}$, ${P}^{\prime \prime}$.

^{17}^{17}17 “Regardons $x$, $y$, $z$, $t\sqrt{-1}$, $\delta x$, $\delta y$, $\delta z$, $\delta t\sqrt{-1}$, ${\delta}_{1}x$, ${\delta}_{1}y$, ${\delta}_{1}z$, ${\delta}_{1}t\sqrt{-1}$, comme les coordonnées de 3 points $P$, ${P}^{\prime}$, ${P}^{\prime \prime}$ dans l’espace à 4 dimensions. Nous voyons que la transformation de Lorentz n’est qu’une rotation de cet espace autour de l’origine, regardée comme fixe. Nous n’aurons donc pas d’autres invariants distincts que les six distances des trois points $P$, ${P}^{\prime}$, ${P}^{\prime \prime}$ entre eux et à l’origine, ou, si l’on aime mieux, que les 2 expressions : ${x}^{2}+{y}^{2}+{z}^{2}-{t}^{2}$, $x\delta x+y\delta y+z\delta z-t\delta t$, ou les 4 expressions de même forme qu’on en déduit en permutant d’une manière quelconque les 3 points $P$, ${P}^{\prime}$, ${P}^{\prime \prime}$” (Poincaré 1906, 168–169).

Here Poincaré formed three quadruplets representing the differential
displacement of two point masses, with respect to a
certain four-dimensional vector space, later called a pseudo-Euclidean
space.^{18}^{18}
18
Poincaré’s three points $P,{P}^{\prime},{P}^{\prime \prime}$ may be interpreted
in modern terminology as follows. Let the spacetime coordinates of
the passive mass point be $A=({x}_{0},{y}_{0},{z}_{0},{t}_{0})$, with ordinary
velocity $\xi =(\delta x/\delta t,\delta y/\delta t,\delta z/\delta t)$, such that at time ${t}_{0}+\delta t$ it occupies the spacetime
point ${A}^{\prime}=({x}_{0}+\delta x,{y}_{0}+\delta y,{z}_{0}+\delta z,{t}_{0}+\delta t)$.
Likewise for the active mass point, $B=({x}_{0}+x,{y}_{0}+y,{z}_{0}+z,{t}_{0}+t)$,
with ordinary velocity ${\xi}_{1}=({\delta}_{1}x/{\delta}_{1}t,{\delta}_{1}y/{\delta}_{1}t,{\delta}_{1}z/{\delta}_{1}t)$, such that at time
${t}_{0}+t+{\delta}_{1}t$, it occupies the spacetime point
${B}^{\prime}=({x}_{0}+x+{\delta}_{1}x,{y}_{0}+y+{\delta}_{1}y,{z}_{0}+z+{\delta}_{1}z,{t}_{0}+t+{\delta}_{1}t)$. Poincaré’s three quadruplets may now be
expressed as position 4-vectors: $P=B-A$, ${P}^{\prime}={B}^{\prime}-B$,
${P}^{\prime \prime}={A}^{\prime}-A$. By introducing such a 4-space,
Poincaré simplified the task of identifying quantities invariant with
respect to the Lorentz transformations, the line interval of the new
space being formally identical to that of a Euclidean 4-space. He
treated his three points $P$, ${P}^{\prime}$, and ${P}^{\prime \prime}$ as 4-vectors, the scalar
products of which are invariant, just as in Euclidean space. In fact,
Poincaré did not employ vector terminology or notation in his study of
gravitation, but provided formal definitions of certain objects later
called 4-vectors.

Poincaré’s habit, and that of the overwhelming majority of his French
colleagues in mathematical physics well into the 1920s, was to express
ordinary vector quantities in Cartesian coordinate notation, and to
forgo notational shortcuts when differentiating, writing these
operations out in full.^{19}^{19}
19
While the first German textbook on
electromagnetism to employ vector notation systematically dates from
1894
(Föppl 1894), the first comparable textbook in French was
published two decades later by Jean-Baptiste Pomey (1861–1943),
instructor of theoretical electricity at the École supérieure
des Postes et Télégraphes in Paris (Pomey 1914).
Although he did not exclude symbols such as $\mathrm{\Delta}$ or $\mathrm{\square}$ from
his scientific papers and lectures, he employed them
parsimoniously.^{20}^{20}
20
The Laplacian was expressed generally as
${\nabla}^{2}={\partial}^{2}/\partial {x}^{2}+{\partial}^{2}/\partial {y}^{2}+{\partial}^{2}/\partial {z}^{2}$, but by Poincaré as $\mathrm{\Delta}$. The
d’Alembertian, $\mathrm{\square}\equiv {\partial}^{2}/\partial {x}^{2}+{\partial}^{2}/\partial {y}^{2}+{\partial}^{2}/\partial {z}^{2}-{\partial}^{2}/\partial {t}^{2}$, became in Poincaré’s notation: $\mathrm{\square}\equiv \mathrm{\Delta}-{d}^{2}/d{t}^{2}$.
Poincaré employed $\mathrm{\square}$ in his lectures on electricity and optics
(Poincaré 1901, 456), and was the first to employ it in a
relativistic context. In line with this practice,
Poincaré did little to promote vector methods from his chair at the
Sorbonne. In twenty volumes of lectures on mathematical physics and
celestial mechanics, there is not a single propadeutic on quaternions
or vector algebra.^{21}^{21}
21
Poincaré’s manuscript lecture notes for
celestial mechanics, however, show that he saw fit to introduce the
quaternionic method to his students (undated notebook on quaternions
and celestial mechanics, 32 pp., private collection, Paris;
hpcd76, 78, 93, Henri Poincaré Archives, Nancy).
Poincaré deplored the “long calculations rendered obscure by
notational complexity” in W. Voigt’s molecular theory of light, and
seems to have been of the opinion that in general, new notation only
burdened the reader.^{22}^{22}
22
Manuscript report of the Ph.D. thesis
submitted by Henri Bouasse, 13 December, 1892, AJ${}^{16}5535$,
Archives nationales, Paris. From Poincaré’s conservative habits
regarding formalism, he appears as an unlikely candidate at best for
the development of a four-dimensional calculus circa 1905;
cf. H. M. Schwartz’s counterfactual conjecture: if Poincaré had
adopted the ordinary vector calculus by the time he wrote his
Rendiconti paper, “he would have in all likelihood
introduced explicitly … the convenient four-dimensional vector
calculus” (1972, 1287, note 7).

The point of forming quadruplets was to obtain a set of Lorentz-invariants corresponding to the ten variables entering into the right-hand side of the new force law, representing the squared distance in space and time of the two bodies and their velocities ($\xi $, $\eta $, $\zeta $, ${\xi}_{1}$, ${\eta}_{1}$, ${\zeta}_{1}$). How did Poincaré obtain his invariants? According to the method cited above, six invariants were to be found from the distances between $P$, ${P}^{\prime}$, ${P}^{\prime \prime}$, and the origin, or from the scalar products of $P$, ${P}^{\prime}$, and ${P}^{\prime \prime}$. These six intermediate invariants were then to be combined to obtain homogeneous invariants depending on the duration of propagation of gravitational action and the velocities of the two point masses. Poincaré skipped over the intermediate step and produced the following four invariants, in terms of squared distance, distance and velocity (twice), and the velocity product:

$$\sum {x}^{2}-{t}^{2},\frac{t-\sum x\xi}{\sqrt{1-\sum {\xi}^{2}}},\frac{t-\sum x{\xi}_{1}}{\sqrt{1-\sum {\xi}_{1}^{2}}},\frac{1-\sum \xi {\xi}_{1}}{\sqrt{\left(1-\sum {\xi}^{2}\right)\left(1-\sum {\xi}_{1}^{2}\right)}}.$$ | (1) |

The Lorentz-invariance and geometric
significance of these quantities
are readily verified.^{23}^{23}
23
The invariants (1) may be expressed in ordinary vector notation, letting $\sum x=\mathbf{x}$, $\sum \xi =\mathbf{v}$, $\sum {\xi}_{1}={\mathbf{v}}_{1}$, and
for convenience, $k=1/\sqrt{1-\sum {\xi}^{2}}$, ${k}_{1}=1/\sqrt{1-\sum {\xi}_{1}^{2}}$,
such that the four quantities (1) read as follows:
${\mathbf{x}}^{2}-{t}^{2}$,
$k(t-\mathrm{\mathbf{x}\mathbf{v}})$,
${k}_{1}(t-{\mathrm{\mathbf{x}\mathbf{v}}}_{1})$,
$k{k}_{1}(1-{\mathrm{\mathbf{v}\mathbf{v}}}_{1})$.
These four invariants
(1), the latter three of which were labeled $A$, $B$, and
$C$, formed the core of Poincaré’s constructive approach to the law of
gravitation. (For convenience, I refer to Poincaré’s four invariants
(1) as his “kinematic” invariants.)

Inspection of the signs of these invariants reveals an inconsistency,
the reason for which is apparent once the intermediate calculations
have been performed. Instead of constructing his four invariants out
of scalar products, Poincaré introduced an inversion for $A$, $B$, and
$C$.^{24}^{24}
24
Poincaré’s four kinematic invariants (1) are
functions of the following six intermediate invariants: $a={x}^{2}+{y}^{2}+{z}^{2}-{t}^{2}$, $b=x\delta x+y\delta y+z\delta z-t\delta t$, $c=x{\delta}_{1}x+y{\delta}_{1}y+z{\delta}_{1}z-t{\delta}_{1}t$, $d=\delta x{\delta}_{1}x+\delta y{\delta}_{1}y+\delta z{\delta}_{1}z-\delta t{\delta}_{1}t$, $e=\delta {x}^{2}+\delta {y}^{2}+\delta {z}^{2}-\delta {t}^{2}$, $f={\delta}_{1}{x}^{2}+{\delta}_{1}{y}^{2}+{\delta}_{1}{z}^{2}-{\delta}_{1}{t}^{2}$. In terms of the latter six invariants, the four
kinematic invariants (1) may be expressed as follows: $\sum {x}^{2}-{t}^{2}=a$, $A=-b/\sqrt{-e}$, $B=-c/\sqrt{-f}$, and
$C=-d/(\sqrt{-e}\sqrt{-f})$. For a slightly different reconstruction
of Poincaré’s kinematic invariants, see
Zahar (1989, 193). This sign inconsistency had no
consequence on his search for a relativistic law of
gravitation, although it affected his final result, and perplexed at
least one of his readers, as I will show in §3.

What Poincaré needed next for his force law was a Lorentz-invariant expression for the force itself. Up to this point, he had neither a velocity 4-vector nor a force 4-vector definition on hand. Presumably, the search for Lorentz-invariant expressions of force led him to define these 4-vectors. Earlier in his memoir (p. 135), Poincaré had determined the Lorentz transformations of force density, but now he was interested in the Lorentz transformations of force at a point. The transformations of force density:

$${X}^{\prime}=k(X+\epsilon T),{Y}^{\prime}=Y,{Z}^{\prime}=Z,{T}^{\prime}=k(T+\epsilon X),$$ | (2) |

where $k$ is the Lorentz factor,
$k=1/\sqrt{1-{\epsilon}^{2}}$, and $\epsilon $ designates frame velocity,
led Poincaré to define a fourth component of force density, $T$, as the
product of the force density vector with velocity, $T=\sum X\xi $.^{25}^{25}
25
This definition was remarked by
Pauli ((Pauli, 1921, 637)).
He gave the same definition for the temporal component of force
at a point: ${T}_{1}=\sum {X}_{1}\xi $.^{26}^{26}
26
The same
subscript denotes the force acting on the passive
mass, $\sum {X}_{1}$, and the velocity of the active mass,
${\xi}_{1}$. Next, dividing force density by force at a point, Poincaré
obtained the charge density $\rho $. Ostensibly from the transformation
for charge density, Poincaré singled out the Lorentz-invariant factor:^{27}^{27}
27
The ratio $\rho /{\rho}^{\prime}$ is equal
to the Lorentz factor, since in Poincaré’s configuration,
$\epsilon =-\xi $. Some writers hastily attribute a 4-current
vector to Poincaré, the form $\rho (\xi $, $\eta $,
$\zeta $, $i)$ being implied by Poincaré’s 4-vector definitions of
force density and velocity.

$$\frac{\rho}{{\rho}^{\prime}}=\frac{1}{k(1+\xi \epsilon )}=\frac{\delta t}{\delta {t}^{\prime}}.$$ | (3) |

The components of a 4-velocity vector followed from the foregoing definitions of position and force density:

The Lorentz transformation … will act in the same way on $\xi $, $\eta $, $\zeta $, $1$ as on $\delta x$, $\delta y$, $\delta z$, $\delta t$, with the difference that these expressions will be multiplied moreover by the same factor $\delta t/\delta {t}^{\prime}=1/k(1+\xi \epsilon )$.

^{28}^{28}28 “La transformation de Lorentz … agira sur $\xi $, $\eta $, $\zeta $, $1$ de la même manière que sur $\delta x$, $\delta y$, $\delta z$, $\delta t$, avec cette différence que ces expressions seront en outre multipliées par le même facteur $\delta t/\delta {t}^{\prime}=1/k(1+\xi \epsilon )$” (Poincaré 1906, 169).

Concerning the latter definition, Poincaré observed a formal analogy between the force and force density 4-vectors, on one hand, and the position and velocity 4-vectors, on the other hand: these pairs of vectors transform in the same way, except that one member is multiplied by $1/k(1+\xi \epsilon )$. While this analogy may seem mathematically transparent, it merits notice, as it appears to have eluded Poincaré at first.

With these four kinematic 4-vectors in hand, Poincaré defined a fifth
quadruplet $Q$ with components of force density $(X,Y,Z,T\sqrt{-1})$. Just as in the previous case, the scalar products of his four quadruplets $P$, ${P}^{\prime}$, ${P}^{\prime \prime}$, and $Q$ were to
deliver four new Lorentz-invariants in terms of the force acting on
the passive mass $({X}_{1},{Y}_{1},{Z}_{1})$:^{29}^{29}
29
The invariants (4)
may be expressed in ordinary vector notation, recalling the
definitions of note 23, and letting
$\sum {X}_{1}={\mathbf{f}}_{1}$, and
${T}_{1}={\mathbf{f}}_{1}\mathbf{v}$:
${k}^{2}{\mathbf{f}}_{1}^{2}(1-{\mathbf{v}}^{2})$,
$k{\mathbf{f}}_{1}(\mathbf{x}-\mathbf{v}t)$,
$k{k}_{1}{\mathbf{f}}_{1}({\mathbf{v}}_{1}-\mathbf{v})$,
${k}^{2}{\mathbf{f}}_{1}(\mathbf{v}-\mathbf{v})$. The fourth
invariant is obviously null in this form.

$$\frac{\sum {X}_{1}^{2}-{T}_{1}^{2}}{1-\sum {\xi}^{2}},\frac{\sum {X}_{1}x-{T}_{1}t}{\sqrt{1-\sum {\xi}^{2}}},\frac{\sum {X}_{1}{\xi}_{1}-{T}_{1}}{\sqrt{1-\sum {\xi}^{2}}\sqrt{1-\sum {\xi}_{1}^{2}}},\frac{\sum {X}_{1}\xi -{T}_{1}}{1-\sum {\xi}^{2}}.$$ | (4) |

The fourth invariant in (4) was always null by definition of ${T}_{1}$, leaving only three invariants, denoted $M$, $N$, and $P$. (In order to distinguish these invariants from the kinematic invariants, I will refer to (4) as Poincaré’s “force” invariants.)

Comparing the signs of the kinematic invariants (1) with
those of the force invariants (4), we see that Poincaré
obtained consistent signs only for the latter invariants. He must not
have computed his force invariants in the same way as his kinematic
invariants, for reasons that remain obscure.
It is not entirely unlikely that in the course of his analysis of the
transformations of velocity and force, Poincaré realized that he could
compute the force invariants directly from the scalar products of four
4-vectors. Two facts, however, argue against this reading. In the
first place, Poincaré did not mention that his force invariants
were the scalar products of position, velocity and force
4-vectors. Secondly, he did not alter the signs of his kinematic
invariants to make them correspond to scalar products of
position and velocity 4-vectors.^{30}^{30}
30
Poincaré’s force
invariants (4) are functions of the following six
intermediate invariants: $m=k({X}_{1}\delta x+{Y}_{1}\delta y+{Z}_{1}\delta z-{T}_{1}\delta t)$, $n=k({X}_{1}{\delta}_{1}x+{Y}_{1}{\delta}_{1}y+{Z}_{1}{\delta}_{1}z-{T}_{1}{\delta}_{1}t)$, $o=k({X}_{1}x+{Y}_{1}y+{Z}_{1}z-{T}_{1}t)$,
$p={k}^{2}({X}_{1}^{2}+{Y}_{1}^{2}+{Z}_{1}^{2}-{T}_{1}^{2})$, $q=\delta {x}^{2}+\delta {y}^{2}+\delta {z}^{2}-\delta {t}^{2}$, and $s={\delta}_{1}{x}^{2}+{\delta}_{1}{y}^{2}+{\delta}_{1}{z}^{2}-{\delta}_{1}{t}^{2}$. Let
the four force invariants (4) be denoted by $M$, $N$, $P$,
and $S$, then $M=p$, $N=o$, $P=n/\sqrt{-s}$, and $S=m/\sqrt{-q}$.
The same force invariants (4) are easily calculated using
4-vectors. Recalling the definitions in
note 23 and
note 29,
let $\Re =(\mathbf{x},\text{i}t)$, $U=k(\mathbf{v},\text{i})$,
${U}_{1}={k}_{1}({\mathbf{v}}_{1},\text{i})$, and ${F}_{1}=k({\mathbf{f}}_{1},\text{i}{\mathbf{f}}_{1}\mathbf{v})$, where $\sqrt{-1}=\text{i}$. Then the force
invariants (4) may be expressed as scalar products of
4-vectors: $M={F}_{1}{F}_{1}$, $N={F}_{1}\Re $, $P={F}_{1}{U}_{1}$, and
$S={F}_{1}U$.
The fact that Poincaré’s kinematic invariants
differ from products of 4-position and 4-velocity vectors leads us to
believe that when forming these invariants he was not thinking
in terms of 4-vectors.^{31}^{31}
31
The kinematic invariants (1)
obtained by Poincaré differ from those obtained from the products of
4-position and 4-velocity, contrary to Zahar’s account
(Zahar 1989, 194). Recalling the 4-vectors $\Re $, $U$,
${U}_{1}$ from note 30, we form the products:
$\Re \Re $, $\Re U$, $\Re {U}_{1}$, and $U{U}_{1}$.
In Poincaré’s notation, the latter four
products are expressed as follows:
$$\sum {x}^{2}-{t}^{2},-\frac{t-\sum x\xi}{\sqrt{1-\sum {\xi}^{2}}},-\frac{t-\sum x{\xi}_{1}}{\sqrt{1-\sum {\xi}_{1}^{2}}},-\frac{1-\sum \xi {\xi}_{1}}{\sqrt{\left(1-\sum {\xi}^{2}\right)\left(1-\sum {\xi}_{1}^{2}\right)}}.$$
These invariants differ from those of Poincaré (1) only by
the sign of $A$, $B$, and $C$, as noted by
Sommerfeld (1910b, 686).

From this point on, Poincaré worked exclusively with arithmetic combinations of three force invariants ($M$, $N$, $P$) and four kinematic invariants ($\sum {x}^{2}-{t}^{2}$, $A$, $B$, $C$) in order to come up with a relativistic law of gravitation. He had no further use, in particular, for the four quadruplets he had identified in the process of constructing these same invariants (corresponding to modern 4-position, 4-velocity, 4-force-density and 4-force vectors), although in the end he expressed his laws of gravitation in terms of 4-force components.

To find a law applicable to the general case of two bodies in relative
motion, Poincaré introduced constraints and approximations designed to
reduce the complexity of his seven invariants and recover the form of
the Newtonian law in the limit of slow motion $({\xi}_{1}\ll 1)$.
Poincaré naturally looked first to the velocity of propagation of
gravitation. He briefly considered an emission theory, where the
velocity of gravitation depends on the velocity of the source.
Although the emission hypothesis was compatible with his invariants,
Poincaré rejected this option because it violated his initial
injunction barring a hyperlight velocity of gravitational
propagation.^{32}^{32}
32
An emission theory was proposed a few years
later by Walter Ritz; see Ritz (1908). That left him with a
propagation velocity of gravitation less than or equal to that of
light, and to simplify his invariants Poincaré set it equal to that of
light in empty space, such that $t=-\sqrt{\sum {x}^{2}}=-r$. This stipulation reduced the total
number of invariants from seven to six.

With the propagation velocity of gravitation decided, Poincaré proceeded to construct a force law for point masses. He tried two approaches, the first of which is the most general. The basic idea of both approaches is to neglect terms in the square of velocity occurring in the invariants, and to compare the resulting approximations with their Newtonian counterparts. In the Newtonian scheme, the coordinates of the active mass point differ from those in the relativistic scheme (cf. note 18); Poincaré took the former to be $({x}_{0}+{x}_{1},{y}_{0}+{y}_{1},{z}_{0}+{z}_{1})$ at the instant of time ${t}_{0}$, where the subscript 0 corresponds to the position of the passive mass point, and the coordinates with subscript 1 are found by assuming uniform motion of the source:

$$x={x}_{1}-{\xi}_{1}r,y={y}_{1}-{\eta}_{1}r,z={z}_{1}-{\zeta}_{1}r,r={r}_{1}-\sum x{\xi}_{1}.$$ | (5) |

In the first approach, Poincaré made use of both the kinematic and
force invariants. Substituting the values (5) into the
kinematic invariants $A$, $B$, and $C$ from (1) and the force
invariants $M$, $N$, and $P$ from (4), neglecting terms in
the square of velocity, Poincaré obtained their sought-after Newtonian
counterparts. Replacing the force vector occurring in the transformed
force invariants by Newton’s law $\left(\sum {X}_{1}=-1/{r}_{1}^{2}\right)$, and
rearranging, Poincaré obtained three quantities in terms of distance
and velocity.^{33}^{33}
33
Using (5), Poincaré found the
transformed force invariants $1/{r}_{1}^{4}$, $-1/{r}_{1}-\sum {x}_{1}(\xi -{\xi}_{1})/{r}_{1}^{2}$, and $\sum {x}_{1}(\xi -{\xi}_{1})/{r}_{1}^{3}$. He then
re-expressed these quantities in terms of two of his original kinematic
invariants, $A$ and $B$, and equated the three resulting kinematic
invariants to their corresponding original force invariants
(4). He now had the solution in hand; three expressions
relate his force invariants (containing the force vector $\sum {X}_{1}$)
to two of his kinematic invariants:

$$M=\frac{1}{{B}^{4}},N=\frac{+A}{{B}^{2}},P=\frac{A-B}{{B}^{3}}.$$ | (6) |

He noted that complementary terms could be entertained for the three relations (6), provided that they were certain functions of his kinematic invariants $A$, $B$, and $C$. Then without warning, he cut short his demonstration, remarking that the gravitational force components would take on imaginary values:

The solution (6) appears at first to be the simplest, nonetheless, it may not be adopted. In fact, since $M$, $N$, $P$ are functions of ${X}_{1}$, ${Y}_{1}$, ${Z}_{1}$, and ${T}_{1}=\mathrm{\Sigma}{X}_{1}\xi $, the values of ${X}_{1}$, ${Y}_{1}$, ${Z}_{1}$ can be drawn from these three equations (6), but in certain cases these values would become imaginary.

^{34}^{34}34 “Au premier abord, la solution (6) paraît la plus simple, elle ne peut néanmoins être adoptée; en effet, comme $M$, $N$, $P$ sont des fonctions de ${X}_{1}$, ${Y}_{1}$, ${Z}_{1}$, et de ${T}_{1}=\mathrm{\Sigma}{X}_{1}\xi $, on peut tirer de ces trois équations (6) les valeurs de ${X}_{1}$, ${Y}_{1}$, ${Z}_{1}$; mais dans certains cas ces valeurs deviendraient imaginaires” (Poincaré 1906, 172).

The quoted remark seems to suggest that for selected values of the
particle velocities, the force turns out to be imaginary. However, the
real difficulty springs from the equation $M=1/{B}^{4}$, which allows for
a repulsive force. The general approach failed to
deliver.^{35}^{35}
35
Replacing $A$ and $B$ in
(6) by their definitions results in the three equations:
$M={k}^{2}{\mathbf{f}}_{1}^{2}(1-{\mathbf{v}}^{2})=1/{k}^{4}{(r+{\mathrm{\mathbf{x}\mathbf{v}}}_{1})}^{4}$,
$N={\mathbf{f}}_{1}(\mathbf{x}+\mathbf{v}r)=-(r+\mathrm{\mathbf{x}\mathbf{v}})/[{k}_{1}^{2}{(r+{\mathrm{\mathbf{x}\mathbf{v}}}_{1})}^{2}]$,
$P=k{k}_{1}{\mathbf{f}}_{1}({\mathbf{v}}_{1}-\mathbf{v})=[k(r+\mathrm{\mathbf{x}\mathbf{v}})-{k}_{1}(r+{\mathrm{\mathbf{x}\mathbf{v}}}_{1})]/[{k}_{1}^{3}{(r+{\mathrm{\mathbf{x}\mathbf{v}}}_{1})}^{3}]$. Equations
$N$ and $P$ imply an attractive force for all values of $\mathbf{v}$
and ${\mathbf{v}}_{1}$, while $M$ leads to the ambiguously-signed
solution: ${\mathbf{f}}_{1}=\pm 1/[{k}^{2}{(r+{\mathrm{\mathbf{x}\mathbf{v}}}_{1})}^{2}]$.
Presumably, the superfluous plus sign in (6) is an
indication of Poincaré’s preoccupation with obtaining a force of
correct sign.

The fact that Poincaré published the preceding derivation may be understood in one of two ways. On the one hand, there is a psychological explanation: Poincaré’s habit, much deplored by his peers, was to present his findings more or less in the order in which he found them. The case at hand may be no different from the others. On the other hand, Poincaré may have felt it worthwhile to show that the general approach breaks down. From the latter point of view, Poincaré’s result is a positive one.

For his second attack on the law of gravitation, Poincaré adopted a
less general approach. He knew where his first approach had become
unsuitable, and consequently, leaving aside his three force
invariants, he fell back on the form of his basic force 4-vector,
which he now wrote in terms of his kinematic invariants, re-expressed
in terms of $r=-t$, ${k}_{0}=1/\sqrt{1-{\xi}^{2}}$, and
${k}_{1}=1/\sqrt{1-{\xi}_{1}^{2}}$.^{36}^{36}
36
$A=-{k}_{0}(r+\sum x\xi )$, $B=-{k}_{1}(r+\sum x{\xi}_{1})$, and $C={k}_{0}{k}_{1}(1-\sum x\xi {\xi}_{1})$. He assumed the
gravitational force on the passive mass (moving with velocity $\xi $,
$\eta $, $\zeta $) to be a function of the distance separating the two
mass points, the velocity of the passive mass point, and the velocity
of the source, with the form:

$$\begin{array}{cc}\hfill {X}_{1}& =x\frac{\alpha}{{k}_{0}}+\xi \beta +{\xi}_{1}\frac{{k}_{1}}{{k}_{0}}\gamma ,\hfill \\ \hfill {Y}_{1}& =y\frac{\alpha}{{k}_{0}}+\eta \beta +{\eta}_{1}\frac{{k}_{1}}{{k}_{0}}\gamma ,\hfill \end{array}\mathit{\hspace{1em}\hspace{1em}}\begin{array}{cc}\hfill {Z}_{1}& =z\frac{\alpha}{{k}_{0}}+\zeta \beta +{\zeta}_{1}\frac{{k}_{1}}{{k}_{0}}\gamma ,\hfill \\ \hfill {T}_{1}& =-r\frac{\alpha}{{k}_{0}}+\beta +\frac{{k}_{1}}{{k}_{0}}\gamma ,\hfill \end{array}$$ | (7) |

where $\alpha $, $\beta $, and $\gamma $ denote functions of the
kinematic invariants.^{37}^{37}
37
Using modern 4-vector notation, and
denoting Poincaré’s gravitational force 4-vector
${F}_{1}={k}_{0}({X}_{1},{Y}_{1},{Z}_{1},i{T}_{1})$, equation (7) may be expressed:
${F}_{1}=\alpha \Re +\beta U+\gamma {U}_{1}$, where
$\Re $ denotes a lightlike 4-vector between the mass points,
$\alpha $, $\beta $, $\gamma $ stand for undetermined functions of the
three kinematic invariants $A$, $B$, and $C$, while $U={k}_{0}(\mathbf{v},i)$, ${U}_{1}={k}_{1}({\mathbf{v}}_{1},i)$ designate the 4-velocities of the
passive and active mass points, respectively. By definition, the
component ${T}_{1}$ is the scalar product of the ordinary force and the
velocity of the passive mass point, ${T}_{1}=\sum {X}_{1}\xi $, such that the
three functions $\alpha $, $\beta $, $\gamma $ satisfy the equation:

$$-A\alpha -\beta -C\gamma =0.$$ | (8) |

Poincaré further assumed $\beta =0$, thereby eliminating a term
depending on the velocity of the passive mass, and fixing the value of
$\gamma $ in terms of $\alpha $. Applying the same slow-motion
approximation and translation (5) as in his
initial approach, Poincaré found ${X}_{1}=\alpha {x}_{1}$, and by comparison
with Newton’s law, $\alpha $ reduces to $-1/{r}_{1}^{3}$. In
terms of the kinematic invariants (1), this relation was
expressed as $\alpha =1/{B}^{3}$, and the law of gravitation (7)
took on the form:^{38}^{38}
38
In ordinary vector form, recalling the
definitions in note 23 and note 29, the spatial part of
Poincaré’s law is expressed as follows:
${\mathbf{f}}_{1}=-[(\mathbf{x}+r{\mathbf{v}}_{1})+\mathbf{v}\times ({\mathbf{v}}_{1}\times \mathbf{x})]/[k{k}_{1}^{3}{(r+{\mathrm{\mathbf{x}\mathbf{v}}}_{1})}^{3}(1-{\mathrm{\mathbf{v}\mathbf{v}}}_{1})]$.
Cf. Zahar ((Zahar, 1989, 199)).

$$\begin{array}{cc}\hfill {X}_{1}& =\frac{x}{{k}_{0}{B}^{3}}-{\xi}_{1}\frac{{k}_{1}}{{k}_{0}}\frac{A}{{B}^{3}C},\hfill \\ \hfill {Y}_{1}& =\frac{y}{{k}_{0}{B}^{3}}-{\eta}_{1}\frac{{k}_{1}}{{k}_{0}}\frac{A}{{B}^{3}C},\hfill \end{array}\mathit{\hspace{1em}\hspace{1em}}\begin{array}{cc}\hfill {Z}_{1}& =\frac{z}{{k}_{0}{B}^{3}}-{\zeta}_{1}\frac{{k}_{1}}{{k}_{0}}\frac{A}{{B}^{3}C},\hfill \\ \hfill {T}_{1}& =-\frac{r}{{k}_{0}{B}^{3}}-\frac{{k}_{1}}{{k}_{0}}\frac{A}{{B}^{3}C}.\hfill \end{array}$$ | (9) |

Inspection of Poincaré’s gravitational force (9) reveals two components: one parallel to the position 4-vector between the passive mass and the retarded source, and one parallel to the source 4-velocity. The law was not unique, Poincaré noted, and it neglected possible terms in the velocity of the passive mass.

Poincaré underlined the open-ended nature of his solution by proposing
a second gravitational force law. Rearranging (9) and
replacing the factor $1/{B}^{3}$ by $C/{B}^{3}$, such that the force depended
linearly on the velocity of the passive mass, Poincaré arrived at a
second law of gravitation:^{39}^{39}
39
This law may be reformulated using
the vectors defined in
note 23 and
note 29, and
neglecting (with Poincaré) the component ${T}_{1}$:
${\mathbf{f}}_{1}=-[(\mathbf{x}+r{\mathbf{v}}_{1})+\mathbf{v}\times ({\mathbf{v}}_{1}\times \mathbf{x})]/[{k}_{1}^{2}{(r+{\mathrm{\mathbf{x}\mathbf{v}}}_{1})}^{3}]$.
Cf. Zahar (1989, 199). Comparable expressions
were developed by
Lorentz (1910, 1239) and
Kottler (1922, 169).

${X}_{1}$ | $={\displaystyle \frac{\lambda}{{B}^{3}}}+{\displaystyle \frac{\eta {\nu}^{\prime}-\zeta {\mu}^{\prime}}{{B}^{3}}},$ | (10) | ||

${Y}_{1}$ | $={\displaystyle \frac{\mu}{{B}^{3}}}+{\displaystyle \frac{\zeta {\lambda}^{\prime}-\xi {\nu}^{\prime}}{{B}^{3}}},$ | |||

${Z}_{1}$ | $={\displaystyle \frac{\nu}{{B}^{3}}}+{\displaystyle \frac{\xi {\mu}^{\prime}-\eta {\lambda}^{\prime}}{{B}^{3}}},$ |

where

$$\begin{array}{cc}\hfill {k}_{1}(x+r{\xi}_{1})& =\lambda ,\hfill \\ \hfill {k}_{1}({\eta}_{1}z-{\zeta}_{1}y)& ={\lambda}^{\prime},\hfill \end{array}\mathit{\hspace{1em}}\begin{array}{cc}\hfill {k}_{1}(y+r{\eta}_{1})& =\mu ,\hfill \\ \hfill {k}_{1}({\zeta}_{1}x-{\xi}_{1}z)& ={\mu}^{\prime},\hfill \end{array}\mathit{\hspace{1em}}\begin{array}{cc}\hfill {k}_{1}(z+r{\zeta}_{1})& =\nu ,\hfill \\ \hfill {k}_{1}({\xi}_{1}y-x{\eta}_{1})& ={\nu}^{\prime}.\hfill \end{array}$$ |

Poincaré neglected to write down the expression for ${T}_{1}$, probably because of its complicated form. (For the sake of simplicity, I refer to (9) and (10) including the latter’s neglected fourth component, as Poincaré’s first and second law.) The unprimed triplet ${B}^{-3}(\lambda ,\mu ,\nu )$ supports what Poincaré termed a “vague analogy” with the mechanical force on a charged particle due to an electric field, while the primed triplet ${B}^{-3}({\lambda}^{\prime},{\mu}^{\prime},{\nu}^{\prime})$ supports an analogy to the mechanical force on a charged particle due to a magnetic field. He identified the fields as follows:

Now $\lambda $, $\mu $, $\nu $, or $\frac{\lambda}{{B}^{3}}$, $\frac{\mu}{{B}^{3}}$, $\frac{\nu}{{B}^{3}}$, is an electric field of sorts, while ${\lambda}^{\prime}$, ${\mu}^{\prime}$, ${\nu}^{\prime}$, or rather $\frac{{\lambda}^{\prime}}{{B}^{3}}$, $\frac{{\mu}^{\prime}}{{B}^{3}}$, $\frac{{\nu}^{\prime}}{{B}^{3}}$, is a magnetic field of sorts.

^{40}^{40}40 “Alors $\lambda $, $\mu $, $\nu $, ou $\lambda /{B}^{3}$, $\mu /{B}^{3}$, $\nu /{B}^{3}$, est une espèce de champ électrique, tandis que ${\lambda}^{\prime}$, ${\mu}^{\prime}$, ${\nu}^{\prime}$, ou plutôt ${\lambda}^{\prime}/{B}^{3}$, ${\mu}^{\prime}/{B}^{3}$, ${\nu}^{\prime}/{B}^{3}$, est une espèce de champ magnétique” (Poincaré 1906, 175).

While Poincaré wrote freely of a “gravity wave” (onde
gravifique), he abstained from speculating on the nature of the
field referred to here. As one of the first
theorists (with FitzGerald and Lorentz) to have employed retarded
potentials in Maxwellian electrodynamics, Poincaré must have
considered the possibility of introducing a corresponding
gravitational 4-potential.^{41}^{41}
41
Whittaker
(1951, 1:394, note 3).
A 4-potential corresponding to Poincaré’s second law
(10) was given by
Kottler (1922, 169). Additional
assumptions are required in order to identify a “gravito-magnetic” field
with a term arising from the Lorentz transformation of force:
$\mathbf{v}\times ({\mathbf{v}}_{1}\times \mathbf{x})$, or the second
term of the 3-vector version of (10) (neglecting the global
factor; see note 39). In particular, it must be assumed
that when the sources of the “gravito-electric” field
${B}^{-3}(\lambda ,\mu ,\nu )$ are at rest, the force on a mass
point $m$ is $\mathbf{f}=m{B}^{-3}(\lambda ,\mu ,\nu )$,
independent of the velocity of $m$. For a detailed discussion, see
Jackson (1975, 578).
But as matters stood when Poincaré submitted
this paper for publication in July, 1905, he was not in a position to
elaborate the physics of fields in four-dimensional terms, since he
possessed neither a 4-potential nor a 6-vector.

Poincaré had realized the objective of formulating a Lorentz-invariant
force of gravitation. As we have seen, he surpassed this objective by
identifying not one but two such force laws. Designed to reduce to
Newton’s law in the first order of approximation in ${\xi}_{1}$ (or particle
velocity divided by the speed of light), Poincaré’s laws could diverge from
Newton’s only in second-order terms. The argument satisfied Poincaré,
who did not report any precise numerical results, explaining that this
would require further investigation. Instead, he noted that the
disagreement would be ten thousand times smaller than a first-order
difference stemming from the assumption of a propagation velocity of
gravitation equal to that of light, “ceteris non mutatis”
(p. 175). His result contradicted Laplace, who had predicted an
observable first-order effect arising from just such an assumption. At
the very least, Poincaré had demonstrated that Laplace’s argument was
not compelling in the context of the new dynamics.^{42}^{42}
42
Poincaré
reviewed Laplace’s argument in his 1906–1907 lectures
(Poincaré 1953, 194). For a contemporary overview of the
question of the propagation velocity of gravitation see
Tisserand (1889, 511).

On several occasions over the next seven years, Poincaré returned to
the question of gravitation and relativity, without ever comparing the
predictions of his laws with observation. During his 1906–1907
Sorbonne lectures, for example, when he developed a general formula
for perihelion advance, Poincaré used a Lagrangian approach, rather
than one or the other of his laws (Poincaré 1953, 238). Student notes
of this course indicate that he stopped short of a numerical
evaluation for the various electron models (perhaps leaving this as an
exercise). However, Poincaré later provided the relevant numbers in a
general review of electron theory.
Lorentz’s theory called for an extra
7" centennial advance by Mercury’s perihelion, a figure
slightly greater than the one for Abraham’s non-relativistic electron
theory.^{43}^{43}
43
Fritz Wacker, a student of Richard Gans in Tübingen,
published similar results in 1906. According to the
best available data, Mercury’s anomalous perihelial advance was
42", prompting Poincaré to remark that another explanation
would have to be found in order to account for the remaining seconds
of arc. Astronomical observations, Poincaré concluded soberly,
provided no arguments in favor of the new electron
dynamics.^{44}^{44}
44
Poincaré (1908, 400).
Poincaré explained to his
students that Mercury’s anomalous advance could plausibly be
attributed to an intra-Mercurial matter belt
(Poincaré 1953, 265),
an idea advanced forcefully by Hugo von Seeliger in 1906
(Roseveare 1982, 78).
In a lecture delivered in
September, 1909, Poincaré revised his estimate of the relativistic
perihelial advance downward slightly to 6"
(Poincaré 1909).

Poincaré capsulized the situation of his new theory in a fable in which Lorentz plays the role of Ptolemy, and Poincaré that of an unknown astronomer appearing sometime between Ptolemy and Copernicus. The unknown astronomer notices that all the planets traverse either an epicycle or a deferent in the same lapse of time, a regularity later captured in Kepler’s second law. The analogy to electron dynamics turns on a regularity discovered by Poincaré in his study of gravitation:

If we were to admit the postulate of relativity, we would find the same number in the law of gravitation and the laws of electromagnetism, which would be the velocity of light; and we would find it again in all the other forces of any origin whatsoever.

^{45}^{45}45 “[S]i nous admettions le postulat de relativité, nous trouverions dans la loi de gravitation et dans les lois électromagnétiques un nombre commun qui serait la vitesse de la lumière; et nous le retrouverions encore dans toutes les autres forces d’origine quelconque” (Poincaré 1906, 131).

This common propagation velocity of gravitational action, of electromagnetic fields, and of any other force, could be understood in one of two ways:

Either everything in the universe would be of electromagnetic origin, or this aspect–shared, as it were, by all physical phenomena–would be a mere epiphenomenon, something due to our methods of measurement.

^{46}^{46}46 “Ou bien il n’y aurait rien au monde qui ne fût d’origine électromagnétique. Ou bien cette partie qui serait pour ainsi dire commune à tous les phénomènes physiques ne serait qu’une apparence, quelque chose qui tiendrait à nos méthodes de mesure” (Poincaré 1906, 131–132).

If the electromagnetic worldview were valid, all particle interactions would be governed by Maxwell’s equations, featuring a constant propagation velocity. Otherwise, the common propagation velocity of forces had to be a result of a measurement convention. In relativity theory, as Poincaré went on to point out, the measurement convention to adopt was one defining lengths as equal if and only if spanned by a light signal in the same lapse of time, as this convention was compatible with the Lorentz contraction. There was a choice to be made between the electromagnetic worldview (as realized in the electron models of Abraham and Bucherer-Langevin) and the postulate of relativity (as upheld by the Lorentz-Poincaré electron theory). Although Poincaré favored the latter theory, he felt that its destiny was to be superseded, just as Ptolemaic astronomy was superseded by Copernican heliocentrism.

The failure of his Lorentz-invariant law of gravitation to explain the
anomalous advance of Mercury’s perihelion probably fed Poincaré’s
dissatisfaction with the Lorentz-Poincaré theory in general, but what
he found particularly troubling at the time was something else
altogether: the discovery of magneto-cathode rays. There is no place
in the Lorentz-Poincaré electron theory for rays that are both neutral
(as Paul Villard reported in June, 1904) and deflected by electric and
magnetic fields, which is probably why Poincaré felt the “entire
theory” to be “endangered” by magneto-cathode
rays.^{47}^{47}
47
Poincaré (1906, 132),
Stein (1987, 397, note 29).
On the history of magneto-cathode rays, see
Carazza & Kragh (1990).

Uncertainty over the empirical adequacy of the Lorentz-Poincaré electron theory may explain why the Rendiconti memoir was Poincaré’s last in the field of electron physics. But is it enough to explain his disinterest in the development of a four-dimensional formalism? One year after the publication of his article on electron dynamics, Poincaré commented:

A translation of our physics into the language of four-dimensional geometry does in fact appear to be possible; the pursuit of this translation would entail great pain for limited profit, and I will just cite Hertz’s mechanics, where we see something analogous. Meanwhile, it seems that the translation would remain less simple than the text and would always have the feel of a translation, and that three-dimensional language seems the best suited to the description of our world, even if one admits that this description may be carried out in another idiom.

^{48}^{48}48 “Il semble bien en effet qu’il serait possible de traduire notre physique dans le langage de la géométrie à quatre dimensions; tenter cette traduction ce serait se donner beaucoup de mal pour peu de profit, et je me bornerai à citer la mécanique de Hertz où l’on voit quelque chose d’analogue. Cependant, il semble que la traduction serait toujours moins simple que le texte, et qu’elle aurait toujours l’air d’une traduction, que la langue des trois dimensions semble la mieux appropriée à la description de notre monde, encore que cette description puisse se faire à la rigueur dans un autre idiome” (Poincaré 1907, 15). See also Walter (1999b, 98), and for a different translation, Galison (1979, 95). On Hertz’s mechanics, see Lützen (1999).

Poincaré clearly saw in his own work the outline of a four-dimensional formalism for physics, yet he saw no future in its development, and this, entirely apart from the question of the empirical adequacy of the Lorentz-Poincaré theory.

Why did Poincaré discount the value of a language tailor-made for relativity? Three sources of disinterest in such a prospect spring to mind, the first of which stems from his conventionalist philosophy of science. Poincaré recognized an important role for notation in the exact sciences, as he famously remarked with respect to Edmond Laguerre’s work on quadratic forms and Abelian functions that

in the mathematical sciences, having the right notation is philosophically as important as having the right classification in the life sciences.

^{49}^{49}49 “[D]ans les Sciences mathématiques, une bonne notation a la même importance philosophique qu’une bonne classification dans les Sciences naturelles” (Poincaré 1898, x).

More than likely, Poincaré was aware of the philosophical implications
of a four-dimensional notation for physics, although he had yet to
make his views public. But given his strong belief in the immanence of
Euclidean geometry’s fitness for physics, he must have considered the
chances for success of such a language to be vanishingly
small.^{50}^{50}
50
Poincaré’s analysis of the concepts of space and time
in relativity theory appeared in 1912 (Poincaré 1912). On the cool
reception among mathematicians of Poincaré’s views on physical
geometry, see Walter (1997).

A second source for Poincaré’s disinterest in four-dimensional formalism is his practice of physics. As mentioned above, Poincaré dispensed with vectorial systems (and most notational shortcuts); he even avoided writing “i” for $\sqrt{-1}$. When considered in conjunction with his conventionalist belief in the suitability of Euclidean geometry for physics, this conservative habit with respect to notation makes Poincaré appear all the less likely to embrace a four-dimensional language for physics.

The third possible source of discontent is Poincaré’s vexing experience with invariants of pseudo-Euclidean 4-space. As shown above, Poincaré’s first approach to the construction of a law of gravitation ended unsatisfactorily, and the failure of Poincaré’s intuition in this instance may well have colored his view of the prospects for a four-dimensional physics.

An immediate consequence of Poincaré’s refusal to work out the form of
four-dimensional physics was that others could readily pick up where
he left off. Roberto Marcolongo (1862–1945), Professor of
Mathematical Physics in Messina, and a leading proponent of vectorial
analysis, quickly discerned in Poincaré’s paper a potential for formal
development. Marcolongo referred, like Poincaré, to a four-dimensional
space with one imaginary axis, but defined the fourth coordinate as
the product of time $t$ and the negative square root of $-1$ (i.e.,
$-t\sqrt{-1}$ instead of $t\sqrt{-1}$). After
forming a 4-vector potential out of the ordinary vector and scalar
potentials, and defining a 4-current vector, he expressed the
Lorentz-covariance of the equations of electrodynamics in matrix form.
No other applications were forthcoming from Marcolongo, and a failure
to produce further 4-vector quantities and functions limited the scope
of his contribution, which went unnoticed outside of
Italy.^{51}^{51}
51
Marcolongo (1906). This paper later gave rise to a
priority claim for a slightly different substitution: $u=\text{i}t$
(Marcolongo to Arnold Sommerfeld, 5 May, 1913, Archives for History
of Quantum Physics 32). On Marcolongo’s paper see also
Maltese (2000, 135). Nothing further on Poincaré’s method
appeared in print until April, 1908, when Hermann Minkowski’s paper on
the four-dimensional formalism and its application to the problem of
gravitation appeared in the Göttinger Nachrichten.

## 2 Hermann Minkowski’s spacetime laws of gravitation

The young Hermann Minkowski, fifth child of an immigrant family of
Russian Jews, attended the Altstädtische Gymnasium in Königsberg
(later Kaliningrad). Shortly after graduation, Minkowski submitted an
essay for the Paris Academy’s 1882 Grand Prize in Mathematical
Sciences. His entry on quadratic forms shared top honors with a
submission by the seasoned British mathematician Henry J. S. Smith,
his senior by thirty-eight
years.^{52}^{52}
52
Rüdenberg (1973),
Serre (1993),
Strobl (1985). The young
mathematician went on to study with Heinrich Weber in Königsberg, and
with Karl Weierstrass and Leopold Kronecker in Berlin. In the years
following the prize competition, Minkowski became acquainted with
Poincaré’s writings on algebraic number theory and quadratic forms,
and in particular, with a paper in Crelle’s Journal containing
some of the results from Minkowski’s prize paper, still in press. To
his friend David Hilbert he confided the “angst and alarm” brought
on by Poincaré’s entry into his field of predilection; with his
“swift and versatile” energy, Poincaré was bound to bring the whole
field to closure, or so it seemed to him at the
time.^{53}^{53}
53
Minkowski to Hilbert, 14 February, 1885, Rüdenberg &
Zassenhaus (1973, 30).
Minkowski’s fears turned out to be for naught, as Poincaré pursued a
different line of research
(Zassenhaus 1975, 446). On Minkowski’s
early work on the geometry of numbers see
Schwermer (1991); on later
developments, see
Krätzel (1989).
From the earliest, formative
years of his scientific career, Minkowski found in Poincaré–his
senior by a decade–a daunting intellectual rival.

While Minkowski had discovered in Poincaré a rival, he was soon to
find that that the Frenchman could also be a teacher, from whom he
could learn new analytical skills and methods. Named Privatdozent in
Bonn in 1887, Minkowski contributed to the abstract journal Jahrbuch der Fortschritte der Mathematik, and in 1892, took on the
considerable task of summarizing the results of the paper for which
Poincaré was awarded the King Oscar II Prize (Minkowski 1890). The
mathematics Poincaré created in his prize paper (the study of
homoclinic points in particular) was highly innovative, and at the
same time, difficult to follow. Among those whom we know had trouble
understanding certain points of Poincaré’s prize memoir were Charles
Hermite, Gustav Mittag-Leffler, and Karl Weierstrass, who happened to
constitute the prize committee.^{54}^{54}
54
See Gray (1992) and the
reception study by Barrow-Green (1997, chap. 6). Minkowski,
however, welcomed the review as a learning opportunity, as he wrote to his
friend and former teacher, Adolf Hurwitz:

Poincaré’s prize paper is also among the works I have to report on for the Fortschritte. I am quite fond of it. It is a fine opportunity for me to get acquainted with problems I have not worried about too much up to now, since I will naturally set a positive goal of making my case well.

^{55}^{55}55 Minkowski to Hurwitz, 5 January, 1892, Cod. Ms. Math. Arch. 78: 188, Handschriftenabteilung, Niedersächsische Staats- und Universitätsbibliothek (NSUB). On Minkowski’s report see also Barrow-Green (1997, 143).

In the 1890s, building on his investigations of the algebraic theory
of quadratic forms, Minkowski developed the geometric analog to this
theory: geometrical number theory. A high point of his efforts in this
new field, and one which contributed strongly to the establishment of
his reputation in mathematical circles, was the publication of Geometrie der Zahlen (1896). The same year,
Minkowski accepted a chair at Zurich Polytechnic, whereby he rejoined
Hurwitz. Minkowski’s lectures on mathematics and mathematical physics
attracted a small following of talented and ambitious students,
including the future physicists Walter Ritz and Albert Einstein, and
the budding mathematicians Marcel Grossmann and Louis
Kollros.^{56}^{56}
56
Minkowski papers, Arc. 4° 1712, Jewish National and
University Library (JNUL); Minkowski to Hilbert, 11 March, 1901,
Rüdenberg & Zassenhaus (1973, 139).

Minkowski’s lectures on mechanics in Zurich throw an interesting light
on his view of symbolic methods in physics at the close of the
nineteenth century. The theory of quaternions, he noted in 1897, was
used nowhere outside of England, due to its “relatively abstract
character and inherent difficulty.”^{57}^{57}
57
Vorlesungen über
analytische Mechanik, Wintersemester 1897/98, p. 29, Minkowski
papers, Arc. 4° 1712, JNUL. Two of its fundamental
concepts, scalars and vectors, had nevertheless gained broad approval
among physicists, Minkowski wrote, and had found “frequent
application especially in the theory of
electricity.”^{58}^{58}
58
Loc. cit. note 57.
The concepts of
scalar and vector mentioned by Minkowski were those introduced by W.
R. Hamilton (1805–1865), the founder of quaternion theory. Even in
Britain, vectors were judged superior to quaternions for use in
physics, giving rise to spirited exchanges in the pages of
Nature during the 1890s, as noted by Bork (1966) and
Crowe (1967, chap. 6). On the introduction of vector analysis
as a standard tool of the physicist during this period, see
Jungnickel & McCormmach (1986, 2:342),
and for a general history, see
Crowe (1967). Applications of quaternions to problems of
physics were advanced in Germany with the publication of Felix Klein
and Arnold Sommerfeld’s Theorie des Kreisels, a work referred
to in Minkowski’s lecture notes of
1898–1899.^{59}^{59}
59
Klein (1897); Vorlesungen über Mechanik,
Wintersemester 1898/99, 47, 59, Minkowski papers, Arc. 4° 1712,
JNUL. Minkowski referred to Klein and Sommerfeld’s text in relation
to the concept of force and its anthropomorphic origins, the kinetic
theory of gas, and the theory of elasticity. Minkowski admired
Klein and Sommerfeld’s text, expressing “great interest” in the
latter to Sommerfeld, along with his approval of the fundamental
significance accorded to the concept of momentum. However, their text
did not make the required reading list for Minkowski’s course in
mechanics.^{60}^{60}
60
Minkowski to Sommerfeld, 30 October,
1898, MSS 1013A, Special Collections, National Museum of American
History. An extensive reading list of mechanics texts is found in
Minkowski’s course notes for the 1903–1904 winter semester,
Mechanik I, 9, Minkowski papers, Arc. 4° 1712, JNUL.

In 1899, at the request of Sommerfeld, who a year earlier had agreed
to edit the physics volumes of Felix Klein’s ambitious Encyclopedia
of the Mathematical Sciences including Applications (hereafter
Encyklopädie), Minkowski agreed to cover a topic in molecular
physics he knew little about, but one perfectly suited to his skills
as an analyst: capillarity.^{61}^{61}
61
Minkowski to Sommerfeld, 30
October, 1898, loc. cit. note 60; Minkowski to Sommerfeld,
18 November, 1899, Nachlass Sommerfeld, Arch HS1977-28/A, 233,
Deutsches Museum München; research notebook, 12 December, 1899,
Arc. 4° 1712, Minkowski papers, JNUL. The article that appeared
seven years later represented his second contribution to physics,
after a short note on theoretical hydrodynamics published in 1888, but
which, ten years later, Minkowski claimed no one had read–save
the abstracter.^{62}^{62}
62
Minkowski (1888,
1907); Minkowski to
Sommerfeld, 30 October, 1898, loc. cit. note 60.

When Minkowski accepted Göttingen’s newly-created third chair of pure
mathematics in the fall of 1902, the pace of his research changed
brusquely. The University of Göttingen at the turn of the last century
was a magnet for talented young mathematicians and
physicists.^{63}^{63}
63
On Göttingen’s rise to preeminence in these
fields, see Manegold (1970),
Pyenson (1985b, chap. 7), and
Rowe (1989,
1992). Minkowski soon was immersed in the
activities of Göttingen’s Royal Society of Science, its mathematical
society, and research seminars. Several faculty members, including
Max Abraham, Gustav Herglotz, Eduard Riecke, Karl Schwarzschild, and
Emil Wiechert, actively pursued theoretical or experimental
investigations motivated by the theory of electrons, and it was not
long before Minkowski, too, took up the theory. During the summer
semester of 1905 he co-led a seminar with Hilbert on electron theory,
featuring reports by Wiechert and Herglotz, and by Max Laue, who had
just finished a doctoral thesis under Max Planck’s
supervision.^{64}^{64}
64
Nachlass Hilbert 570/9, Handschriftenabteilung,
NSUB; Pyenson (1985b, chap. 5).

Along with seminars on advanced topics in physics and analytical
mechanics, Göttingen featured a lively mathematical society, with
weekly meetings devoted to presentations of work-in-progress and
reports on scientific activity outside of Göttingen. The electron
theory was a frequent topic of discussion in this venue. For instance,
the problem of gravitational attraction was first addressed by
Schwarzschild in December, 1904, in a report on Alexander Wilkens’
recent paper on the compatibility of Lorentz’s electron theory with
astronomical observations.^{65}^{65}
65
Jahresbericht der deutschen
Mathematiker-Vereinigung 14, 61.

A focal point of sorts for the mathematical society, Poincaré’s
scientific output fascinated Göttingen scientists in general, and
Minkowski in particular, as mentioned above.^{66}^{66}
66
Although
Poincaré spoke on celestial mechanics in Göttingen in 1895
(Rowe 1992, 475), and was invited back in 1902, he did not
return until 1909, a few months after Minkowski’s sudden death. See
Hilbert to Poincaré, 6 November, 1908 (Dugac 1986, 209); Klein
to Poincaré, 14 Jan., 1902 (Dugac 1989, 124–125). Sponsored by
the Wolfskehl Fund, Poincaré’s 1909 lecture series took place during
“Poincaré week”, in the month of April. His lectures were
published the following year (Poincaré 1910) in a collection
launched in 1907, based on an idea of Minkowski’s
(Klein 1907, IV).
Minkowski reported to the mathematical
society on Poincaré’s publications on topology, automorphic functions,
and capillarity, devoting three talks in 1905–1906 to Poincaré’s
1888–1889 Sorbonne lectures on this subject
(Poincaré 1895). Others
reporting on Poincaré’s work were Conrad Müller on Poincaré’s St. Louis
lecture on the current state and future of mathematical physics (31
January, 1905), Hugo Broggi on probability (27 October, 1905), Ernst
Zermelo on a boundary-value problem (12 December, 1905), Erhard
Schmidt on the theory of differential equations (19 December, 1905),
Max Abraham on the Sorbonne lectures (6 February, 1906) and Paul Koebe
on the uniformization theorem (19 November, 1907). One gathers from
this list that the Göttingen mathematical society paid attention to
Poincaré’s contributions to celestial mechanics, mathematical physics,
and pure mathematics, all subjects intersecting with the ongoing
research of its members. It also appears that no other member of the
mathematical society was quite as assiduous in this respect as
Minkowski.^{67}^{67}
67
Jahresbericht der deutschen
Mathematiker-Vereinigung 14:128, 586; 15:154–155; 17:5.

When Einstein’s relativity paper appeared in late September, 1905, it
drew the attention of the Bonn experimentalist Walter Kaufmann, a
former Göttingen Privatdozent and friend of Max Abraham, but neither
Abraham nor any of his colleagues rushed to report on the new ideas to
the mathematical society.^{68}^{68}
68
On Kaufmann’s cathode-ray
deflection experiments, see Miller (1981, 226) and
Hon (1995). Readings of Kaufmann’s articles are discussed at
length by Richard Staley (1998, 270). Poincaré’s
long memoir on the dynamics of the electron, published in January,
1906, fared better, although nearly two years went by before Minkowski
found an occasion to comment on Poincaré’s gravitation theory, and to
present his own related work-in-progress. Minkowski’s typescript has
been conserved, and is the source referred to here.^{69}^{69}
69
Undated
typescript of a lecture on a new form of the equations of
electrodynamics, Math. Archiv 60:3, Handschriftenabteilung, NSUB.
This typescript differs significantly from the
posthumously-published version (1915).

On the occasion of the 5 November, 1907, meeting of the mathematical society,
Minkowski began his review of Poincaré’s work by observing that
gravitation remained an “important question” in relativity theory,
since it was not yet known “how the law of gravitation is arranged
for in the realm of the principle of
relativity.”^{70}^{70}
70
“Es entsteht die grosse
Frage, wie sich denn das Gravitationsgesetz in das Reich des
Relativitätsprinzipes einordnen lässt”
(p. 15). The basic problem of gravitation
and relativity, in other words, had not been solved by Poincaré.
Eliding mention of Poincaré’s two laws, Minkowski recognized in his
work only one positive result: by considering gravitational attraction
as a “pure mathematical problem,” he said, Poincaré had found
gravitation to propagate with the speed of light, thereby overturning
the standard Laplacian argument to the contrary.^{71}^{71}
71
Actually,
Poincaré postulated the lightlike propagation velocity of
gravitation, as mentioned above.

Minkowski expressed dissatisfaction with Poincaré’s
approach, allowing that Poincaré’s was “only one of many” possible
laws, a fact stemming from its construction out of Lorentz-invariants.
Consequently, Poincaré’s investigation “had by no means a definitive
character.”^{72}^{72}
72
“Poincaré weist ein
solches Gesetz auf, indem er auf die Betrachtung von Invarianten der
Lorentzschen Gruppe eingeht, doch ist das Gesetz nur eines unter
vielen möglichen, und die betreffenden Untersuchungen tragen in
keiner Weise einen definitiven Charakter”
(p. 16).
See also Pyenson (1973, 233).
A critical remark of this sort often introduces an alternative theory,
but in this instance none was forthcoming, and as I will show in what
follows, there is ample reason to doubt that Minkowski was actually in
a position to improve on Poincaré’s investigation. Nonetheless, at the
end of his talk Minkowski set forth the possibility of elaborating his
report.

Minkowski’s lecture was not devoted entirely to Poincaré’s
investigation of Lorentz-invariant gravitation. The purpose of his
lecture, according to the published abstract, was to present a new
form of the equations of electrodynamics leading to a mathematical
redescription of physical laws in four areas: electricity, matter,
mechanics, and gravitation.^{73}^{73}
73
Jahresbericht der deutschen
Mathematiker-Vereinigung 17 (1908), Mitt. u. Nachr., 4–5.
These laws were to be expressed in terms of the differential equations
used by Lorentz as the foundation of his successful theory of
electrons (1904a), but in a form taking greater advantage of
the invariance of the quadratic form ${x}^{2}+{y}^{2}+{z}^{2}-{c}^{2}{t}^{2}$. Physical
laws, Minkowski stated, were to be expressed with respect to a
four-dimensional manifold, with coordinates ${x}_{1}$, ${x}_{2}$, ${x}_{3}$,
${x}_{4}$, where units were chosen such that $c=1$, the ordinary Cartesian
coordinates $x$, $y$, and $z$, went over into the first three, and the
fourth was defined to be an imaginary time coordinate, ${x}_{4}=it$.
Implicitly, then, Minkowski took as his starting point the
four-dimensional vector space described in the last section of Poincaré’s
memoir on the dynamics of the electron.

Minkowski acknowledged, albeit obliquely, a certain continuity between Poincaré’s memoir and his own program to reform the laws of physics in four-dimensional terms. By formulating the electromagnetic field equations in four-dimensional notation, Minkowski said he was revealing a symmetry not realized by his predecessors, not even by Poincaré himself (Walter 1999b, 98). While Poincaré had not sought to modify the standard form of Maxwell’s equations, Minkowski felt it was time for a change. The advantage of expressing Maxwell’s equations in the new notation, Minkowski informed his Göttingen colleagues, was that they were then “easier to grasp” (p. 11).

His reformulation naturally began in the electromagnetic domain, with an expression for the potentials. He formed a 4-vector potential denoted ($\psi $) by taking the ordinary vector potential over for the first three components, and setting the fourth component equal to the product of $i$ and the scalar potential. The same method was applied to obtain a four-component quantity for current density: for the first three components, Minkowski took over the convection current density vector, $\varrho \U0001d534$, or charge density times velocity, and defined the fourth component to be the product of $i$ and the charge density. Rewriting the potential and current density vectors in this way, Minkowski imposed what is now known as the Lorenz condition, $\mathrm{Div}(\psi )=0$, where $\mathrm{Div}$ is an extension of ordinary divergence. This led him to the following expression, summarizing two of the four Maxwell equations:

$$\mathrm{\square}{\psi}_{j}=-{\varrho}_{j}\mathit{\hspace{1em}}(j=1,\mathrm{\hspace{0.25em}2},\mathrm{\hspace{0.25em}3},\mathrm{\hspace{0.25em}4}),$$ | (11) |

where $\mathrm{\square}$ is the d’Alembertian, employed earlier by Poincaré (cf. note 20).

Of the formal innovations presented by Minkowski to the mathematical
society, the most remarkable was what he called a Traktor, a
six-component entity used to represent the electromagnetic
field.^{74}^{74}
74
The same term was employed by Cayley to denote a line
which meets any given lines, in a paper of 1869. He defined the six
components via the 4-vector potential, using a two-index notation:
${\psi}_{jk}=\partial {\psi}_{k}/\partial {x}_{j}-\partial {\psi}_{j}/\partial {x}_{k}$, noting the antisymmetry relation
${\psi}_{kj}=-{\psi}_{jk}$, and zeros along the diagonal ${\psi}_{jj}=0$. In this way, the Traktor components ${\psi}_{14}$, ${\psi}_{24}$,
${\psi}_{34}$, ${\psi}_{23}$, ${\psi}_{31}$, ${\psi}_{12}$ match up with the
field quantities $-i{\U0001d508}_{x}$, $-i{\U0001d508}_{y}$,
$-i{\U0001d508}_{z}$, ${\U0001d525}_{x}$, ${\U0001d525}_{y}$, ${\U0001d525}_{z}$.^{75}^{75}
75
When written out in full, one obtains, for example,
${\psi}_{23}=\partial {\psi}_{3}/\partial {x}_{2}-\partial {\psi}_{2}/\partial {x}_{2}={\U0001d525}_{x}$. Minkowski later renamed the
Traktor a Raum-Zeit-Vektor II. Art
(Minkowski 1908, § 5), but it is better known as either a
6-vector, an antisymmetric 6-tensor, or an antisymmetric,
second-rank tensor. As the suite of synonyms suggests, this object
found frequent service in covariant formulations of
electrodynamics.

The Traktor first found application when Minkowski turned to his
second topic: the four-dimensional view of matter. Ignoring the
electron theories of matter of Lorentz and Joseph Larmor, Minkowski
focused uniquely on the macroscopic electrodynamics of moving
media.^{76}^{76}
76
For a comparison of the Lorentz and Larmor theories,
see Darrigol (1994). For this subject he introduced a
“Polarisationstraktor”, $(p)$, along with a 4-current-density,
$(\sigma )$, defined by the current density vector $\mathbf{i}$ and the
charge density $\varrho $: $(\sigma )={i}_{x},{i}_{y},{i}_{z},\text{i}\varrho $
(typescript, p. 9).
Recalling (11), Minkowski wrote Maxwell’s source equations in
covariant form:

$$\frac{\partial {p}_{1j}}{\partial {x}_{1}}+\frac{\partial {p}_{2j}}{\partial {x}_{2}}+\frac{\partial {p}_{3j}}{\partial {x}_{3}}+\frac{\partial {p}_{4j}}{\partial {x}_{4}}={\sigma}_{j}-{\varrho}_{j}.$$ | (12) |

Minkowski’s relativistic extension of Maxwell’s theory was all the
simpler in that it elided the covariant expression of the
constitutive equations, which involves 4-velocity.^{77}^{77}
77
On the
four-dimensional transcription of Ohm’s law see
Arzeliès & Henry (1959, 65–67). While none of his formulas invoked
4-velocity, Minkowski acknowledged that his theory required a
“velocity vector of matter $(w)={w}_{1},{w}_{2},{w}_{3},{w}_{4}$” (typescript, p. 10).

In order to express the “visible velocity of matter in any location,” Minkowski needed a new vector as a function of the coordinates $x$, $y$, $z$, $t$ (typescript, p. 7). Had he understood Poincaré’s 4-velocity definition, he undoubtedly would have employed it at this point. Instead, following the same method of generalization from three to four components successfully applied in the case of 4-vector potential, 4-current density, and 4-force density, Minkowski took over the components of the velocity vector $\U0001d534$ for the spatial elements of the quadruplet designated ${w}_{1}$, ${w}_{2}$, ${w}_{3}$, ${w}_{4}$:

$${\U0001d534}_{x},{\U0001d534}_{y},{\U0001d534}_{z},\text{i}\sqrt{1-{\U0001d534}^{2}}.$$ | (13) |

There are two curious aspects to Minkowski’s definition. First of all,
its squared magnitude does not vanish when ordinary velocity vanishes;
even a particle at rest with respect to a reference frame is described
in that frame by a 4-velocity vector of nonzero length. This is also
true of Poincaré’s 4-velocity definition, and is a feature of
relativistic kinematics. Secondly, the components of Minkowski’s
quadruplet do not transform like the coordinates ${x}_{1}$, ${x}_{2}$, ${x}_{3}$,
${x}_{4}$, and consequently lack what he knew to be an essential
property of a 4-vector.^{78}^{78}
78
Minkowski mentions this very
property on p. 6.

The most likely source for Minkowski’s blunder is Poincaré’s paper. We
recall that Poincaré’s derivation of his kinematic invariants
ignored 4-vectors, and what is more, his paper
features a misleading misprint, according to which the spatial part of
a 4-velocity vector is given to be the ordinary velocity
vector.^{79}^{79}
79
The passage in question may be translated as follows:
“Next we consider $X$, $Y$, $Z$, $T\sqrt{-1}$, as the coordinates
of a fourth point $Q$; the invariants will then be functions of the
mutual distances of the five points $O$, $P$, ${P}^{\prime}$, ${P}^{\prime \prime}$, $Q$, and among
these functions we must retain only those that are 0th degree
homogeneous with respect, on one hand, to $X$, $Y$, $Z$, $T$, $\delta x$, $\delta y$, $\delta z$, $\delta t$ (variables that can be further
replaced by ${X}_{1}$, ${Y}_{1}$, ${Z}_{1}$, ${T}_{1}$, $\xi $, $\eta $, $\zeta $, 1),
and on the other hand, with respect to ${\delta}_{1}x$, ${\delta}_{1}y$,
${\delta}_{1}z$, 1 (variables that can be further replaced by ${\xi}_{1}$,
${\eta}_{1}$, ${\zeta}_{1}$, 1)”
(Poincaré 1906, 170).
The misprint is in
the next-to-last set of variables, where instead of 1 we should have
${\delta}_{1}t$. Other sources of error can easily be imagined, of
course.^{80}^{80}
80
One other obvious source for Minkowski’s error is
Lorentz’s transformation of charge density: ${\varrho}^{\prime}=\varrho /\beta {l}^{3}$,
where $1/\beta =\sqrt{1-{v}^{2}/{c}^{2}}$, and $l$ is a constant later set
to unity (Lorentz 1904a, 813),
although this formula was carefully corrected by Poincaré.
It is strange that Minkowski did not check
the transformation properties of his 4-velocity definition, but given
its provenance, he probably had no reason to doubt its
soundness.

Minkowski’s mistake strongly suggests that at the time of his lecture,
he did not yet conceive of particle motion in terms of a worldline
parameter. Such an approach to particle motion would undoubtedly have
spared Minkowski the error, since it renders trivial the definition of
4-velocity.^{81}^{81}
81
Let the differential parameter $d\tau $ of a
worldline be expressed in Minkowskian coordinates by $d{\tau}^{2}=-(d{x}_{1}^{2}+d{x}_{2}^{2}+d{x}_{3}^{2}+d{x}_{4}^{2})$. The 4-velocity vector ${U}_{\mu}$ is naturally
defined to be the first derivative with respect to this parameter,
${U}_{\mu}=d{x}_{\mu}/d\tau $ ($\mu =1$, 2, 3, 4). As matters stood in
November, 1907, however, Minkowski could proceed no further with his
project of reformulation.^{82}^{82}
82
The incongruity noted by
Pyenson (1985b, 84) between Minkowski’s announcement of a
four-dimensional physics on one hand, and on the other hand, a
trifle of 4-vector definitions and expressions, is to be understood
as a indication of Minkowski’s gradual ascent of the learning curve
of four-dimensional physics. The development of four-dimensional
mechanics was hobbled by Minkowski’s spare stock of 4-vectors even more
than that of electrodynamics. Although Minkowski defined a
force-density 4-vector, the fourth component of which he correctly
identified as the energy equation, he did not go on to define 4-force
at a point.^{83}^{83}
83
Minkowski defined the spatial components of the
empty space force density 4-vector ${\U0001d51b}_{j}$ in terms of the
ordinary force density components $\U0001d51b$, $\U0001d51c$,
$\u2128$, and their product with velocity: $\U0001d504=\U0001d51b{\U0001d534}_{x},\U0001d51c{\U0001d534}_{y},\u2128{\U0001d534}_{z}$, such that
${\U0001d51b}_{j}=\U0001d51b,\U0001d51c,\u2128,\text{i}\U0001d504$. He also expressed the force density 4-vector as the
product of 4-current-density and the Traktor: ${\U0001d51b}_{j}={\varrho}_{1}{\psi}_{j1}+{\varrho}_{2}{\psi}_{j2}+{\varrho}_{3}{\psi}_{j3}+{\varrho}_{4}{\psi}_{j4}$. Once again, the definition of a force 4-vector
at a point would have been trivial, had Minkowski possessed a correct
4-velocity definition. No more than a review of Planck’s recent
investigation (Planck 1907), Minkowski’s discussion of
mechanics involved no 4-vectors at all. Likewise for the subsequent
section on gravitation, which reviewed Poincaré’s theory, as shown
above. Without a valid 4-vector for velocity, Minkowski’s
electrodynamics of moving media was severely hobbled; without a point
force 4-vector, his four-dimensional mechanics and theory of
gravitation could go nowhere.

The difficulty encountered by Minkowski in formulating a four-dimensional approach to physics is surprising in light of the account he gave later of the background to his discovery of spacetime (Minkowski 1909). Minkowski presented his spacetime view of relativity theory as a simple application of group methods to the differential equations of classical mechanics. These equations were known to be invariant with respect to uniform translations, just as the squared sum of differentials $d{x}^{2}+d{y}^{2}+d{z}^{2}$ was known to be invariant with respect to rotations and translations of Cartesian axes in Euclidean 3-space, and yet no one, he said, had thought of compounding the two corresponding transformation groups. When this is done properly (by introducing a positive parameter $c$), one ends up with a group Minkowski designated ${G}_{c}$, with respect to which the laws of physics are covariant. (The group ${G}_{c}$ is now known as the Poincaré group.) Presumably, the four-dimensional approach appeared simple to Minkowski in hindsight, several months after his struggle with 4-velocity.

In summary, while Minkowski formulated the idea of a four-dimensional language for physics based on the form-invariance of the Maxwell equations under the transformations of the Lorentz group, his development of this program beyond electrodynamics was hindered by a misunderstanding of the four-dimensional counterpart of an ordinary velocity vector. This was to be only a temporary obstacle.

On 21 December, 1907, Minkowski presented to the Royal Society
of Science in Göttingen a memoir entitled “The Basic Equations for
Electromagnetic Processes in Moving Bodies,” which I will refer to
for brevity as the Grundgleichungen.^{84}^{84}
84
Minkowski’s
manuscript was delivered to the printer on 21 February, 1908,
corrected, and published on 5 April, 1908
(Journal für die “Nachrichten” der
Gesellschaft der Wissenschaften zu Göttingen,
mathematische-naturwissenschaftliche Klasse 1894–1912, Scient. 66,
Nr. 1, 471, Archiv der Akademie der Wissenschaften zu
Göttingen). I thank Tilman Sauer for
pointing out this source to me.
Minkowski’s memoir revisits in detail most of the topics introduced in
his 5 November lecture to the mathematical society, but employs none
of the jargon of spaces, geometries, and manifolds. What it emphasizes
instead–in agreement with its title–is the achievement of the first
theory of electrodynamics of moving bodies in full conformance to the
principle of relativity. Also underlined is a second result described
as “very surprising”: the laws of mechanics follow
from the postulate of relativity and the law of energy conservation
alone. On the four-dimensional world and the new form of the equations
of electrodynamics, both topics headlined in his November lecture,
Minkowski remained coy. Curiously, the introduction mentions nothing
about a new formalism, even though all but one of fourteen sections
introduce and employ new notation or calculation rules (not counting
the appendix).

The added emphasis on the laws of mechanics in Minkowski’s
introduction, on the other hand, reflects Minkowski’s recent discovery
of correct definitions of 4-velocity and 4-force, along with geometric
interpretations of these entities. It was in the
Grundgleichungen that Minkowski first employed the term
“spacetime” (Raumzeit).^{85}^{85}
85
While the published version
of Minkowski’s 5 November lecture refers on one occasion to a
“Raumzeitpunkt” (Minkowski 1915, 934),
the term occurs nowhere in
the archival typescript. The source of this addition is unknown. A
manuscript annotation of the first page of the typescript bears
Sommerfeld’s initials, and indicates that he compared parts of the
typescript to the proofs, as Lewis Pyenson
(1985b, 82) points out. Pyenson errs, however, in attributing to
Sommerfeld the authorship of the remaining annotations, which were
all penned in Minkowski’s characteristic cramped
hand.
For example, he introduced 4-current density as
the exemplar of a “spacetime vector of the first kind”
(§ 5), and used it to derive a velocity 4-vector.
Identifying ${\varrho}_{1}$, ${\varrho}_{2}$, ${\varrho}_{3}$, ${\varrho}_{4}$ with
$\varrho {\U0001d534}_{x}$, $\varrho {\U0001d534}_{y}$, $\varrho {\U0001d534}_{z}$, $i\varrho $, just as he had done in his lecture of 5 November, Minkowski wrote the transformation to a primed system moving
with uniform velocity $$:

$${\varrho}^{\prime}=\varrho \left(\frac{-q{\U0001d534}_{z}+1}{\sqrt{1-{q}^{2}}}\right),{\varrho}^{\prime}{\U0001d534}_{{z}^{\prime}}^{\prime}=\varrho \left(\frac{{\U0001d534}_{z}-q}{\sqrt{1-{q}^{2}}}\right),{\varrho}^{\prime}{\U0001d534}_{{x}^{\prime}}^{\prime}=\varrho {\U0001d534}_{x},{\varrho}^{\prime}{\U0001d534}_{{y}^{\prime}}^{\prime}=\varrho {\U0001d534}_{y}.$$ | (14) |

Observing that this transformation did not alter the expression ${\varrho}^{2}(1-{\U0001d534}^{2})$, Minkowski announced an “important remark” concerning the relation of the primed to the unprimed velocity vector (§4). Dividing the 4-current density by the positive square root of the latter invariant, he obtained a valid definition of 4-velocity,

$$\frac{{\U0001d534}_{x}}{\sqrt{1-{\U0001d534}^{2}}},\frac{{\U0001d534}_{y}}{\sqrt{1-{\U0001d534}^{2}}},\frac{{\U0001d534}_{z}}{\sqrt{1-{\U0001d534}^{2}}},\frac{i}{\sqrt{1-{\U0001d534}^{2}}},$$ | (15) |

the squared magnitude of which is equal to $-1$. Minkowski seemed satisfied with this definition, naming it the spacetime velocity vector (Raum-Zeit-Vektor Geschwindigkeit).

The significance of the spacetime velocity vector, Minkowski observed, lies in the relation it establishes between the coordinate differentials and matter in motion, according to the expression

$$\sqrt{-(d{x}_{1}^{2}+d{x}_{2}^{2}+d{x}_{3}^{2}+d{x}_{4}^{2})}=dt\sqrt{1-{\U0001d534}^{2}}.$$ | (16) |

The Lorentz-invariance of the right-hand side of (16), signaled earlier by both Poincaré and Planck, now described the relation of the sum of the squares of the coordinate differentials to the components of 4-velocity.

The latter relation plays no direct role in Minkowski’s subsequent development of the electrodynamics of moving media, and in this it is unlike the 4-velocity definition. Rewriting the right-hand side of (16) as the ratio of the coordinate differential $d{x}_{4}$ to the temporal component of 4-velocity, ${w}_{4}$, Minkowski defined the spacetime integral of (16) as the “proper time” (Eigenzeit) pertaining to a particle of matter. The introduction of proper time streamlined Minkowski’s 4-vector expressions, for instance, 4-velocity was now expressed in terms of the coordinate differentials, the imaginary unit, and the differential of proper time, $d\tau $:

$$\frac{dx}{d\tau},\frac{dy}{d\tau},\frac{dz}{d\tau},i\frac{dt}{d\tau}.$$ | (17) |

Along with the notational simplification realized by the introduction of proper time, Minkowski signaled a geometric interpretation of 4-velocity. Since proper time is the parameter of a spacetime line (or as he later called it, a worldline), it follows that 4-velocity is equal to the slope of a worldline at a given spacetime point, much like ordinary three-velocity is described by the slope of a displacement curve in classical kinematics. What Minkowski pointed out, in other words, is that 4-velocity is tangent to a worldline at a given spacetime point (p. 108).

In order to develop his mechanics, Minkowski needed a workable definition of mass. He adapted Einstein’s and Planck’s notion of rest mass to the arena of spacetime by considering that a particle of matter sweeps out a hypertube in spacetime. Conservation of particle mass $m$ was then expressed as invariance of the product of rest mass density with the volume slices of successive constant-time hypersurfaces over the length of the particle’s worldline, such that $dm/d\tau =0$. Minkowski did not consider the case of variable rest mass density, which arises, for instance, in the case of heat exchange.

Minkowski’s decision to adopt a constant rest mass density is linked
to his view of the electrodynamics of moving media. Recall that he had
introduced a six-vector in his 5 November lecture to represent the
field. The product of the field and excitation six-vectors, he noted,
leads to an interesting 4 by 4 matrix, combining the Maxwell stresses,
Poynting vector, and electromagnetic energy density. He did not assign
a name to this object, known later as the energy-momentum tensor, and
often viewed as one of Minkowski’s greatest achievements in
electrodynamics.^{86}^{86}
86
While Minkowski’s tensor is traceless, it is
also asymmetric, a fact which led to criticism and rejection by
leading theorists of the day. His asymmetric tensor was later
rehabilitated; for a technical discussion with reference to the
original papers, see Møller (1972, 219). In the absence of
matter, his tensor assumes a symmetric form; in this form, it was
hailed by theorists. Of special interest to Minkowski was the fact
that the 4-divergence of this matrix, denoted $\mathrm{lor}S$, is a
4-vector, $K$:^{87}^{87}
87
Minkowski defined the energy-momentum
tensor $S$ in two ways: as the product of six-vectors, $fF=S-L$,
where $L$ is the Lagrange density, and in component form via the
equations for Maxwell stresses, the Poynting vector, and
electromagnetic energy density (Minkowski 1908, 96).

$$K=\mathrm{lor}S.$$ | (18) |

This 4-divergence (18) was used to define the “ponderomotive” force density, or generalized force per unit volume, neither mechanical nor non-mechanical in the pure sense of these terms. The 4-vector $K$ is not normal, in general, to the velocity $w$ of a given volume element, so to ensure that the ponderomotive force acts orthogonally to $w$, Minkowski added a component containing a velocity term:

$$K+(w\overline{K})w.$$ | (19) |

The parentheses in (19) indicate a scalar product, and
$\overline{K}$ stands for the transpose of $K$. By defining the
ponderomotive force density in this way, Minkowski effectively opted
for an equation of motion in which 4-acceleration is normal to
4-velocity.^{88}^{88}
88
Minkowski’s alternative between a 4-force
definition and the “natural” spacetime equations of motion was
underlined by Pauli (1921, 664). It appears that Minkowski
let this view of force and acceleration guide his development of
spacetime mechanics. In the latter domain, he formed a 4 by 4 matrix
$S$ in the force density and energy of an elastic media with the same
transformation properties as the energy-momentum tensor $S$ of
(18), and used the 4-divergence of this tensor to express the
equations of motion of a volume element of constant rest mass density
$\nu $ (p. 106):

$$\nu \frac{d{w}_{h}}{d\tau}={K}_{h}+\kappa {w}_{h}\mathit{\hspace{1em}\hspace{1em}}(h=1,2,3,4).$$ | (20) |

The factor $\kappa $ was determined by the definition of 4-velocity to be equal to the scalar product $(K\overline{w})$, much like the definition of ponderomotive force (19). In sum, it may be supposed that the non-orthogonality with respect to a given volume element of the 4-divergence of Minkowski’s asymmetric energy-momentum tensor for moving media led Minkowski to introduce a velocity term to his definition of ponderomotive force. This definition was then ported to spacetime mechanics, where for the sake of consistency, Minkowski held rest mass density constant in the equations of motion (20).

Minkowski’s stipulation of constant rest mass density was eventually challenged by Max Abraham (1909, 739) and others, for reasons that do not concern us here. Despite its obvious drawbacks, it greatly simplified the tasks of outlining the mechanics of spacetime and developing a theory of gravitation. For example, it permitted him to define the equations of motion of a particle in terms of the product of rest mass and 4-acceleration, where the latter is the derivative of 4-velocity with respect to proper time. Since 4-velocity is orthogonal to 4-acceleration, for constant proper mass it is also orthogonal to a 4-vector Minkowski called a “driving force” (bewegende Kraft, p. 108). Minkowski wrote four equations defining this force:

$$m\frac{d}{d\tau}\frac{dx}{d\tau}={R}_{x},m\frac{d}{d\tau}\frac{dy}{d\tau}={R}_{y},m\frac{d}{d\tau}\frac{dz}{d\tau}={R}_{z},m\frac{d}{d\tau}\frac{dt}{d\tau}={R}_{t}.$$ | (21) |

The first three expressions differ from Planck’s equations of
motion, in that Planck defined force as change in momentum,
instead of mass times acceleration. It was only a few months later
that Minkowski explicitly defined four-momentum as the product of
4-velocity with proper mass.^{89}^{89}
89
Planck (1906, eqn. 6),
Minkowski (1909, §4).
In the latter lecture, Minkowski proposed
the modern definition of kinetic energy as the temporal component of
4-momentum times ${c}^{2}$, or $m{c}^{2}dt/d\tau $. The “spatial” part of the
driving force (21) was referred to by Lorentz
(1910, 1237) as a “Minkowskian force”
(Minkowskische Kraft), differing from the Newtonian force by
a Lorentz factor. Lorentz complemented the Minkowskian force with a
“Minkowskian mass” (Minkowskische Masse). By
dividing Minkowski’s first three equations by a Lorentz factor, one
obtains Planck’s equations. Minkowski’s fourth equation, ${R}_{t}$,
formally dependent on the other three, expresses the law of energy
conservation.^{90}^{90}
90
Minkowski’s argument may be summarized as
follows. From the definition of a 4-vector, the following orthogonality
relation holds for the driving force $R$:
$${R}_{x}\frac{dx}{d\tau}+{R}_{y}\frac{dy}{d\tau}+{R}_{z}\frac{dz}{d\tau}={R}_{t}\frac{dt}{d\tau}.$$
(22)
Integration of the rest-mass density over the hypersurface normal to
the worldline of the mass point results in the driving force
components (21), but if the integration is to be performed
instead over a constant-time hypersurface, proper time is replaced
by coordinate time, such that the fourth equation reads:
$md/dt(dt/d\tau )={R}_{t}d\tau /dt$. From (22) we obtain an
expression for ${R}_{t}$, which we multiply by $d\tau /dt$:
$$m\frac{d}{dt}\left(\frac{dt}{d\tau}\right)={\U0001d534}_{x}{R}_{x}\frac{d\tau}{dt}+{\U0001d534}_{y}{R}_{y}\frac{d\tau}{dt}+{\U0001d534}_{z}{R}_{z}\frac{d\tau}{dt}.$$
(23)
Minkowski reasoned that since the right-hand side of (23)
describes the rate at which work is done on the particle, the
left-hand side must be the rate of change of the particle’s kinetic
energy, such that (23) represents the law of energy
conservation. He immediately related (23) to the kinetic
energy of the particle:
$$m\left(\frac{dt}{d\tau}-1\right)=m\left(\frac{1}{\sqrt{1-{\U0001d534}^{2}}}-1\right)=m(\frac{1}{2}{|\U0001d534|}^{2}+\frac{3}{8}{|\U0001d534|}^{4}+\mathrm{\cdots}).$$
(24)
Minkowski did not justify the latter expression, but in virtue of his definition of proper time, $d\tau =dt\sqrt{1-{\U0001d534}^{2}}$, the left-hand side of
(23) may be rewritten as
$m(d/dt)(1/\sqrt{1-{\U0001d534}^{2}})$, such that upon
integration the particle’s kinetic energy is
$m/\sqrt{1-{\U0001d534}^{2}}+C$, where $C$ is a constant. For
agreement with the Newtonian expression of kinetic
energy in case of small particle velocities ($\U0001d534\ll 1$), we let
$C=-m$, which accords both with (24) and the definition of
kinetic energy given in a later lecture
(cf. note 89).
From energy conservation and the relativity
postulate alone, Minkowski concluded, one may derive the equations of
motion. This is the single “surprising” result of his investigation
of relativistic mechanics, referred to at the outset of his paper (see
above).

If Minkowski found few surprises in spacetime mechanics, many of his
readers were taken aback by his four-dimensional approach. For
example, the first physicists to comment on his work, Albert Einstein and
Jakob Laub, rewrote Minkowski’s expressions in ordinary vector
notation, sparing the reader the “sizable demands” (ziemlich
große Anforderungen) of Minkowski’s mathematics
(Einstein 1908, 532). They did not specify the
nature of the demands, but the abstracter of their paper pointed to
the “special knowledge of the calculation methods” required for
assimilation of Minkowski’s equations.^{91}^{91}
91
Jahrbuch über
die Fortschritte für Mathematik 39, 1908, 910. In other words,
for Minkowski’s readers, his novel matrix calculus was the principal
technical obstacle to overcome. Where Poincaré pushed rejection of
formalism to an extreme, Minkowski pulled in the other direction,
introducing a formalism foreign to the practice of physics. What
motivated this brash move is unclear, and his
choice is all the more curious because he knowingly defied the German
trend of vector notation in electrodynamics.^{92}^{92}
92
This trend is
described by Darrigol (1993, 270). The sharp contrast between the
importance assigned to vector methods in France and Germany may be
linked to the status accorded to applied mathematics in these two
nations, as discussed by H. Gispert in her review of the French
version of Klein’s Encyklopädie (Gispert 2001). As
mentioned above, Minkowski was ill-disposed toward quaternions,
although he admitted in print that they could be brought into use for
relativity instead of matrix calculus. He spoke here from experience,
as manuscript notes reveal that he used quaternions (in addition to
Cartesian-coordinate representation and ordinary vector analysis) to
investigate the electrodynamics of moving media.^{93}^{93}
93
At one point
during his calculations Minkowski seemed convinced of the utility of
this formalism, remarking that electrodynamics is “predestined for
application of quaternionic calculations” (Math. Archiv 60:6, 21,
Handschriftenabteilung, NSUB). In the end, however, he felt that
for his purposes quaternions were “too limited and cumbersome”
(zu eng und schwerfällig, p. 79).

As far as notation is concerned, Minkowski’s treatment of differential
operations broke cleanly with then-current practice. It also broke
with the precedent of his 5 November lecture, where he had introduced,
albeit parsimoniously, both $\mathrm{\square}$ and $\mathrm{Div}$ (see above). For the Grundgleichungen he adopted a
different approach, extending the $\nabla $ to four dimensions, and
labeling the resulting operator lor, already
encountered above in (18). The name is short for Lorentz, and
the effect is the operation: $\left|\begin{array}{cccc}\hfill \partial /\partial {x}_{1},\hfill & \hfill \partial /\partial {x}_{2},\hfill & \hfill \partial /\partial {x}_{3},\hfill & \hfill \partial /\partial {x}_{4}\hfill \end{array}\right|$. When applied to a 6-vector, lor results in a
4-vector, in what Minkowski described as an appropriate translation of
the matrix product rule (p. 89); it also mimics the effect of the
ordinary $\nabla $. Transforming as a 4-vector, lor is liberally
employed in the second part of the Grundgleichungen, to the
exclusion of any and all particular 4-vector functions.^{94}^{94}
94
A
precedent for Minkowski’s exclusive use of lor may be found in
Gibbs & Wilson (1901), where $\nabla $ is similarly preferred to vector
functions. The use of lor made for a presentation of
electrodynamics elegant in the extreme, at the expense of legibility
for German physicists more used to thinking in terms of gradients,
divergences, and curls (or rotations).

Minkowski’s equations of electrodynamics departed radically in form
with those of the old electrodynamics, shocking the thought patterns
of physicists, and creating a phenomenon of rejection that took
several years–and a new formalism–to overcome.^{95}^{95}
95
Cf. Max von
Laue’s remark that physicists understood little of Minkowski’s work
because of its unfamiliar mathematical expression
(Von Laue 1951, 515), and Chuang Liu’s account of the
difficulty experienced by Max Abraham and Gunnar Nordström in
applying Minkowski’s formalism (Liu 1991, 66). While
Minkowski’s calculus is a straightforward extension of Cayley’s
formalism (for a summary, see Cunningham 1914, chap. 8), the
latter formalism was itself unfamiliar to physicists. Why did
Minkowski break with this tradition? Did he feel that the new physics
of spacetime required a clean break with nineteenth-century practice?
Perhaps, but he must have recognized that the old methods would prove
resistant to change. His own subsequent practice shows as much: after
writing the Grundgleichungen Minkowski did not bother with lor
during his private explorations of the formal side of electrodynamics,
preferring the coordinate method.^{96}^{96}
96
Math. Archiv 60:5,
Handschriftenabteilung, NSUB. This 82-page set of notes dates from
23 May to 6 October, 1908. A posthumously published paper on the
electron-theoretical derivation of the basic equations of
electrodynamics for moving media, while purported to be from
Minkowski’s Nachlass, was written entirely by Max Born, as he
acknowledged
(Minkowski & Born 1910, 527). In the latter publication lor
makes only a brief appearance.

He also relied largely–but not exclusively–on a Cartesian-coordinate approach during his preliminary investigations of the subjects treated in the Grundgleichungen. His surviving research notes, made up almost entirely of symbolic calculations, shed an interesting light on Minkowski’s path to both a theory of the electrodynamics of moving media, and a theory of gravitation, or more generally to his process of discovery. Notably, where the subjects of mechanics and gravitation are relegated to the appendix of the Grundgleichungen, these notes show that Minkowski pursued questions of electrodynamics and gravitation in parallel, switching from one topic to the other three times in the course of 163 carefully numbered pages. At least fifteen of these pages are specifically concerned with gravitation; the notes are undated, but those concerning gravitation are certainly posterior to the typescript of the 5 November lecture, because unlike the latter, they feature valid definitions of 4-velocity and 4-force.

Minkowski’s attempt to capture gravitational action in terms of a 4-scalar potential is of particular interest. We recall that Minkowski had expressed Maxwell’s equations in terms of a 4-vector potential (11) during his lecture of 5 November, and on this basis, it was natural for him to investigate the possibility of representing gravitational force on a point mass in a fashion analogous to that of the force on a point charge moving in an electromagnetic field. In his scratch notes, Minkowski defined a 4-scalar potential $\mathrm{\Phi}$, in terms of which he initially devised the law of motion:

$$\begin{array}{cc}\hfill \frac{d}{d\tau}\frac{1}{\sqrt{1-{v}^{2}}}-\frac{\partial \mathrm{\Phi}}{\partial t}& =0,\hfill \\ \hfill \frac{d}{d\tau}\frac{-{x}^{\prime}}{\sqrt{1-{v}^{2}}}-\frac{\partial \mathrm{\Phi}}{\partial x}& =0,\hfill \end{array}\mathit{\hspace{1em}\hspace{1em}}\begin{array}{cc}\hfill \frac{d}{d\tau}\frac{-{y}^{\prime}}{\sqrt{1-{v}^{2}}}-\frac{\partial \mathrm{\Phi}}{\partial y}& =0,\hfill \\ \hfill \frac{d}{d\tau}\frac{-{z}^{\prime}}{\sqrt{1-{v}^{2}}}-\frac{\partial \mathrm{\Phi}}{\partial z}& =0,\hfill \end{array}$$ | (25) |

where constants are neglected, $\tau $ denotes proper time, and primes
indicate differentiation with respect to coordinate time $t$ (i.e.,
${x}^{\prime}=dx/dt$).^{97}^{97}
97
Math. Archiv 60:6, 10,
Handschriftenabteilung, NSUB. This generalization of the Newtonian
potential to a 4-scalar potential appears to be one of the first paths
explored by Minkowski in his study of gravitation, but his investigation
is inconclusive. In particular, there is no indication in these
notes of a recognition on Minkowski’s part that a four-scalar
potential conflicts with the postulates of invariant rest mass and
light velocity.^{98}^{98}
98
This “peculiar” consequence of Minkowski’s
spacetime mechanics was underlined by Maxwell’s German translator,
the Berlin physicist Max B. Weinstein (1914, 42). In Minkowski
spacetime, 4-acceleration is orthogonal to 4-velocity: ${U}_{\mu}d{U}_{\mu}/d\tau =0$, $\mu =1$, 2, 3, 4, where $\tau $ is the proper
time. We assume a 4-scalar potential $\mathrm{\Phi}$ such that the gravitational
4-force ${F}_{\mu}=-m\partial \mathrm{\Phi}/\partial {x}_{\mu}$. If we consider
a point mass with 4-velocity ${U}_{\mu}$ subjected to a 4-force ${F}_{\mu}$
derived from this potential, we have ${U}_{\mu}{F}_{\mu}=-{U}_{\mu}m\partial \mathrm{\Phi}/\partial {x}_{\mu}$. Writing 4-velocity as
$d{x}_{\mu}/d\tau $, and substituting in the latter expression, we obtain
$${U}_{\mu}{F}_{\mu}=-m\frac{d{x}_{\mu}}{d\tau}\frac{\partial \mathrm{\Phi}}{\partial {x}_{\mu}}=-m\frac{d\mathrm{\Phi}}{d\tau}=0,$$
and consequently, $d\mathrm{\Phi}/d\tau =0$, which means that the law of
motion describes the trajectory of the passive mass $m$ only in the
trivial case of constant $\mathrm{\Phi}$ along its worldline. Nor is there
any evidence that he considered suspending either one of these
postulates.

Likewise, in the Grundgleichungen there is no question of adopting either a variable mass density or a gravitational 4-potential. Once he had established the foundations of spacetime mechanics, Minkowski took up the case of gravitational attraction. The problem choice is significant, in that the same question had been treated at length by Poincaré (although not to Minkowski’s satisfaction, as mentioned above). Implicitly, Minkowski encouraged readers to compare methods and results. Explicitly, proceeding in what he described (in a footnote) as a “wholly different way” from Poincaré, Minkowski said he wanted to make “plausible” the inclusion of gravitation in the scheme of relativistic mechanics (p. 109). It will become clear in what follows that his project was more ambitious than the modest elaboration of a plausibility argument, as it was designed to validate his spacetime mechanics.

The point of departure for Minkowski’s theory of gravitation was quite different from that of Poincaré, because the latter’s results were integrated into the former’s formalism. For example, where Poincaré initially assumed a finite propagation velocity of gravitation no greater than that of light, only to opt in the end for a velocity equal to that of light, Minkowski assumed implicitly from the outset that this velocity was equal to that of light. Similarly, Poincaré initially supposed the gravitational force to be Lorentz covariant, only to opt in the end for an analog of the Lorentz force, where Minkowski required implicitly from the outset that all forces transform like the Lorentz force.

Combining geometric and symbolic arguments, Minkowski’s exposition of his theory of gravitation introduces a new geometric object, the three-dimensional “ray form” (Strahlgebilde) of a given spacetime point, known today as a light hypercone (or lightcone). For a fixed spacetime point $B*=(x*,y*,z*,t*)$, the lightcone of $B*$ is defined by the sets of spacetime points $B=(x,y,z,t)$ satisfying the equation

$${(x-x*)}^{2}+{(y-y*)}^{2}+{(z-z*)}^{2}={(t-t*)}^{2},t-t*\geqq 0.$$ | (26) |

For all the spacetime points $B$ of this lightcone, $B*$ is what
Minkowski called $B$’s lightpoint. Any worldline intersects the
lightcone in one spacetime point only, Minkowski observed, such that
for any spacetime point $B$ on a worldline there exists one and only
one lightpoint $B*$. Minkowski remarked in a later lecture that
the lightcone divides four-dimensional space into three regions:
timelike, spacelike and lightlike.^{99}^{99}
99
Minkowski introduced the
terms zeitartig and raumartig in
(1909).

Using this novel insight to the structure of four-dimensional space,
in combination with the 4-vector notation set up in earlier in his
memoir, Minkowski presented and applied his law of gravitational
attraction in two highly condensed pages. Minkowski’s geometric
argument employs non-Euclidean relations that were unfamiliar to
physicists, yet he provided no diagrams. Visually-intuitive arguments
had fallen into disfavor with mathematicians by this time, with the
rise of the axiomatic approach to geometry favored by David Hilbert
(Rowe 1997),
yet Minkowski never renounced the use of figures in
geometry; he employed them in earlier works on number geometry, and
went on to publish spacetime diagrams in the sequel to the
Grundgleichungen.^{100}^{100}
100
There is little agreement on where
to situate Minkowski’s work on relativity along a line running from
the intuitive to the formal. Peter Galison
(1979, 89), for example, underlines Minkowski’s
visual thinking (i.e., reasoning that appeals to figures or
diagrams), while
Leo Corry (1997, 275;
2004, chap. 4) considers Minkowski’s work in the context of
Hilbert’s axiomatic program for physics. For the purposes of my
reconstruction, I refer to a spacetime diagram (Figure 1) of the sort Minkowski employed in the sequel (reproduced in Figure
3).^{101}^{101}
101
Two spatial dimensions are suppressed in
Figure 1, and lightcones are represented by broken lines
with slope equal to $\pm 1$, the units being chosen so that the
propagation velocity of light is unity ($c=1$). In this model of
Minkowski space, orthogonal coordinate axes appear oblique in
general, for example, the spatial axes $x*y*z*$ are
orthogonal to the tangent $B*C*$ at spacetime point
$B*$ of the central line of the filament $F*$ described by
a particle of proper mass $m*$.

On the assumption that the force of gravitation is a 4-vector normal to the 4-velocity of the passive mass $m$, Minkowski derived his law of attraction in the following way. The trajectories of two particles of mass $m$ and $m*$ correspond to two spacetime filaments $F$ and $F*$, respectively. Minkowski’s arguments refer to worldlines he called central lines (Hauptlinien) of these filaments, which pass through points on the successive constant-time hypersurfaces delimited by the respective particle volumes. The central lines of the filaments $F$ and $F*$ are shown in Figure 1. An infinitesimal element of the central line of $F$ is labeled $BC$, and the two lightpoints corresponding to the endpoints $B$ and $C$ are labeled $B*$ and $C*$ on the central line of $F*$. From the origin of the rest frame $O$, a 4-vector parallel to $B*C*$ intersects at ${A}^{\prime}$ the three-dimensional hypersurface defined by the equation $-{x}^{2}-{y}^{2}-{z}^{2}+{t}^{2}=1$. Finally, a spacelike 4-vector $BD*$ extends from $B$ to a point $D*$ on the worldline tangent to the central line of $F*$ at $B*$.

Referring to the latter configuration of seven spacetime points, two central lines, a lightcone and a calibration hypersurface, Minkowski expressed the spatial components of the driving force of gravitation exerted by $m*$ on $m$ at $B$,

$$mm*{\left(\frac{O{A}^{\prime}}{B*D*}\right)}^{3}BD*.$$ | (27) |

Minkowski’s gravitational driving force is composed of the latter 4-vector (27) and a second 4-vector parallel to $B*C*$ at $B$, such that the driving force is always orthogonal to the 4-velocity of the passive mass $m$ at $B$. (For reasons of commodity, I will refer to this law of force as Minkowski’s first law.)

The form of Minkowski’s first law of gravitation is comparable to that
of his ponderomotive force for moving media (19), in that the
driving force has two components, only one of which depends on the
motion of the test particle. In the gravitational case, however,
Minkowski did not write out the 4-vector components in terms of matrix
products. Instead, he relied on spacetime geometry and the definition
of a 4-vector. The only way physicists could understand (27)
was by reformulating it in terms of ordinary vectors referring to a
conveniently chosen inertial frame, and even then, they could not rely
on Minkowski’s description alone, as it is incomplete.^{102}^{102}
102
The
4-vector $O{A}^{\prime}$ in (27) has unit magnitude by definition in
all inertial frames, while $B*D*$ is a timelike 4-vector
tangent to the central line of $F*$ at $B*$. Consequently,
$B*D*$ may be taken to coincide with the temporal axis
of a frame instantaneously at rest with $m*$ at $B*$, such
that it has only one nonzero component: the difference in proper
time between the points $B*$ and $D*$. It is assumed that
the rest frame may be determined unambiguously for a particle in
arbitrary motion, as asserted without proof by Minkowski in a later
lecture (Minkowski 1909, § III); subsequently, Max Born
(1909, 26) remarked that any motion may be
approximated by what he called hyperbolic motion, and noted that
such motion is characterized by an acceleration of constant
magnitude (as measured in an inertial frame). If we locate the
origin of this frame at $B*$, and let $D*=(0,0,0,t)$,
then $B*D*=(0,0,0,\text{i}t)$, and ${(B*D*)}^{3}=-\text{i}{t}^{3}$. Likewise in this same frame, $A={A}^{\prime}=(0,0,0,1)$, and $O{A}^{\prime}=OA=(0,0,0,\text{i})$. Minkowski understood the term $(O{A}^{\prime}/B*D*)$
as the ratio (Verhältnis) of two parallel 4-vectors, an
operation familiar from the calculus of quaternions, but one not
defined for 4-vectors. While modern vector systems ignore vector
division, in Hamilton’s quaternionic calculus the quotient of
vectors is unambiguously defined; see, for example,
Tait (1882, chap. 2).
Accordingly, the quotient in
(27) is the ratio of lengths, $(O{A}^{\prime}/B*D*)=1/t$,
and the cubed ratio is ${t}^{-3}$. The point $B$ lies on the same
constant-time hypersurface as $D*$, so we assign it the value
$(x,y,z,t)=(\mathbf{r},t)$. This assignment determines the value of
the 4-vector $BD*$: $B*D*=(-x,-y,-z,0)=(-\mathbf{r},0)$. Since $B*$ is a lightpoint of $B$, we can
apply (26) to obtain ${x}^{2}+{y}^{2}+{z}^{2}={t}^{2}={r}^{2}$, and consequently,
${t}^{3}={r}^{3}$. Substituting for ${t}^{3}$ results in ${(O{A}^{\prime}/B*D*)}^{3}=1/{t}^{3}=1/{r}^{3}$. The 4-vector $B*D*$ is spacelike, such
that its projection on the constant-time hypersurface orthogonal to
$B*D*$ at $D*$ is the ordinary vector $(-x,-y,-z)=-\mathbf{r}$. In terms of ordinary vectors and scalars measured
in the rest frame of $m*$, Minkowski’s expression (27) is
equivalent to Newton’s law (neglecting the gravitational constant):
$$-mm*\frac{\mathbf{r}}{{r}^{3}}$$
(28)
Neither (27) nor (28) contains any velocity-dependent
terms, while the timelike component of Minkowski’s first law depends on
the velocity of the passive mass. Newton’s law (28) thus
coincides with Minkowski’s first law only in the case of relative rest.

Even without spacetime diagrams or a transcription into ordinary
vector notation, the formal analogy of (27) to Newton’s law
is readily apparent, and this is probably why Minkowski wrote it this
way. In doing so, however, he passed up an opportunity to employ the
new matrix machinery at his disposal. Had he seized this opportunity,
he would have gained a simple, self-contained, coordinate-free
expression of the law of gravitation, and provided readers with a more
elaborate example of his calculus in action, but the latter desiderata
must not have been among his primary objectives.^{103}^{103}
103
Minkowski’s
driving force may be expressed in his notation as a function of
scalar products of 4-velocities and 4-position:
$$-mm*\frac{(w\overline{w}*)\Re -(w\overline{\Re})w*}{{(\Re \overline{w}*)}^{3}(w\overline{w}*)}.$$
Here I let $w$ and $w*$ designate 4-velocity at the passive and
active mass points, while $\Re $ is the associated 4-position,
the parentheses denote a scalar product, and the bar indicates
transposition.

Minkowski was not yet finished with his law of gravitation. Unlike Poincaré, after writing his law of gravitation, Minkowski went on to apply it to the particular case of uniform rectilinear motion of the source $m*$. He considered the latter in a comoving frame, in which the temporal axis is chosen to coincide with the tangent to the central line of $F*$ at $B*$ (cf. the situation described in note 102). Referring to the reconstructed spacetime diagram in Figure 2, the temporal axis is represented by a vertical line $F*$, such that the origin is established in a frame comoving with $m*$. To the retarded position of $m*$, denoted $B*$, Minkowski assigned the coordinates (0, 0, 0, $\tau *$), and to the position $B$ of the passive mass $m$ he assigned the coordinates ($x$, $y$, $z$, $t)$. The geometry of this configuration fixes the location of $D*$ at (0, 0, 0, $t$), from which the 4-vectors $BD*=(-x$, $-y$, $-z$, $0)$ and $B*D*=(0,0,0,\text{i}(t-\tau *))$ are determined. In this case, Minkowski pointed out, (26) reduces to:

$${x}^{2}+{y}^{2}+{z}^{2}={(t-\tau *)}^{2}.$$ | (29) |

Substituting the above values of $BD*$ and $B*D*$
into Minkowski’s formula (27), the spatial
components of the 4-acceleration of the passive mass $m$ at $B$ due to
the active mass $m*$ at $B*$ turn out to be:^{104}^{104}
104
The
intermediate calculations can be reconstructed as follows. Let the
driving force be designated ${F}_{\mu}$, $\mu =1,2,3,4$. Since
${(O{A}^{\prime}/B*D*)}^{3}={t}^{-3}$, and $BD*=(-x$, $-y$,
$-z$, $0)$, equations (21) and (27) yield: ${F}_{1}/m={d}^{2}x/d{\tau}^{2}=-m*x/{(t-\tau *)}^{3}$, ${F}_{2}/m={d}^{2}y/d{\tau}^{2}=-m*y/{(t-\tau *)}^{3}$, ${F}_{3}/m={d}^{2}z/d{\tau}^{2}=-m*z/{(t-\tau *)}^{3}$.

$$\frac{{d}^{2}x}{d{\tau}^{2}}=-\frac{m*x}{{(t-\tau *)}^{3}},\frac{{d}^{2}y}{d{\tau}^{2}}=-\frac{m*y}{{(t-\tau *)}^{3}},\frac{{d}^{2}z}{d{\tau}^{2}}=-\frac{m*z}{{(t-\tau *)}^{3}}.$$ | (30) |

From (30) and (29), the corresponding temporal
component at $B$ may be determined:^{105}^{105}
105
Minkowski omitted the
intermediate calculations, which may be reconstructed in modern
notation as follows.
Let the 4-velocity of the passive mass point be designated ${U}_{\mu}=(dx/d\tau ,dy/d\tau ,dz/d\tau ,idt/d\tau )$, while the first three
components of its 4-acceleration, designated ${A}_{\mu}$, at $B$ due to
the source $m*$ are given by (30). From the
orthogonality of 4-velocity and 4-acceleration we have:
$${U}_{\mu}{A}_{\mu}=-\frac{dx}{d\tau}\frac{m*x}{{(t-\tau *)}^{3}}-\frac{dy}{d\tau}\frac{m*y}{{(t-\tau *)}^{3}}-\frac{dz}{d\tau}\frac{m*z}{{(t-\tau *)}^{3}}-\frac{idt}{d\tau}\frac{i{d}^{2}t}{d{\tau}^{2}}=0.$$
(31)
Rearranging (31) results in an expression for the temporal
component of 4-acceleration:
$$\frac{{d}^{2}t}{d{\tau}^{2}}=-\frac{m*}{{(t-\tau *)}^{3}}\left(\frac{xdx}{dt}+\frac{ydy}{dt}+\frac{zdz}{dt}\right).$$
(32)
Differentiating (29) with respect to $dt$ results in
$xdx/dt+ydy/dt+zdz/dt=(t-\tau *)d(t-\tau *)/dt$,
the right-hand side of which we substitute in (32) to obtain
(33).

$$\frac{{d}^{2}t}{d{\tau}^{2}}=-\frac{m*}{{(t-\tau *)}^{2}}\frac{d(t-\tau *)}{dt}.$$ | (33) |

Inspecting (30), it appears that the only difference between these acceleration components
and those corresponding to Newtonian attraction is a
replacement in the latter of coordinate time $t$ by proper time
$\tau $.^{106}^{106}
106
A young Polish physicist in Göttingen, Felix Joachim
de Wisniewski later studied this case, but with equations differing
from (30) by a Lorentz factor (Wisniewski 1913a, 388). In a
postscript to the second installment of his paper
(Wisniewski 1913b, 676), he employed Minkowski’s matrix notation,
becoming, with Max Born, one of the rare physicists to adopt this
notation.

The formal similarity of (30) to the Newtonian law of motion under a central force probably suggested to Minkowski that his law induces Keplerian trajectories. With the knowledge gained from (30), to the effect that the only difference between classical and relativistic trajectories is that arising from the substitution of proper time for coordinate time, Minkowski demonstrated the compatibility of his relativistic law of gravitation with observation using only Kepler’s equation and the definition of 4-velocity.

Writing Kepler’s equation in terms of proper time yields:

$$n\tau =E-e\mathrm{sin}E,$$ | (34) |

where $n\tau $ denotes the mean anomaly, $e$ the eccentricity, and $E$ the eccentric anomaly. Minkowski referred to (34) and to the norm of a 4-velocity vector:

$${\left(\frac{dx}{d\tau}\right)}^{2}+{\left(\frac{dy}{d\tau}\right)}^{2}+{\left(\frac{dz}{d\tau}\right)}^{2}={\left(\frac{dt}{d\tau}\right)}^{2}-1,$$ | (35) |

in order to determine the difference between the mean anomaly in
coordinate time $nt$ and the mean anomaly in proper time $n\tau $.
From (35), Minkowski deduced:^{107}^{107}
107
The intermediate
calculations were omitted by Minkowski, but figure among his
research notes (Math. Archiv 60:6, 126–127, Handschriftenabteilung,
NSUB). Following the method outlined by Otto
Dziobek (1888, 12), Minkowski began with the energy integral of
Keplerian motion:
$${\left(\frac{dt}{dW}\right)}^{2}-1=\frac{2}{{\mathrm{\ell}}^{2}}\left(\frac{M}{R}-C\right),$$
(36)
where $\mathrm{\ell}$ denotes the velocity of light, $M$ is the sum of the
masses times the gravitational constant, $M={k}^{2}(m+m*)$, $R$ is the
radius, and $C$ is a constant. The left-hand side of (36)
is the same as the right-hand side of (35) for $W=\tau $.
In order to express $dt/dW$ (which is to say $dt/d\tau $) in terms of
$E$, Minkowski considered a conic section in polar coordinates, with
focus at the origin: $R=a(1-{e}^{2})/(1+e\mathrm{cos}\phi )=a(1-e\mathrm{cos}E)$, where $a$ denotes the semi-major axis, and $\phi $ is the
true anomaly. By eliminating $\phi $ in favor of $E$ and $e$, and
differentiating (34), Minkowski obtained an expression
equivalent to (37).

$${\left(\frac{dt}{d\tau}\right)}^{2}-1=\frac{m*}{a{c}^{2}}\frac{1+e\mathrm{cos}E}{1-e\mathrm{cos}E}.$$ | (37) |

Solving (37) for the coordinate time $dt$, expanding to terms in ${c}^{-2}$, and multiplying by $n$ led Minkowski to the expression:

$$ndt=nd\tau \left(1+\frac{1}{2}\frac{m*}{a{c}^{2}}\frac{1+e\mathrm{cos}E}{1-e\mathrm{cos}E}\right).$$ | (38) |

Recalling (34), Minkowski managed to express the
difference between the mean anomaly in coordinate time and proper
time:^{108}^{108}
108
I insert the eccentricity $e$ in the second term on the
right-hand side, correcting an obvious omission in Minkowski’s paper
(1908, 111, eqn. 31).

$$nt+\text{const.}=\left(1+\frac{1}{2}\frac{m*}{a{c}^{2}}\right)n\tau +\frac{m*}{a{c}^{2}}e\mathrm{sin}E.$$ | (39) |

Evaluating the relativistic factor $m*/a{c}^{2}$ for solar mass and the Earth’s semi-major axis to be ${10}^{-8}$, Minkowski found the deviation from Newtonian orbits to be negligible in the solar system. On this basis, he concluded that

a decision against such a law and the proposed modified mechanics in favor of the Newtonian law of attraction with Newtonian mechanics would not be deducible from astronomical observations.

^{109}^{109}109 “…eine Entscheidung gegen ein solches Gesetz und die vorgeschlagene modifizierte Mechanik zu Gunsten des Newtonschen Attraktionsgesetzes mit der Newtonschen Mechanik aus den astronomischen Beobachtungen nicht abzuleiten sein” (Minkowski 1908, 111).

According to the quoted remark, there was more at stake here for Minkowski than just the empirical adequacy of his law of gravitational attraction, as his claim is for parity between Newton’s law and classical mechanics, on one hand, and the system composed of the law of gravitation and spacetime mechanics on the other hand. This new system, Minkowski claimed, was verified by astronomical observations at least as well as the classical system formed by the Newtonian law of attraction and Newtonian mechanics.

Instead of comparing his law with one or the other of Poincaré’s laws, Minkowski noted a difference in method, as mentioned above. In light of Minkowski’s emphasis on the methodological difference with Poincaré, and the hybrid geometric-symbolic nature of Minkowski’s exposition, it is clear that the point of reexamining the problem of relativity and gravitation in the Grundgleichungen was not simply to make plausible the inclusion of gravitation in a relativistic framework. Rather, since gravitational attraction was the only example Minkowski provided of his formalism in action, his line of argument served to validate his four-dimensional calculus, over and above the requirements of plausibility.

From the latter point of view, Minkowski had grounds for satisfaction, although one imagines that he would have preferred to find that his law diverged from Newton’s law just enough to account for the observed anomalies. It stands to reason that if Minkowski had been fully satisfied with his first law, he would not have proposed a second law in his next paper–which turned out to be the last he would finish for publication. The latter article developed out of a well-known lecture entitled “Space and Time” (Raum und Zeit), delivered in Cologne on 21 September, 1908, to the mathematics section of the German Association of Scientists and Physicians in its annual meeting (Walter 1999a, 49).

In the final section of his Cologne lecture, Minkowski took up the
Lorentz-Poincaré theory, and showed how to determine the field due to a
point charge in arbitrary motion. On this occasion, just as in his
earlier discussion of gravitation in the Grundgleichungen,
Minkowski referred to a spacetime diagram, but this time he provided
the diagram (Figure 3). Identifying the 4-vector potential
components for the source charge on this diagram, Minkowski remarked
that the Liénard-Wiechert law was a consequence of just these
geometric relations.^{110}^{110}
110
Minkowski’s explanation of the
construction of his spacetime diagram (Figure 3) may be
paraphrased in modern terminology as follows. Suppressing the
$z$-axis, we associate two worldlines with two point charges ${e}_{1}$
and $e$. The worldline of ${e}_{1}$ passes through the point at which we
wish to determine the field, ${P}_{1}$. To find the retarded position of
the source $e$, we draw the retrograde lightcone (with broken lines)
from ${P}_{1}$, which intersects the worldline of $e$ at $P$, where
there is a hyperbola of curvature $\varrho $ with three
infinitely-near points lying on the worldline of $e$; it has its
center at $M$. The coordinate origin is established at $P$, by
letting the $t$-axis coincide with the tangent to the worldline. A
line from ${P}_{1}$ intersects this axis orthogonally at point $Q$; it
is spacelike, and if its projection on a constant-time hypersurface
has length $r$, the length of the 4-vector $PQ$ is $r/c$. The
4-vector potential has magnitude $e/r$ and points in the direction
of $PQ$ (i.e., parallel to the 4-velocity of $e$ at $P$). The
$x$-axis lies parallel to $Q{P}_{1}$, such that $N$ is the intersection
of a line through $M$ normal to the $x$-axis.

Minkowski then described the driving force between two point charges. Adopting dot notation for differentiation with respect to proper time, he wrote the driving force exerted on an electron of charge ${e}_{1}$ at point ${P}_{1}$ by an electron of charge $e$:

$$-e{e}_{1}\left({\dot{t}}_{1}-\frac{{\dot{x}}_{1}}{c}\right)\U0001d50e,$$ | (40) |

where ${\dot{t}}_{1}$ and ${\dot{x}}_{1}$ are 4-velocity components of the test
charge ${e}_{1}$ and $\U0001d50e$ is a certain 4-vector. This was the
first such description of the electrodynamic driving force due to a
4-vector potential, the simplicity of which, Minkowski claimed,
compared favorably with the earlier formulations of Schwarzschild and
Lorentz.^{111}^{111}
111
Minkowski noted four conditions on $\U0001d50e$: it
is normal to the 4-velocity of ${e}_{1}$ at ${P}_{1}$, $c{\U0001d50e}_{t}-{\U0001d50e}_{x}=1/{r}^{2}$, ${\U0001d50e}_{y}=\ddot{y}/({c}^{2}r)$, and
${\U0001d50e}_{z}=0$, where $r$ is the spacelike distance between the
test charge ${e}_{1}$ at ${P}_{1}$ and the advanced position $Q$ of the
source $e$, and $\ddot{y}$ is the $y$-component of e’s 4-acceleration
at P. For a derivation of the 4-potential and 4-force
corresponding to Minkowski’s presentation, see
Pauli (1921, 644–645).

In the same celebratory tone, Minkowski finished his article with a discussion of gravitational attraction. The “reformed mechanics”, he claimed, dissolved the disturbing disharmonies between Newtonian mechanics and electrodynamics. In order to provide an example of this dissolution, he asked how the Newtonian law of attraction would sit with his principle of relativity. Minkowski continued:

I will assume that if two point masses $m$, ${m}_{1}$ describe worldlines, a driving force vector is exerted by $m$ on ${m}_{1}$, exactly like the one in the expression just given for the case of electrons, except that instead of $-e{e}_{1}$, we must now put in $+m{m}_{1}$.

Applying the substitution suggested by Minkowski to (40), we obtain:

$$m{m}_{1}\left({\dot{t}}_{1}-\frac{{\dot{x}}_{1}}{c}\right)\U0001d50e,$$ | (41) |

where the coefficients $m$ and ${m}_{1}$ refer to proper masses. Minkowski’s new law of gravitation (41) fully expresses the driving force, unlike the formula (27) of his first law, which describes only one component. In addition, the 4-vectors are immediately identifiable from the notation alone. (In order to distinguish the law given in the Grundgleichungen from that of the Cologne lecture (41), I will call (41) Minkowski’s second law.)

Since (40) was obtained from Lorentz-Poincaré theory via a
4-vector potential, the law of gravitation (41) ostensibly
implied a 4-vector potential as well; in other words, following the
example set by Poincaré’s second law (10), Minkowski appealed
in turn to a Maxwellian theory of gravitation similar to those of
Heaviside, Lorentz, and Gans.^{112}^{112}
112
See § 1,
as well as Heaviside (1893), and
Gans (1905).
Theories in which the gravitational field is determined by equations
having the form of Maxwell’s equations were later termed vector
theories of gravitation by Max Abraham (1914, 477). For a more
recent version of such a theory, see Coster & Shepanski (Coster and Shepanski (1969)). Although
Minkowski made no effort to attach his law to these field theories, it
was understood by Sommerfeld to be a formal consequence of just such a
theory, as I will show in the next section.

What were the numerical consequences of this new law? Minkowski spared the reader the details, noting only that in the case of uniform motion of the source, the only divergence from a Keplerian orbit would stem from the replacement of coordinate time by proper time. He indicated that the numbers for this case had been worked out earlier, and his conclusion with respect to this new law was naturally the same: combined with the new mechanics, it was supported by astronomical observations to the same extent as the Newtonian law combined with classical mechanics.

Curiously enough, Minkowski offered no explanation of the need for a
second law of attraction. Furthermore, by proposing two laws instead
of one, Minkowski tacitly acknowledged defeat; despite his criticism
of Poincaré’s approach (see above), he could
hardly claim to have solved unambiguously the problem of gravitation.
It may also seem strange that Minkowski discarded the differences
between his new law (41) and the one he had proposed
earlier.^{113}^{113}
113
Minkowski’s neglect of the differences between his
two theories may explain why historians have failed to distinguish
them. The principal difference between the two laws stems from the
presence of acceleration effects in the second law. By 1905 it was
known that accelerated electrons radiate energy, such that by formal
analogy, a Maxwellian theory of gravitation should have featured
accelerated point masses radiating “gravitational” energy. For a
brief overview of research performed in the first two decades of the
twentieth century on the energy radiated from accelerated electrons,
see Whittaker (1951, 2:246).

Minkowski revealed neither the motivation behind a second law of gravitation, nor why he neglected the differences between his two laws, but there is a straightforward way of explaining both of these mysteries. First, we recall the circumstances of his Cologne lecture, the final section of which Minkowski devoted to the theme of restoring unity to physics. What he wanted to stress on this occasion was that mechanics and electrodynamics harmonized in his four-dimensional scheme of things:

In the mechanics reformed according to the world postulate, the disturbing disharmonies between Newtonian mechanics and modern electrodynamics fall out on their own.

^{114}^{114}114 “In der dem Weltpostulate gemäß reformierten Mechanik fallen die Disharmonien, die zwischen der Newtonschen Mechanik und der modernen Elektrodynamik gestört haben, von selbst aus” (Minkowski 1909, § 5).

To support this view, Minkowski had to show that his reformed mechanics was a synthesis of classical mechanics and electrodynamics. A Maxwellian theory of gravitation fit the bill quite well, and consequently, Minkowski brought out his second law of gravitation (41). Clearly, this was not the time to point out the differences between his two laws. On the contrary, it was the perfect occasion to observe that a law of gravitation derived from a 4-vector potential formally identical to that of electrodynamics was observationally indistinguishable from Newton’s law. Naturally, Minkowski seized this opportunity.

Sadly, Minkowski did not live long enough to develop his ideas on gravitation
and electrodynamics; he died on 12 January, 1909, a few days after
undergoing an operation for appendicitis. At the time, no objections
to a field theory of gravitation analogous to Maxwell’s
electromagnetic theory were known, apart from Maxwell’s own
sticking-points. However, additional objections to this approach were raised by Max Abraham in
1912, after which the Maxwellian approach withered on the vine, as Gustav
Mie and others pursued unified theories of electromagnetism and
gravitation.^{115}^{115}
115
Abraham showed that a mass set into oscillation
would be unstable due to the direction of energy flow
(Norton 1992, 33).
On the early history of unified field
theories, see the reference in note 12.

Minkowski’s first law of gravitation fared no better than his second law, but the four-dimensional language in which his two laws were couched had a bright future. The first one to use Minkowski’s formal ideas to advantage was Sommerfeld, as we will see next.

## 3 Arnold Sommerfeld’s hyper-Minkowskian laws of gravitation

Neither Poincaré’s nor Minkowski’s work on gravitation and relativity drew comment until 25 October, 1910, when the second installment of Arnold Sommerfeld’s vectorial version of Minkowski’s calculus, entitled “Four-dimensional vector analysis” (Vierdimensionale Vektoranalysis), appeared in the Annalen der Physik (Sommerfeld 1910b). Sommerfeld’s contribution differs from those of Poincaré and Minkowski in that it is openly concerned with the presentation of a new formalism, much as its title indicates. In this section, I discuss Sommerfeld’s interest in vectors, the salient aspects of his 4-vector formalism, and his portrayal of Poincaré’s and Minkowski’s laws of gravitation.

Sommerfeld displayed a lively interest in vectors, beginning with his
editorship of the physics volume of Klein’s six-volume
Encyklopädie in the summer of 1898.^{116}^{116}
116
Sommerfeld’s work
on the Encyklopädie is discussed in an editorial note to his
scientific correspondence
(Eckert & Märker 2001, 40). He imposed a
certain style of vector notation on his contributing authors,
including typeface, terminology, symbolic representation of
operations, units and dimensions, and the choice of symbols for
physical quantities. Articles 12 to 14 of the physics volume appeared
in 1904, and were the first to implement the notation scheme backed by
Sommerfeld, laid out the same year in the Physikalische
Zeitschrift.^{117}^{117}
117
Reiff & Sommerfeld (1904),
Lorentz (1904c,
1904b),
Sommerfeld (1904a).
The scheme proposed by Sommerfeld differed from that published in
articles 12 to 14 of the Encyklopädie only in that the
operands of scalar and vector products were no longer separated by a
dot. While Sommerfeld belonged to the Vector Commission formed at
Felix Klein’s behest in 1902, it was clear to him as early as 1901
that the article on Maxwell’s theory (commissioned to Lorentz) would
serve as a “general directive” for future work in
electrodynamics.^{118}^{118}
118
Sommerfeld to Lorentz, 21 March, 1901,
Eckert & Märker (2001, 191).
On Sommerfeld’s participation on the
Commission see Reich (1996) and
Eckert & Märker (2001, 144).
His intuition turned out to be correct: the principal “vector” of
influence was Lorentz’s Article 13 (Lorentz 1904b), featuring
sections on vector notation and algebra, which set a de facto
standard for vector approaches to electrodynamics.

As mentioned above, only one effort to
extend Poincaré’s four-dimensional approach beyond the domain of
gravitation was published prior to Minkowski’s
Grundgleichungen. By 1910, the outlook for relativity theory
had changed due to the authoritative support of Planck and Sommerfeld,
the announcement of experimental results favoring Lorentz’s electron
theory, and the broad diffusion (in 1909) of Minkowski’s Cologne
lecture. Dozens of physicists and mathematicians began to take an
interest in relativity, resulting in a leap in relativist
publications.^{119}^{119}
119
For bibliometric data, and discussions of
Sommerfeld’s role in the rise of relativity theory, see
Walter (1999a, 68–73;
1999b, 96, 108).

The principal promoter of Minkowskian relativity, Sommerfeld must have
felt by 1910 that it was the right moment to introduce a
four-dimensional formalism. He was not alone in feeling this way, for
three other formal approaches based on Minkowski’s work appeared in
1910. Two of these were 4-vector systems, similar in some respects to
Sommerfeld’s, and worked out by Max Abraham and the
American physical chemist Gilbert Newton Lewis, respectively. A third,
non-vectorial approach was proposed by the Zagreb mathematician
Vladimir Varičak. Varičak’s was a real, four-dimensional,
coordinate-based approach relying on hyperbolic geometry. Sommerfeld
probably viewed this system as a potential rival to his own approach;
although he did not mention Varičak, he wrote that a
non-Euclidean approach was possible but could not be recommended
(Sommerfeld 1910a, 752, note 1). Of the three alternatives to
Sommerfeld’s system, the non-Euclidean style pursued by Varičak
and others was the only one to obtain even a modest following. An
investigation of the reasons for the contemporary neglect of these
alternative four-dimensional approaches is beyond the purview of our
study; for what concerns us directly, none of these methods was
applied to the problem of gravitation.^{120}^{120}
120
See
Abraham (1910),
Lewis (1910a,
1910b),
Varičak (1910). On Varičak’s
contribution see Walter (1999b).

Sommerfeld’s paper, like those of Abraham, Lewis, and Varičak, emphasized formalism, and in this it differed from the Grundgleichungen, as mentioned above. Like the latter work, it focused attention on the problem of gravitation. Following the example set by both Poincaré and Minkowski, Sommerfeld capped his two-part Annalen paper with an application to gravitational attraction, which consisted of a reformulation, comparison and commentary of their work in his own terms. Not only was Sommerfeld’s comparison of Poincaré’s and Minkowski’s laws of gravitation the first of its kind, it also proved to be the definitive analysis for his generation.

Sommerfeld’s four-dimensional vector algebra and analysis offered no
new 4-vector or 6-vector definitions, but it introduced a suite of
4-vector functions, notation, and vocabulary. The most far-reaching
modification with respect to Minkowski’s calculus was the elimination
of lor
in favor of extended versions of
ordinary vector functions. In Sommerfeld’s notational scheme, the
ordinary vector functions div, rot, and grad (used by Lorentz in his
Encyklopädie article on Maxwell’s theory) were replaced by
4-vector counterparts marked by a leading capital letter: Div, Rot,
and Grad. These three functions were joined by a 4-vector divergence
marked by German typeface, $\U0001d507\U0001d526\U0001d533$. Sommerfeld chose to retain
$\mathrm{\square}$ (cf. note 20), while noting the equivalence to his
4-vector functions: $\mathrm{\square}=$ Div Grad. The principal advantage of the
latter functions was that their meaning was familiar to physicists. In
the same vein, Sommerfeld supplanted Minkowski’s unwieldy terminology
of “spacetime vectors of the first and second type”
(Raum-Zeit-Vektoren I${}^{\text{\mathit{t}\mathit{e}\mathit{r}}}$ und II${}^{\text{\mathit{t}\mathit{e}\mathit{r}}}$
Art) with the more succinct “four-vector” (Vierervektor)
and “six-vector” (Sechservektor). The result was a compact
and transparent four-dimensional formalism differing as little as
possible from the ordinary vector algebra employed in the physics volume
of the Encyklopädie.^{121}^{121}
121
Not all of Sommerfeld’s
notational choices were retained by later investigators; Laue, for
instance, preferred a notational distinction between 4-vectors and
6-vectors. For a summary of notation used by Minkowski, Abraham,
Lewis, and Laue, see
Reich (1994).

To show how his formalism performed in action, Sommerfeld first took up the geometric interpretation and calculation of the electrodynamic 4-vector potential and 4-force. In the new notation, Sommerfeld wrote the electrodynamic 4-force $\U0001d50e$ between two point charges $e$ and ${e}_{0}$ in terms of three components in the direction of the lightlike 4-vector $\Re $, the source 4-velocity $\U0001d505$, and the 4-acceleration $\dot{\U0001d505}$:

$4\pi {\U0001d50e}_{\Re}$ | $={\displaystyle \frac{e{e}_{0}}{c{(\Re \U0001d505)}^{2}}}\left({\displaystyle \frac{{c}^{2}-(\Re \dot{\U0001d505})}{(\Re \U0001d505)}}({\U0001d505}_{0}\U0001d505)+({\U0001d505}_{0}\dot{\U0001d505})\right)\Re ,$ | (42) | ||

$4\pi {\U0001d50e}_{\U0001d505}$ | $={\displaystyle \frac{-e{e}_{0}}{c{(\Re \U0001d505)}^{2}}}{\displaystyle \frac{{c}^{2}-(\Re \dot{\U0001d505})}{(\Re \U0001d505)}}({\U0001d505}_{0}\Re )\U0001d505,$ | |||

$4\pi {\U0001d50e}_{\dot{\U0001d505}}$ | $={\displaystyle \frac{-e{e}_{0}}{c{(\Re \U0001d505)}^{2}}}{\displaystyle \frac{{c}^{2}-(\Re \dot{\U0001d505})}{(\Re \U0001d505)}}({\U0001d505}_{0}\Re )\dot{\U0001d505},$ |

where parentheses indicate scalar products. Sommerfeld was careful to note the equivalence between (42) and what he called Minkowski’s “geometric rule” (40).

In the ninth and final section of his paper, Sommerfeld took up the law of electrostatics and the classical law of gravitation. The former was naturally considered to be a special case of (42), with two point charges relatively at rest. The same was true for the law of gravitation, as Sommerfeld noted that Minkowski had proposed a formal variant of (40) as a law of gravitational attraction (what I call Minkowski’s second law, (41)). Sommerfeld’s expression of the electrodynamic 4-force is unwieldy, but takes on a simpler form in case of uniform motion of the source ($\dot{\U0001d505}=0$). Neglecting the $4\pi $ factor, and substituting $-m{m}_{0}$ for $+e{e}_{0}$, Sommerfeld expressed the corresponding version of Minkowski’s second law:

$$-m{m}_{0}c\frac{({\U0001d505}_{0}\U0001d505)\Re -({\U0001d505}_{0}\Re )\U0001d505}{{(\Re \U0001d505)}^{3}}.$$ | (43) |

The latter law is compact and self-contained, in that its interpretation depends only on the definitions and rules of the algebraic formalism. In this sense, (43) improves on the Minkowskian (41), even if it represents only a special case of the latter law.

Once Sommerfeld had expressed Minkowski’s second law in his own terms, he turned to Poincaré’s two laws. The transformation of Poincaré’s first law was more laborious than the transformation of Minkowski’s second law. First of all, Sommerfeld transcribed Poincaré’s first law (9) into his 4-vector notation, while retaining the original designation of invariants. This step itself was not simple: in order to cast Poincaré’s kinematic invariants as scalar products of 4-vectors, Sommerfeld had to adjust the leading sign of (9), to obtain:

$$\frac{{k}_{0}\U0001d50e}{m{m}^{\prime}}=-\frac{1}{{B}^{3}C}\left(C\Re -\frac{1}{c}A\U0001d505\right).$$ | (44) |

Sommerfeld noted the “correction” of what he
called an “obvious sign error” in (9).^{122}^{122}
122
“Mit
Umkehr des bei Poincaré offenbar versehentlichen Vorzeichens”
(Sommerfeld 1910b, 686, note 1). The difference is due to Poincaré’s
irregular derivation of the kinematic invariants (1), as
mentioned
above, although from Sommerfeld’s remark
it is not clear that he saw it this way.

The transformation of Poincaré’s second law (10) was less straightforward. It appears that instead of deriving a 4-vector expression as in the previous case, Sommerfeld followed Poincaré’s lead by eliminating the Lorentz-invariant factor $C$ from the denominator on the right-hand side of the first law (44), which results in the equation:

$$\frac{{k}_{0}\U0001d50e}{m{m}^{\prime}}=-\frac{1}{{B}^{3}}\left(C\Re -\frac{1}{c}A\U0001d505\right).$$ | (45) |

Sommerfeld expressed Poincaré’s kinematic invariants $A$, $B$, and $C$ as scalar products:

$$A=-\frac{1}{c}(\Re {\U0001d505}_{0}),B=-\frac{1}{c}(\Re \U0001d505),C=-\frac{1}{{c}^{2}}({\U0001d505}_{0}\U0001d505).$$ | (46) |

He also replaced the mass term ${m}^{\prime}$ in (44) and (45) by the product of rest mass ${m}_{0}$ and the Lorentz factor ${k}_{0}$, i.e., ${m}^{\prime}={m}_{0}{k}_{0}$. At this point, he could express Poincaré’s two laws exclusively in terms of constants, scalars, and 4-vectors:

$m{m}_{0}{c}^{3}{\displaystyle \frac{({\U0001d505}_{0}\U0001d505)\Re -({\U0001d505}_{0}\Re )\U0001d505}{{(\Re \U0001d505)}^{3}({\U0001d505}_{0}\U0001d505)}},$ | (47) | ||

$-m{m}_{0}c{\displaystyle \frac{({\U0001d505}_{0}\U0001d505)\Re -({\U0001d505}_{0}\Re )\U0001d505}{{(\Re \U0001d505)}^{3}}}$ | (48) |

In the latter form, Sommerfeld’s (approximate) version of Minkowski’s second law (43) matches exactly his (exact) version of Poincaré’s second law (48). Sommerfeld pointed out this equivalence, and noted again that the difference between (47) and (48) amounted to a single factor, in the scalar product of 4-velocities: $C=-({\U0001d505}_{0}\U0001d505)/{c}^{2}$. (All six Lorentz-invariant laws of gravitation of Poincaré, Minkowski, and Sommerfeld are presented in Table 1.) Sommerfeld summed up his result by saying that when the acceleration of the active mass is neglected, Minkowski’s special formulation of Newton’s law (41) is subsumed by Poincaré’s indeterminate formulation. In other words, the approximate form of Minkowski’s second law was captured by Poincaré’s remark that his first law (9) could be multiplied by an unlimited number of Lorentz-invariant quantities (within certain constraints).

The message of the basic equivalence of Poincaré’s pair of laws to
Minkowski’s pair echoes the latter’s argument in his Cologne lecture,
to the effect that spacetime mechanics removed the disharmonies of
classical mechanics and electrodynamics (see
above). This message was reinforced by Sommerfeld’s
graphical representation of the 4-vector components of these laws in a
spacetime diagram, reproduced in Figure 4. The 4-vector
relations in (47) and (48) are shown in the figure;
the worldline of the active mass $m$ appears on the left-hand side of
the diagram, and the line $OL$ (which coincides with $\Re $) lies on
the retrograde lightcone from the origin $O$ on the worldline of the
passive mass ${m}_{0}$. All three 4-vectors in (47) and
(48), $\Re $, ${\U0001d505}_{0}$, and ${\U0001d505}_{0}$ are
represented in the diagram, along with an angle $\psi $ corresponding
to the Lorentz-invariant $C=\mathrm{cos}\psi $ distinguishing (47) and
(48).^{123}^{123}
123
Sommerfeld explained Figure 4
roughly as follows: two skew 4-velocities $\U0001d505$ and
${\U0001d505}_{0}$ determine a three-dimensional space, containing all
the lines shown. Points $OLSAP$ are coplanar, while the triangles
$OQT$ and $OTS$, and the parallelogram $LQTS$ all generally lie in
distinct planes. In particular, $T$ lies outside the plane of
$OLSAP$, and $OT$ is orthogonal to ${\U0001d505}_{0}$. The broken
vertical line $l$ represents the temporal axis of a frame with
origin $O$; a spacelike plane orthogonal to $l$ at $O$ intersects
the worldline of $m$ at point $A$. The spacelike 4-vector ${\Re}^{\prime}$ is
orthogonal to $\U0001d505$, while $\U0001d516$ is orthogonal to
${\U0001d505}_{0}$; both ${\Re}^{\prime}$ and $\U0001d516$ intersect the
origin, while $\U0001d505$ and ${\U0001d505}_{0}$ together form an
angle $\psi $.

So far, Sommerfeld had dealt with three of the four laws of gravitation, leaving out only Minkowski’s first law. Since Minkowski’s presentation of his first law was a purely geometric affair, Sommerfeld had no choice but to reconstruct his argument with reference to a spacetime diagram describing the components of (27) in terms of the angle $\psi $ and a fourth 4-vector, $\U0001d516$. He showed the numerator in (47) and (48) to be equal to the product $({\U0001d505}_{0}\U0001d505)\U0001d516$, and expressed the denominator of (48) in terms of the length ${R}^{\prime}$ of the 4-vector ${\Re}^{\prime}$ in Figure 4, to obtain the formula:

$$\U0001d50e=m{m}_{0}\mathrm{cos}\psi \frac{\U0001d516}{{R}^{\prime 3}},$$ | (49) |

which he showed to be equivalent to (47). Eliminating the factor $C=\mathrm{cos}\psi $ from the latter equation, Sommerfeld obtained an expression for (48) in terms of $\U0001d516$:

$$\U0001d50e=\frac{m{m}_{0}\U0001d516}{{R}^{\prime 3}}.$$ | (50) |

The latter two driving force equations, (49) and (50), were thus rendered geometrically by Sommerfeld, facilitating the comprehension of their respective vector-symbolic expressions (47) and (48).

In general, the driving force of (49) is weaker,
ceteris paribus, than that of (50) due to the cosine
in the former, but Sommerfeld did not develop these results
numerically, noting only that the four laws were equally valid from an
empirical standpoint.^{124}^{124}
124
This view was confirmed independently
by the Dutch astronomer W. de Sitter, who worked out the numbers for
the one-body problem (De Sitter 1911).
De Sitter found the second
law to require a post-Newtonian centennial advance in Mercury’s
perihelion of 7", while the first law required no
additional advance. His figure for the second law agrees with the
one given by Poincaré. He noted that Poincaré’s analysis allowed for several other laws, but
that in all cases, one sticking-point remained: there was no answer to
the question of how to localize momentum in the gravitational field.

By rewriting Poincaré’s and Minkowski’s laws in his new 4-vector
formalism, Sommerfeld effectively rationalized their contributions for
physicists. The goal of his paper, announced at the outset, was to
display the “remarkable simplification of electrodynamic concepts and
calculations” resulting from “Minkowski’s profound spacetime
conception.”^{125}^{125}
125
“In dieser und einigen
anschließenden Studien möchte ich darstellen, wie merkwürdig sich die
elektrodynamischen Begriffe und Rechnungen vereinfachen, wenn man
sich dabei von der tiefsinnigen Raum-Zeit-Auffassung Minkowskis
leiten läßt”
(Sommerfeld 1910a, 749).
Actually, Sommerfeld’s comparison of Poincaré’s and Minkowski’s laws
of gravitation was designed to show his formalism in an
attractive light. In realizing this comparison in his own formalism,
Sommerfeld smoothed out the idiosyncrasies of Poincaré’s method,
inappropriately lending him a 4-vector approach. He felt that Poincaré
had “already employed 4-vectors” (Sommerfeld 1910b, 685),
although as shown in the first section, Poincaré’s use of
four-dimensional entities was tightly circumscribed by the objective
of formulating Lorentz-invariants. In Thomas Kuhn’s optical metaphor
(Kuhn 1970, 112), Sommerfeld read Poincaré’s theory
through a Minkowskian lens; in other words, he read it as a spacetime
theory. For Sommerfeld, no less than for Minkowski, the discussion of
gravitation and relativity was modulated by the programmatic objective
of promoting a four-dimensional formalism. Satisfying this objective
without ignoring Poincaré’s work, however, meant rationalizing
Poincaré’s contribution.^{126}^{126}
126
Faced with a similar situation in
his Cologne lecture of September, 1908, Minkowski simply neglected
to mention Poincaré’s contribution; see
Walter (1999a, 56).

Sommerfeld’s reading of Minkowski’s second law contrasts with its
muted exposition in the original text
above, in that he gave it pride of place with
respect to the other three laws. This change in emphasis on
Sommerfeld’s part reflects his own research interests in
electrodynamics, and his outlook on the future direction of
physics.^{127}^{127}
127
Sommerfeld later preferred Gustav Mie’s field theory
of gravitation. Such an approach was more promising than that of Poincaré
and Minkowski, which grasped gravitation “to some extent as
action at a distance” (Sommerfeld 1913, 73). But what
originally motivated him to propose a 4-dimensional formalism? The
inevitability of a 4-dimensional vector algebra as a standard tool of
the physicist was probably a foregone conclusion for him by 1910, such
that the promotion of the ordinary vector notation used in the
Encyklopädie obliged him to propose essentially the same
notation for 4-vectors. Sommerfeld referred modestly to his work as an
“explanation of Minkowskian ideas” (Sommerfeld 1910a, 749), but
as he explained to his friend Willy Wien, co-editor with Planck of the
Annalen der Physik, Minkowski’s original 4-vector scheme had
evolved. “The geometrical systematics” Sommerfeld announced, “is
now hyper-Minkowskian.”^{128}^{128}
128
“Die geometrische Systematik ist
jetzt hyper-minkowskisch” (Sommerfeld to Wien, 11 July, 1910,
Eckert & Märker (2001, 388).
In the same letter to Wien, Sommerfeld
confessed that his paper had required substantial effort, and he
expressed doubt that it would prove worthwhile. Sommerfeld displayed
either pessimism or modesty here, but in fact his effort was richly
rewarded, as his streamlined four-dimensional algebra and analysis
quickly won both Einstein’s praise and the confidence of his
contemporaries.^{129}^{129}
129
Einstein to Sommerfeld, July, 1910,
Klein (1993, 243–247);
Eckert & Märker (2001, 386–388). In light
of Einstein and Laub’s earlier dismissal of Minkowski’s formalism
(see above), Sommerfeld naturally supposed
that Einstein would disapprove of his system, prompting the protest:
“Wie können Sie denken, dass ich die Schönheit einer solchen
Untersuchung nicht zu schätzen wüsste?”

Sommerfeld’s work was eagerly read by young theoretical physicists raised in the heady atmosphere of German vectorial electrodynamics. One of the early adepts of Sommerfeld’s formalism was Philipp Frank (1884–1966), who was then a Privatdozent in Vienna. By way of introduction to his 1911 study of the Lorentz-covariance of Maxwell’s equations, Frank described the new four-dimensional algebra as a combination of “Sommerfeld’s intuitiveness with Minkowski’s mathematical elegance” (Frank 1911, 600). He recognized, however, that of late, physicists had been overloaded with outlandish symbolic systems and terminology, and promised to stay within the boundaries of Sommerfeld’s system, at least as far as this was possible.

Physicists were indeed inundated in 1910–1911 with a bewildering array of new symbolic systems, including an ordinary vector algebra (Burali-Forti & Marcolongo 1910), and a quaternionic calculus (Conway 1911), in addition to the hyperbolic-coordinate system and three 4-vector formalisms already mentioned. By 1911, 4-vector and 6-vector operations featured prominently in the pages of the Annalen der Physik. Out of the nine theoretical papers concerning relativity theory published in the Annalen that year, five made use of a four-dimensional approach to physics, either in terms of 4-vector operations, or by referring to spacetime coordinates. Four out of five authors of “four-dimensional” papers cited Minkowski’s or Sommerfeld’s work; the fifth referred to Max Laue’s new relativity textbook (Laue 1911). This timely and well-written little book went far in standardizing the terminology and notation of four-dimensional algebra, such that by January of 1912, Max Abraham preferred the Sommerfeld-Laue notation to his own for the exposition of his theory of gravitation (Abraham 1910, 1912a, 1912b).

While young theorists were quick to pick up on the Sommerfeld-Laue calculus, textbook writers did not follow the trend. Of the four textbooks to appear on relativity in 1913–1914, only the second edition of Laue’s book (Laue 1913) employed this formalism. Ebenezer Cunningham presented a 4-dimensional approach based on Minkowski’s work, but explicitly rejected Sommerfeld’s “quasi-geometrical language”, which conflicted with his own purely algebraic presentation (Cunningham 1914, 99). A third textbook by Ludwik Silberstein (1914), a former student of Planck, gave preference to a quaternionic presentation, while the fourth, by Max B. Weinstein (1913), opted for Cartesian coordinates. Curiously enough, Weinstein dedicated his work to the memory of Minkowski. Apparently disturbed by this profession of fidelity, Max Born, who had briefly served as Minkowski’s assistant, deplored the form of Weinstein’s approach to relativity:

[Minkowski] put perhaps just as much value on his presentation as on its content. For this reason, I do not believe that entrance to his conceptual world is facilitated when it is overwhelmed by an enormous surfeit of formulas.

^{130}^{130}130 “[Minkowski] hat auf seine Darstellung vielleicht ebenso viel Wert gelegt, wie auf ihren Inhalt. Darum glaube ich nicht, daß der Zugang zu seiner Gedankenwelt erleichtert wird, wenn sie von einer ungeheuren [sic] Fülle von Formeln überschüttet wird” (Born 1914).

By this time, Born himself had dropped Minkowski’s formalism in favor
of the Sommerfeld-Laue approach, such that the target of his criticism
was Weinstein’s disregard for 4-dimensional methods in general, and
not the neglect of Minkowski’s matrix calculus.^{131}^{131}
131
By the end of
1911 Born had already acknowledged that, despite its “formal simplicity
and greater generality compared to the tradition of vectorial
notation,” Minkowski’s calculus was “unable to hold its ground in
mathematical physics” (Born 1912, 175). What Born was
pointing out here was that it had become highly impractical to study
the theory of relativity without recourse to a 4-dimensional
formalism. This may explain why Laue’s was the only one of the four
textbooks on relativity to be reedited, reaching a sixth edition in 1955.

In summary, the language developed by Sommerfeld for the expression of the laws of gravitation of Poincaré and Minkowski endured, while the laws themselves remained tentative at best. This much was clear as early as 1912, when Jun Ishiwara reported from Japan on the state of relativity theory. This theory, Ishiwara felt, had shed no light on the problem of gravitation, with a single exception: Minkowski and Sommerfeld’s “formal mathematical treatment” (Ishiwara 1912, 588). The trend from Poincaré to Sommerfeld was one of increasing reliance on formal techniques catering to Lorentz-invariance; in the space of five years, the physical content of the laws of gravitation remained stable, while their formal garb evolved from Cartesian to hyper-Minkowskian.

$$\begin{array}{ccc}\hfill {\mathbf{\text{Poincar\xe9 (1906)}}}^{\text{a}}\hfill & \hfill {\mathbf{\text{Minkowski (1908)}}}^{\text{b}}\hfill & \hfill {\mathbf{\text{Sommerfeld (1910)}}}^{\text{c}}\hfill \\ \hfill \begin{array}{cc}\hfill {X}_{1}& =\frac{x}{{k}_{0}{B}^{3}}-{\xi}_{1}\frac{{k}_{1}}{{k}_{0}}\frac{A}{{B}^{3}C}\hfill \\ \hfill {Y}_{1}& =\frac{y}{{k}_{0}{B}^{3}}-{\eta}_{1}\frac{{k}_{1}}{{k}_{0}}\frac{A}{{B}^{3}C}\hfill \\ \hfill {Z}_{1}& =\frac{z}{{k}_{0}{B}^{3}}-{\zeta}_{1}\frac{{k}_{1}}{{k}_{0}}\frac{A}{{B}^{3}C}\hfill \\ \hfill {T}_{1}& =-\frac{r}{{k}_{0}{B}^{3}}-\frac{{k}_{1}}{{k}_{0}}\frac{A}{{B}^{3}C}\hfill \end{array}\hfill & \hfill m{m}^{*}{\left(\frac{O{A}^{\prime}}{{B}^{*}{D}^{*}}\right)}^{3}B{D}^{*}\hfill & \hfill m{m}_{0}{c}^{3}\frac{({\U0001d505}_{0}\U0001d505)\Re -({\U0001d505}_{0}\Re )\U0001d505}{{(\Re \U0001d505)}^{3}({\U0001d505}_{0}\U0001d505)}\hfill \\ & & \\ \hfill \begin{array}{cc}\hfill {X}_{1}& =\frac{\lambda}{{B}^{3}}-\frac{\eta {\nu}^{\prime}-\zeta {\mu}^{\prime}}{{B}^{3}}\hfill \\ \hfill {Y}_{1}& =\frac{\mu}{{B}^{3}}-\frac{\zeta {\lambda}^{\prime}-\xi {\nu}^{\prime}}{{B}^{3}}\hfill \\ \hfill {Z}_{1}& =\frac{\nu}{{B}^{3}}-\frac{\xi {\mu}^{\prime}-\eta {\lambda}^{\prime}}{{B}^{3}}\hfill \end{array}\hfill & \hfill m{m}_{1}\left({\dot{t}}_{1}-\frac{{\dot{x}}_{1}}{c}\right)\U0001d50e\hfill & \hfill -m{m}_{0}c\frac{({\U0001d505}_{0}\U0001d505)\Re -({\U0001d505}_{0}\Re )\U0001d505}{{(\Re \U0001d505)}^{3}}\hfill \\ & & \end{array}$$ |

${}^{\text{a}}$Mass terms are neglected, such that the right-hand side of each equation is implicitly multiplied by the product of the two masses. When both sides of the four equations are multiplied by the factor ${k}_{0}$, they express components of a 4-vector, ${k}_{0}({X}_{1},{Y}_{1},{Z}_{1},i{T}_{1})$. The constants ${k}_{0}$ and ${k}_{1}$ are defined as: ${k}_{0}=1/\sqrt{1-\sum {\xi}^{2}}$ and ${k}_{1}=1/\sqrt{1-\sum {\xi}_{1}^{2}}$. $A$, $B$, and $C$ denote the last three Lorentz-invariants in (1): $A=\frac{t-{\scriptscriptstyle \sum x\xi}}{\sqrt{1-{\scriptscriptstyle \sum {\xi}^{2}}}}$, $B=\frac{t-{\scriptscriptstyle \sum x{\xi}_{1}}}{\sqrt{1-{\scriptscriptstyle \sum {\xi}_{1}^{2}}}}$, $C=\frac{1-{\scriptscriptstyle \sum \xi {\xi}_{1}}}{\sqrt{\left(1-{\scriptscriptstyle \sum {\xi}^{2}}\right)\left(1-{\scriptscriptstyle \sum {\xi}_{1}^{2}}\right)}}$, where $\sum \xi $ and $\sum {\xi}_{1}$ designate the ordinary velocities of the passive and active mass points, with components $\xi $, $\eta $, $\zeta $, and ${\xi}_{1}$, ${\eta}_{1}$, ${\zeta}_{1}$. The time $t$ is set equal to the negative distance between the passive mass point and the retarded position of the active mass point, $t=-\sqrt{\sum {x}^{2}}=-r$. Poincaré’s second law is shown in the bottom row; he neglected to write the fourth component ${T}_{1}$, determined from the first three by the orthogonality condition ${T}_{1}=\sum {X}_{1}\xi $. The new variables in the bottom row are:

$\lambda $ | $={k}_{1}\left(x+r{\xi}_{1}\right),$ | $\mu $ | $={k}_{1}\left(y+r{\eta}_{1}\right),$ | $\nu $ | $={k}_{1}\left(z+r{\zeta}_{1}\right),$ | ||

${\lambda}^{\prime}$ | $={k}_{1}\left({\eta}_{1}z-{\zeta}_{1}y\right),$ | ${\mu}^{\prime}$ | $={k}_{1}\left({\zeta}_{1}x-{\xi}_{1}z\right),$ | ${\nu}^{\prime}$ | $={k}_{1}\left({\xi}_{1}y-x{\eta}_{1}\right).$ |

${}^{\text{b}}$The formula in the top row describes the first three components of the driving force; the fourth component is obtained analytically. The constants $m$ and $m*$ designate the passive and active proper mass, respectively, while the remaining letters stand for spacetime points, as reconstructed in Figure 1. The formula in the bottom row represents the driving force of gravitation as described, but not formally expressed, by Minkowski (1909). The constants $m$ and ${m}_{1}$ designate the active and passive proper mass, ${\dot{t}}_{1}$ and ${\dot{x}}_{1}$ are 4-velocity components of the passive mass, $c$ is the speed of light and $\U0001d50e$ is a 4-vector, for the definition of which see note 111.

${}^{\text{c}}$The constants ${m}_{0}$ and $m$ designate the passive and active proper mass, respectively, $c$ denotes the speed of light, ${\U0001d505}_{0}$ and $\U0001d505$ represent the corresponding 4-velocities, and $\Re $ stands for the lightlike interval between the mass points.

* * *

## 4 Conclusion: On the emergence of the four-dimensional view

After a century-long process of accommodation to the use of tensor calculus and spacetime diagrams for analysis of physical interactions, the mathematical difficulties encountered by the pioneers of 4-dimensional physics are hard to come to terms with. Not only is the oft-encountered image of flat-spacetime physics as a trivial consequence of Einstein’s special theory of relativity and Felix Klein’s geometry consistent with such accommodation, it reflects Minkowski’s own characterization of the background of the four-dimensional approach. However, this description ought not be taken at face value, being better understood as a rhetorical ploy designed to induce mathematicians to enter the nascent field of relativistic physics (Walter 1999a). When the principle of relativity was formulated in 1905, even for one as adept as Henri Poincaré in the application of group methods, the path to a four-dimensional language for physics appeared strewn with obstacles. Much as Poincaré had predicted (above), the construction of this language cost Minkowski and Sommerfeld considerable pain and effort.

Clear-sighted as he proved to be in this regard, Poincaré did not foresee the emergence of forces that would accelerate the construction and acquisition of a four-dimensional language. With hindsight, we can identify five factors favoring the use and development of a four-dimensional language for physics between 1905 and 1910: the elaboration of new concepts and definitions, the introduction of a graphic model of spacetime, the experimental confirmation of relativity theory, the vector-symbolic movement, and problem-solving performance.

In the beginning, the availability of workable four-dimensional concepts and definitions regulated the analytic reach of a four-dimensional approach to physics. Poincaré’s discovery of the 4-vectors of velocity and force in the course of his elaboration of Lorentz-invariant quantities, and Minkowski’s initial misreading of Poincaré’s definitions underline how unintuitive these notions appeared to turn-of-the-century mathematicians. The lack of a 4-velocity definition visibly hindered Minkowski’s elaboration of spacetime mechanics and theory of gravitation. It is remarkable that even after Minkowski presented the notions of proper time, worldline, rest-mass density, and the energy-momentum tensor, putting the spacetime electrodynamics and mechanics on the same four-dimensional footing, his approach failed to convince physicists. Nevertheless, all of these discoveries extended the reach of the four-dimensional approach, in the end making it a viable candidate for the theorist’s toolbox.

Next, Minkowski’s visually-intuitive spacetime diagram played a decisive role in the emergence of the four-dimensional view. While the spacetime diagram reflects some of the concepts mentioned above, its utility as a cognitive tool exceeded by far that of the sum of its parts. In Minkowski’s hands, the spacetime diagram was more than a tool, it was a model used to present both of his laws of gravitation. Beyond their practical function in problem-solving, spacetime diagrams favored the diffusion in wider circles of both the theory of relativity and the four-dimensional view of this theory, in particular among non-mathematicians, by providing a visually intuitive means of grasping certain consequences of the theory of relativity, such as time dilation and Lorentz contraction. Minkowski’s graphic model of spacetime thus enhanced both formal and intuitive approaches to special relativity.

In the third place, the ultimate success of the four-dimensional view hinged on the empirical adequacy of the theory of relativity. It is remarkable that the conceptual groundwork, and much of the formal elaboration of the four-dimensional view was accomplished during a time when the theory of relativity was less well corroborated by experiment than its rivals. The reversal of this situation in favor of relativity theory in late 1908 favored the reception of the existing four-dimensional methods, and provided new impetus both for their application and extension, and for the development of alternatives, such as that of Sommerfeld.

The fourth major factor influencing the elaboration of a four-dimensional view of physics was the vector-symbolic movement in physics and mathematics at the turn of the twentieth century (McCormmach 1976, xxxi). The participants in this movement, in which Sommerfeld was a leading figure, believed in the efficacy of vector-symbolic methods in physics and geometry, and sought to unify the plethora of notations employed by various writers. The movement’s strength varied from country to country; it was largely ignored in France, for example, in favor of the coordinate-based notation favored by Poincaré and others. Poincaré’s pronounced disinterest in the application and development of a four-dimensional calculus for physics was typical of contemporary French attitudes toward vector-symbolic methods. In Germany, on the other hand, electrodynamicists learned Maxwell’s theory from the mid-1890s in terms of curl $\U0001d525$ and div $\U0001d508$. In Zürich and Göttingen during this period, Minkowski instructed students – including Einstein – in the ways of the vector calculus. Unlike Poincaré, Minkowski was convinced that a four-dimensional language for physics would be worth the effort spent on its elaboration, yet he ultimately abandoned the vector-symbolic model in favor of an elegant and sophisticated matrix calculus. This choice was deplored by physicists (including Einstein), and mooted by Sommerfeld’s conservative extension of the standard vector formalism into an immediately successful 4-vector algebra and analysis. In sum, the vector-symbolic movement functioned alternatively as an accelerator of the elaboration of four-dimensional calculi (existing systems served as templates), and as a regulator (penalizing Minkowski’s neglect of standard vector operations).

The fifth and final parameter affecting the emergence of the four-dimensional view of physics was problem-solving performance. From the standpoint of ease of calculation, any four-dimensional vector formalism at all compared well to a Cartesian-coordinate approach, as Weinstein’s textbook demonstrated; the advantage of ordinary vector methods over Cartesian coordinates was less pronounced. As we have seen, Poincaré applied his approach to the problem of constructing a Lorentz-invariant law of gravitational attraction, and was followed in turn by Minkowski and Sommerfeld, both of whom also provided examples of problem-solving. In virtue of the clarity and order of Sommerfeld’s detailed, coordinate-free comparison of the laws of gravitation of Poincaré and Minkowski, his 4-vector algebra appeared to be the superior four-dimensional approach, just when physicists and mathematicians were turning to relativity in greater numbers.

Acknowledgments

This study was inaugurated with the encouragement and support of
Jürgen Renn, during a stay at the Max-Planck-Institut für
Wissenschaftsgeschichte (Berlin) in 1998. I am especially grateful
to Urs Schoepflin and the MPIWG library staff for their expert
assistance. The themes explored here were presented at the
Mathematisches Forschunginstitut Oberwolfach (Jan., 2000), the
University of Heidelberg (June, 2000), the University of Paris
7–REHSEIS (April, 2001), the joint AMS/SMF meeting in Lyons (July,
2001), and the Eighth International Conference on the History of
General Relativity (Amsterdam, June, 2002). Several scholars shared
their ideas with me, shaping the final form of the paper. In
particular, I am indebted to Olivier Darrigol for insightful comments
on an early draft. The paper has been improved thanks to a careful
reading by Shaul Katzir, and discussions with John Norton and
Philippe Lombard. The responsibility for any remaining infelicities
is my own.

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