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2.1 Poincare’s theory of gravitation.

In his letters to Lorentz, Poincaré noted that while he had concocted an electron model that was both stable and relativistic, in the new theory he was unable to preserve the “unity of time”, i.e., a definition of duration valid in both the ether and in moving frames.

From a physical point of view, all frames are moving frames.
The biggest problem is that the speed of light in each frame is the same and constant.
If the observer is considered at rest, how is the speed of any moving object calculated?

The final section of Poincare's memoir is devoted to a topic he had neglected to broach with Lorentz, and that Einstein had neglected altogether: gravitation.

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If the principle of relativity was to be universally valid, Poincaré reasoned, then Newton’s law of gravitation would have to be modified.

The question to ask is: What do you want to understand?
The answer should be something like: What causes the rotation of the stars in our galaxy, the Milky way.

More and more theorists recognized the advantages of vector analysis, and also of a unified vector notation for mathematical physics.

Why using the concept: mathematical physics?

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In particular, Poincaré’s graphical model of light propagation does not display relativity of simultaneity for inertial observers, since it represents a single frame of motion.

Why this problem?

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This sequence of events raises the question of what led Poincaré to embrace the notion of time deformation in moving frames, and to repurpose his light ellipse?

The concept time deformation is physical a very difficult problem.

He didn’t say, but there is a plausible explanation at hand, that I will return to later, as it rests on events in the history of relativity from 1907 to 1908 to be discussed in the next section.

2.2 Minkowski’s path to spacetime

2.3 Spacetime diagrams

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Annalen der Physik, edited by Paul Drude until his suicide in 1906, then by Max Planck and Willy Wien, bear witness to this evolution. Even in the pages of the Annalen der Physik, however, notation was far from standardized, leading several theorists to deplore the field’s babel of symbolic expressions. Among the theorists who regretted the multiplication of systems of nota- tion was Poincaré, who employed ordinary vectors in his own teaching and publications on electrodynamics, while ignoring the notational innovations of Lorentz and others. In particular, Poincaré saw no future for a four- dimensional vector calculus. Expressing physical laws by means of such a calculus, he wrote in 1907, would entail “much trouble for little profit” [11, 438]. This was not a dogmatic view, and in fact, some years later he acknowledged the value of a four-dimensional approach in theoretical physics [12, 210]. He was already convinced that there was a place for (3 + n)-dimensional geometries at the university. As Poincaré observed in the paper Gaston Darboux read in his stead at the International Congress of Mathematicians in Rome, in April, 1908, university students were no longer taken aback by geometries with “more than three dimensions” [13, 938]. Relativity theory, however, was another matter for Poincaré. Recently- rediscovered manuscript notes by Henri Vergne of Poincaré’s lectures on rela- tivity theory in 1906–1907 reveal that Poincaré introduced his students to the Lorentz Group, and taught them how to form Lorentz-invariant quantities with real coordinates. He also taught his students that the sum of squares (2.3) is invariant with respect to the transformations of the Lorentz Group. Curiously, Poincaré did not teach his students that a Lorentz transforma- tion corresponded to a rotation about the origin in a four-dimensional vector space with one imaginary coordinate. He also neglected to show students the handful of four-vectors he had defined in the summer of 1905. Apparently for Poincaré, knowledge of the Lorentz Group and the formation of Lorentz- invariant quantities was all that was needed for the physics of relativity. In other words, Poincaré acted as if one could do without an interpretation of the Lorentz transformation in four-dimensional geometry. If four-dimensional geometry was superfluous to interpretation of the Lorentz transformation, the same was not true for plane geometry. Evidence of this view is found in Vergne’s notes, which feature a curious figure that I’ll call a light ellipse, redrawn here as Figure 2.1. Poincaré’s light ellipse is given to be the meridional section of an ellipsoid of rotation representing the locus of a spherical light pulse at an instant of time. It works as follows: an observer at rest with respect to the ether measures the radius of a spherical light pulse at an instant of absolute time t (as determined by clocks at rest

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Figure 2.1: Poincaré’s light ellipse, after manuscript notes by Henri Vergne, 1906–1907 (Henri Poincaré Archives). Labels H and A are added for clarity.

with respect to the ether). The observer measures the light pulse radius with measuring rods in uniform motion of velocity v. These flying rods are Lorentz-contracted, while the light wave is assumed to propagate spherically in the ether. Consequently, for Poincaré, the form of a spherical light pulse measured in this fashion is that of an ellipsoid of rotation, elongated in the direction of motion of the flying rods. (A derivation of the equation of Poincaré’s light ellipse is provided along these lines in [14].)
The light ellipse originally concerned ether-fixed observers measuring a locus of light with clocks at absolute rest, and rods in motion. Notably, in his first discussion of the light ellipse, Poincaré neglected to consider the point of view of observers in motion with respect to the ether. In particular, Poincaré’s graphical model of light propagation does not display relativity of simultaneity for inertial observers, since it represents a single frame of motion. Nonetheless, Poincaré’s light ellipse was applicable to the case of observers in uniform motion, as he showed himself in 1909. In this case, the radius vector of the light ellipse represents the light-pulse radius at an instant of “apparent” time t
′ , as determined by comoving, light-synchronized clocks,

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and comoving rods corrected for Lorentz-contraction. Such an interpretation implies that clock rates depend on frame velocity, as Einstein recognized in 1905 in consequence of his kinematic assumptions about ideal rods and clocks [5, 904], and which Poincaré acknowledged in a lecture in G¨ ottingen on 28 April, 1909, as an effect epistemically akin to Lorentz-contraction, induced by clock motion with respect to the ether [15, 55].
Beginning in August 1909, Poincaré repurposed his light ellipse diagram to account for the apparent dilation of periods of ideal clocks in motion with respect to the ether [16, 174]. This sequence of events raises the question of what led Poincaré to embrace the notion of time deformation in moving frames, and to repurpose his light ellipse?

the notion of time deformation are physical very complex.

He didn’t say, but there is a plausible explanation at hand, that I will return to later, as it rests on events in the history of relativity from 1907 to 1908 to be discussed in the next section

2.2 Minkowski’s path to spacetime From the summer of 1905 to the fall of 1908, the theory of rela

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Figure 2.2: A reconstruction of Minkowski’s 5 Nov. 1907 presentation of relativistic velocity space, with a pair of temporal axes, one spatial axis, a unit hyperbola and its asymptotes. These two surfaces, a pseudo-hypersphere of unit imaginary radius (2.6), and its real counterpart, the two-sheeted unit hyperboloid (2.7), give rise to well-known models of hyperbolic space, popularized by Helmholtz in the late nineteenth century [31, Vol. 2]. The upper sheet (t> 0) of the unit hyperboloid (2.7) models hyperbolic geometry; for details, see [32]. The conjugate diameters of the hyperboloid (2.7) give rise to a geometric image of the Lorentz transformation. Any point on (2.7) can be considered to be at rest, i.e., it may be taken to lie on a t-diameter, as shown in Figure 2.2. This change of axes corresponds to an orthogonal transformation of the time and space coordinates, which is a Lorentz transformation (letting c = 1). In other words, the three-dimensional hyperboloid (2.7) embedded in four-dimensional pseudo-Euclidean space affords an interpretation of the Lorentz transformation. Although Minkowski did not spell out his geometric interpretation, he prob- ably recognized that a displacement on the hypersurface (2.7) corresponds to a rotation ψ about the origin, such that frame velocity v is described by a hyperbolic function, v = tanh ψ. However, he did not yet realize that his hypersurfaces represent the set of events occurring at coordinate time t ′ =1

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of all inertial observers, the worldlines of whom pass through the origin of coordinates (with a common origin of time). According to (2.7), this time is imaginary, a fact which may have obscured the latter interpretation. How do we know that Minkowski was still unaware of worldlines in spacetime? Inspection of Minkowski’s definition of four-velocity vectors reveals an error, which is both trivial and interesting: trivial from a mathematical standpoint, and interesting for what it says about his knowledge of the structure of spacetime, and the progress he had realized toward his goal of replacing the Euclidean geometry of phenomenal space with the geometry of a four- dimensional non-Euclidean manifold. When faced with the question of how to define a four-velocity vector, Min- kowski had the option of adopting the definition given by Poincar´ e in 1905. Instead, he rederived his own version, by following a simple rule. Minkowski defined a four-vector potential, four-current density, and four-force density, all by simply generalizing ordinary three-component vectors to their four- component counterparts. When he came to define four-velocity, he took over the components of the ordinary velocity vector w for the spatial part of four- velocity, and added an imaginary fourth component, i . (2.8)
Since the components of Minkowski’s quadruplet do not transform like the coordinates of his vector space x
, they lack what he knew to be a four-vector property. Minkowski’s error in defining four-velocity indicates that he did not yet grasp the notion of four-velocity as a four-vector tangent to the worldline of a particle [8]. If we grant ourselves the latter notion, then we can let the square of the differential parameter dτ of a given worldline be dτ 2 = -(dx 2 1 + dx ), such that the 4-velocity w μ may be defined as the first derivative with respect to τ , w μ = dx μ /dτ (µ = 1, 2, 3, 4).
In addition to a valid four-velocity vector, Minkowski was missing a four-force vector, and a notion of proper time. In light of these significant lacunæ in his knowledge of the basic mathematical objects of four-dimensional physics, Minkowski’s triumphant description of his four-dimensional formalism as “virtually the greatest triumph ever shown by the application of mathematics” [28, 373] is all the more remarkable, and bears witness to the depth of Minkowski’s conviction that he was on the right track.
Sometime after Minkowski spoke to the G¨ ottingen Mathematical Society, he repaired his definition of four-velocity, and perhaps in connection with this, he came up with the constitutive elements of his theory of spacetime.

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In particular, he formulated the idea of proper time as the parameter of a hyperline in spacetime, the light-hypercone structure of spacetime, and the spacetime equations of motion of a material particle. He expressed his new theory in a sixty-page memoir [33] published in the G¨ ottinger Nachrichten on 5 April, 1908.
His memoir, entitled “The basic equations for electromagnetic processes in moving bodies” made for challenging reading. It was packed with new no- tation, terminology, and calculation rules, it made scant reference to the scientific literature, and offered no figures or diagrams. Minkowski defined a single differential operator, named lor in honor of Lorentz, which streamlined his expressions, while rendering them all the more unfamiliar to physicists used to the three-dimensional operators of ordinary vector analysis. Along the same lines, Minkowski rewrote velocity, denoted q, in terms of the tangent of an imaginary angle iψ, q = -i tan iψ, (2.9)
where q< 1. From his earlier geometric interpretation of hyperbolic velocity space, Minkowski kept the idea that every rotation of a t-diameter corresponds to a Lorentz transformation, which he now expressed in terms of iψ: (2.10)
Minkowski was undoubtedly aware of the connection between the compo- sition of Lorentz transformations and velocity composition, but he did not mention it. In fact, Minkowski neither mentioned Einstein’s law of velocity addition, nor expressed it mathematically.
While Minkowski made no appeal in “The basic equations” to the hyperbolic geometry of velocity vectors, he retained the hypersurface (2.7) on which it was based, and provided a new interpretation of its physical significance. This interpretation represents an important clue to understanding how Minkowski discovered the worldline structure of spacetime. The appendix to “The basic equations” rehearses the argument according to which one may choose any point on (2.7) such that the line from this point to the origin forms a new time axis, and corresponds to a Lorentz transformation. He further defined a “spacetime line” to be the totality of spacetime points corresponding to any particular point of matter for all time t.
With respect to the new concept of a spacetime line, Minkowski noted that its direction is determined at every spacetime point. Here Minkowski intro- duced the notion of “proper time” (Eigenzeit ), τ , expressing the increase of

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coordinate time dt for a point of matter with respect to dτ : which silently corrects the flawed definition of this fourth component of four- velocity (2.8) delivered by Minkowski in his November 5 lecture. Although Minkowski did not connect four-velocity to Einstein’s law of ve- locity addition, others did this for him, beginning with Sommerfeld, who expressed parallel velocity addition as the sum of tangents of an imaginary angle [34]. Minkowski’s former student Philipp Frank reexpressed both ve- locity and the Lorentz transformation as hyperbolic functions of a real angle [35]. The Serbian mathematician Vladimir Variˇ cak found relativity theory to be ripe for application of hyperbolic geometry , and recapitulated several

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which silently corrects the flawed definition of this fourth component of four- velocity (2.8) delivered by Minkowski in his November 5 lecture.
Although Minkowski did not connect four-velocity to Einstein’s law of velocity addition, others did this for him, beginning with Sommerfeld, who expressed parallel velocity addition as the sum of tangents of an imaginary angle [34]. Minkowski’s former student Philipp Frank reexpressed both velocity and the Lorentz transformation as hyperbolic functions of a real angle [35]. The Serbian mathematician Vladimir Varicak found relativity theory to be ripe for application of hyperbolic geometry, and recapitulated several relativistic formulæ in terms of hyperbolic functions of a real angle [36]. A small group of mathematicians and physicists pursued this “non-Euclidean style” of Minkowskian relativity, including Variˇ cak, Alfred Robb, Emile Borel, Gilbert Newton Lewis, and Edwin Bidwell Wilson [37].
The definition of four-velocity was formally linked by Minkowski to the hy- perbolic space of velocity vectors in “The basic equations”, and thereby to the lightcone structure of spacetime. Some time before Minkowski came to study the Lorentz transformation in earnest, both Einstein and Poincaré understood light waves in empty space to be the only physical objects immune to Lorentz contraction. Minkowski noticed that when light rays are considered as worldlines, they divide spacetime into three regions, corresponding to the spacetime region inside a future-directed (t> 0) hypercone (“Nachkegel ”), the region inside a past-directed (t< 0) hypercone (“Vorkegel ”), and the region outside any such hypercone pair. The propagation in space and time of a spherical light wave is described by a hypercone, or what Minkowski called a lightcone (“Lichtkegel ”).
One immediate consequence for Minkowski of the lightcone structure of spacetime concerned the relativity of simultaneity. In a section of “The basic equations” entitled “The concept of time”, Minkowski [33, § 6] showed that Einstein’s relativity of simultaneity is not absolute. While the relativ- ity of simultaneity is indeed valid for two or three simultaneous “events” (Ereignisse ), the simultaneity of four events is absolute, so long as the four spacetime points do not lie on the same spatial plane. Minkowski’s demon- stration relied on the Einstein simultaneity convention, and employed both light signals and spacetime geometry. His result showed the advantage of employing his spacetime geometry in physics, and later writers – including

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Poincaré – appear to have agreed with him, by attributing to spacetime ge- ometry the discovery of the existence of a class of events for a given observer that can be the cause of no other events for the same observer [12, 210]. Another signature result of Minkowski’s spacetime geometry was the geo- metric derivation of a Lorentz-covariant law of gravitation. Like Poincaré, Minkowski proposed two four-vector laws of gravitation, exploiting analo- gies to Newtonian gravitation and Maxwellian electrodynamics, respectively. Minkowski presented only the Newtonian version of the law of gravitation in “The basic equations”, relating the states of two massive particles in arbri- trary motion, and obtaining an expression for the spacelike component of the four-force of gravitation. Although his derivation involved a new spacetime geometry, Minkowski did not illustrate graphically his new law, a decision which led some physicists to describe his theory as unintelligible. Accord- ing to Minkowski, however, his achievement was a formal one, inasmuch as Poincaré had formulated his theory of gravitation by proceeding in what he described as a “completely different way” [8, 225].
Few were impressed at first by Minkowski’s innovations in spacetime geome- try and four-dimensional vector calculus. Shortly after “The basic equations” appeared in print, two of Minkowski’s former students, Einstein and Laub, discovered what they believed to be an infelicity in Minkowski’s definition of ponderomotive force density [38]. These two young physicists were more im- pressed by Minkowski’s electrodynamics of moving media than by the novel four-dimensional formalism in which it was couched, which seemed far too laborious. Ostensibly as a service to the community, Einstein and Laub reex- pressed Minkowski’s theory in terms of ordinary vector analysis [39, Doc. 51]. Minkowski’s reaction to the latter work is unknown, but it must have come to him as a disappointment. According to Max Born, Minkowski always aspired . . . to find the form for the presentation of his thoughts that cor- responded best to the subject matter. [40]
The form Minkowski gave to his theory of moving media in “The basic equations” had been judged unwieldy by a founder of relativity theory, and in the circumstances, decisive action was called for if his formalism was not to be ignored. In September 1908, during the annual meeting of the German Association of Scientists and Physicians in Cologne, Minkowski took action, by affirming the reality of the four-dimensional “world”, and its necessity for physics [41]. The next section focuses on the use to which Minkowski put spacetime diagrams in his Cologne lecture, and how these diagrams relate to Poincaré’s light ellipse.

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2.3. SPACETIME DIAGRAMS 17 Figure 2.3: The lightcone structure of spacetime

One way for Minkowski to persuade physicists of the value of his spacetime approach to understanding physical interactions was to appeal to their vi- sual intuition [30]. From the standpoint of visual aids, the contrast between Minkowski’s two publications on spacetime is remarkable: where “The ba- sic equations” is bereft of diagrams and illustrations, Minkowski’s Cologne lecture makes effective use of diagrams in two and three dimensions. For in- stance, Minkowski employed a two-dimensional spacetime diagrams to illus- trate FitzGerald-Lorentz contraction of an electron, and the lightcone struc- ture of spacetime (see Figure 2.3).
Minkowski’s lecture in Cologne, entitled “Space and time”, offered two dia- grammatic readings of the Lorentz transformation, one of his own creation, the other he attributed to Lorentz and Einstein. One of these readings was supposed to represent the kinematics of the theory of relativity of Lorentz and Einstein. In fact, Minkowski’s reading captured Lorentzian kinemat- ics, but distorted those of Einstein, and prompted corrective action from Philipp Frank, Guido Castelnuovo, and Max Born [42]. The idea stressed by Minkowski was that in the (Galilean) kinematics employed in Lorentz’s elec- tron theory, time being absolute, the temporal axis on a space-time diagram may be rotated freely about the coordinate origin in the upper half-plane (t> 0), as shown in Figure 2.4. The location of a point P may be described with respect to frames S and S ′ , corresponding to axes (x,t) and (x ′ ,t ′ ), respectively, according to the transformation: x ′ = x - vt, t ′ = t. In contradistinction to the latter view, the theory proposed by Minkowski required a certain symmetry between the spatial and temporal axes. This constraint on symmetry sufficed for a geometric derivation of the Lorentz transformation. Minkowski described his spacetime diagram (Figure 2.5) as an illustration of the Lorentz transformation, and provided an idea of a demonstration in “Space and time”. A demonstration was later supplied by Sommerfeld, in an editorial note to his friend’s lecture [43], which appeared in an anthology of papers on the theory of relativity edited by Otto Blumenthal

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Figure 2.4: A reconstruction of Minkowski’s depiction of the kinematics of Lorentz and Einstein.

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Figure 2.5: Minkowski’s spacetime diagram, after [1]. [44, 37]. Minkowski’s spacetime map was not the only illustration of relativistic kinematics available to scientists in the first decade of the twentieth century.
Theorists pursuing the non-Euclidean style of Minkowskian relativity had recourse to models of hyperbolic geometry on occasion. The Poincaré half- plane and disk models of hyperbolic geometry were favored by Variˇ cak in this context, for example. Poincaré himself did not employ such models in his investigations of the principle of relativity, preferring his light ellipse. Of these three types of diagram, the light ellipse, spacetime map, and hyperbolic map, only the spacetime map attracted a significant scientific following.
The relation between the spacetime map and the hyperbolic maps was underlined by Minkowski, as shown above in relation to surfaces (2.6) and (2.7). There is also a relation between the light ellipse and the spacetime map, al- though this may not have been apparent to either Poincaré or Minkowski.
Their published appreciations of each other’s contributions to relativity field the barest of acknowledgments, suggesting no substantial intellectual indebtedness on either side. The diagrams employed in the field of relativity by Poincaré and Minkowski differ in several respects, but one difference in particular stands out. On the one hand, the light ellipse represents spatial relations in a plane defined as a

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Figure 2.6: Spacetime model of Poincaré’s light ellipse (1906) in a spacelike plane (t =const.).

meridional section of an ellipsoid of rotation. A Minkowski diagram, on the other hand, involves a temporal axis in addition to a spatial axis (or two, for a three-dimensional spacetime map). This difference does not preclude representation of a light ellipse on a Minkowski diagram, as shown in Figures 2.6 and 2.7, corresponding, respectively, to the two interpretations of the Lorentz transformation offered by Poincaré before and after 1909.
In Poincaré’s pre-1909 interpretation of the Lorentz transformation, the ra- dius vector of the light ellipse corresponds to light points at an instant of time t as read by clocks at rest in the ether frame. The representation of this situation on a Minkowski diagram is that of an ellipse contained in a spacelike plane of constant time t (Figure 2.6). The ellipse center coincides with spacetime point B =(vt, 0,t), and the points E, B, F , and A lie on the major axis, such that BH is a semi-minor axis of length ct. The light ellipse intersects the lightcone in two points, corresponding to the endpoints of the minor axis, H and I . There are no moving clocks in this reading, only mea- suring rods in motion with respect to the ether. (The t -axis is suppressed in Figure 2.6 for clarity). The abstract nature of Poincaré’s early interpretation of the light ellipse is apparent in the Minkowskian representation, in that there are points on the light ellipse that lie outside the lightcone, and are

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Figure 2.7: Spacetime model of Poincaré’s light ellipse (1909) in a spacelike plane (t ′ =const.). physically inaccessible to an observer at rest in the ether. In Poincaré’s post-1909 repurposing of the light ellipse, the light pulse is measured with comoving clocks, such that the corresponding figure on a Minkowski diagram is an ellipse in a plane of constant time t ′ . The latter x ′ y ′ -plane intersects the lightcone at an oblique angle, as shown in Figure 2.7, such that their intersection is a Poincaré light ellipse. (The y-axis and the y ′ -axis are suppressed for clarity).
Both before and after 1909, Poincaré found that a spherical light pulse in the ether would be described as a prolate ellipsoid in inertial frames. Mean- while, for Einstein and others who admitted the spatio-temporal relativity of inertial frames, the form of a spherical light pulse remained spherical in all inertial frames. In Poincaré’s scheme of things, only for ether-fixed observers measuring wavefronts with clocks and rods at rest is the light pulse a sphere; in all other inertial frames the light pulse is necessarily shaped like a prolate ellipsoid.
Comparison of Poincaré’s pre-1909 and post-1909 readings of the light el- lipse shows the ellipse dimensions to be unchanged. What differs in the Minkowskian representations of these two readings is the angle of the space- like plane containing the light ellipse with respect to the lightcone. The

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complementary representation is obtained in either case by rotating the light ellipse through an angle ψ = tanh −1 v about the line parallel to the y-axis passing through point B.
We are now in a position to answer the question raised above, concerning the reasons for Poincaré’s embrace of time deformation in 1909. From the stand- point of experiment, there was no pressing need to recognize time dilation in 1909, although in 1907 Einstein figured it would be seen as a transverse Doppler effect in the spectrum of canal rays [45]. On the theoretical side, Minkowski’s spacetime theory was instrumental in convincing leading ether- theorists like Sommerfeld and Max Abraham of the advantages of Einstein’s theory. Taken in historical context, Poincaré’s poignant acknowledgment in G¨ ottingen of time deformation (and subsequent repurposing of his light el- lipse) reflects the growing appreciation among scientists, circa 1909, of the Einstein-Minkowski theory of relativity [46].

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