Relativity with or Without Light and Maxwell

Abstract

The complex relationship between Einstein’s second postulate and the Maxwell electromagnetic theory is elucidated. A simple deduction of the main results of the Ignatowski approach to the theory of relativity is given. The peculiar status of the principle of relativity among the Maxwellians is illustrated.

1. Introduction

Recent papers published in pedagogical journals [1,2,3,4,5,6,7,8,9] testify that the problem of how to teach special relativity continues to be a fascinating topic, discussed at various levels of sophistication. This is no wonder because the concepts of where, when and why are essential for our existence as conscious beings, and with the theory of relativity our instinctive world view is at stake. As Schrödinger notes in Mind and Matter (1958), special relativity “meant […] a liberation from the unbreakable rule of ‘before and after’.” [10].

The historical path to special relativity stricto sensu began with the second postulate introduced by Einstein in 1905 [11]. Immediately after the appearance of [11], he noted that the second postulate is contained in Maxwell’s equations [12], suggesting thus the approach recently elaborated by Aguirregabiria et al. [1,13].

In the present paper, which was initially inspired by references [1,2,3,4,5,6,7,8,9,10,11,12,13], we venture to illuminate some dark corners and to simplify some involved points in relativistic arguments. In Section 2 we discuss the complex relationship between Maxwell’s equations and the second postulate. We point out a conceptual analogy between Newtonian absolute time and absolute Einsteinian time, and we emphasize the logical primacy of the second postulate over the principle of relativity. Section 3 is an attempt to deduce the main results of the ‘relativity without light’ approach, introduced by Vladimir Ignatowski in 1910 [14,15], in a simple way. We provide detailed reasoning and calculations leading to Lorentz-like transformations and clarify some contentious points in standard arguments. The concluding Section 4 contains a praise of the second postulate and a historical remark on the peculiar status of the principle of relativity among the Maxwellians in the late nineteenth century. We give a formal demonstration that the very introduction of $t^{\prime }\ne t$ reduces Ignatowski’s approach to Einstein’s 1905 framework, and also we point out that some familiar relativistic generalizations concerning clock retardation require amendment.

2. Maxwell’s Equations, ‘Inertia-Time’ and ‘Light-Time’

Starting from the postulated validity of Maxwell’s equations in an inertial frame S means that the meaning of ‘time’ has already been settled in the frame S, according to the standard definition based on Galileo’s principle of inertia (‘inertia-time’). As is well known, inertia–time and the validity of Maxwell’s equations in the frame S imply that one can also introduce ‘light-time’ in that frame, based on rectilinear propagation of electromagnetic waves {Following Maxwell (1864), “light itself […] is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.” [16]} in vacuum at the same speed $c\equiv {\left({ϵ}_0{\mu }_0\right)}^{-1/2}$ in all directions; light–time is identical with inertia–time in S. Applying now the principle of relativity (assuming also that ${ϵ}_0$ and ${\mu }_0$ are frame-independent), it follows that one should look for coordinate transformations between two inertial frames S and $S^{\prime }$, in uniform motion with respect to one another, that are consistent with light–time. (While inertia–time and light–time are identical in any frame, inertia–time puts no constraints on the transformations sought except that the validity of Galileo’s principle of inertia in S should imply its validity in $S^{\prime }$.). Clearly, the transformations cannot be of the Galilean type, involving $t^{\prime }=t$, since such transformations destroy invariance of c {The fact that ‘now’ is not the same for all inertial observers is a miracle of a different kind than the one that the motion of the moon about the earth and the falling of an apple have a common root. (Newton proved the latter in the third book of Principia.). According to Newton, “relative, apparent and common time is any sensible and external measure (whether accurate or unequable) of duration by means of motion;” “duration or the perseverance of the existence of things” is Newton’s synonym for his “absolute, true and mathematical time.” Einstein’s postulated uniform propagation of light in vacuum, an ideal time-keeper (Silberstein’s term) identical for all inertial observers and inaccessible to experimental verification, appears to be a perfect analogon of Newton’s absolute time which “of itself and of its own nature, without relation to anything external, flows equably,” and the parts of which “make no impressions on the senses” (“non incurrunt in sensus;” all quotations of Newton are from the first Scholium of Principia [17]). Newton, and all physicists before Einstein (including Voigt, Larmor, Lorentz and Poincaré [18,19,20,21]), took it for granted that there was only one ‘time,’ absolute Newtonian time, for all observers in motion with respect to one another. Einstein was bold enough to venture that each inertial observer has their own absolute Einsteinian time.}. The attributed properties of uniformity of space and time imply that the transformations must be linear; employing also invariance of the speed of light c and (the attributed) isotropy of space, one obtains the Lorentz transformations {Einstein’s original derivation of the Lorentz transformations [11], while cumbersome, is perfectly correct, without involving Galilean transformations, as clarified by Martínez [22]. Einstein for the first time gave the explanation of how homogeneity of space and time imply the linearity of the transformations in [23], pp. 136–137; in the same paper he gave for the first time a definition of a clock ([23], p. 21) that involves “the principle of sufficient reason.”}.

The above argument seems to be condensed in the footnote from Einstein’s second relativity paper: “The principle of the constancy of the velocity of light used there (in [11]) is of course contained in Maxwell’s equations.” [12]. The laconic footnote probably reveals Einstein’s original train of thought leading him to special relativity: from Maxwell’s equations to the constancy of the speed of light, and through the principle of relativity and properties of space and time, to the Lorentz transformations.

However, in 1905, instead of starting from Maxwell’s equations Einstein chose a different path [11]. Why? He gave an explanation thirty years later, at the beginning of his penultimate published attempt to derive the mass–energy relation: “The special theory of relativity grew out of the Maxwell electromagnetic equations. So it came about that even in the derivation of the mechanical concepts and their relations the consideration of those of the electromagnetic field has played an essential role. The question as to the independence of those relations is a natural one because the Lorentz transformation, the real basis of the special relativity theory, in itself has nothing to do with the Maxwell theory and because we do not know the extent to which the energy concepts of the Maxwell theory can be maintained in the face of the data of molecular physics.” [24].

However, as noted in [25], Einstein in 1905 would have had to derive the Lorentz transformations following the template of thermodynamics, making them independent of the Maxwell theory, even if he had been completely ignorant of light quanta or Planck’s 1900 derivation of the black-body radiation formula. Namely, a definition of time that is as simple as possible must conceptually precede any discussion about “the laws according to which the states of physical systems change” (“die Gesetze nach denen sich die Zustände der physikalischen Systeme ändern” [11]), including the domain of the validity of Maxwell’s equations. Attempting to keep the Lorentz transformations divorced from the Maxwell electromagnetic theory, Einstein took a necessary condition of the Maxwell theory, replacing “electromagnetic waves” with “light,” to be his principle of the constancy of the speed of light (the notorious “second postulate”). This principle, which appears to be ‘simplicity itself,’ combined with the (meta-)principle of relativity, is the crux of special relativity. Einstein’s ‘definite velocity V’ of propagation of light in vacuum (he used the symbol ‘V’ instead of ‘c’ in [11]), appearing in the introduction and in Section 2 of [11], does not have the familiar meaning of a derived quantity, $V=$ (light path)/time interval, where ‘time interval’ is a segment of inertia–time. Instead, Einstein’s V is a primitive quantity (of course, all other speeds are derived quantities), and ‘time interval’ is by definition equal to (one-way light path)/V. Thus, the speed of light in vacuum is by definition equal to the universal constant V, identical for all inertial observers, in perfect disagreement with our instinctive Galilean mentality {As is recalled in [26], Einstein’s second postulate has a rather intricate content. It postulates that, relative to an inertial frame, “the one-clock two-way speed of light in vacuum V (a measurable physical quantity and, as measurements reveal, a universal constant) is constant and independent of the velocity of light source and equals the one-way two-clock speed of light in vacuum (an immeasurable quantity). […] It should be pointed out that throughout the relativity paper Einstein used the same symbol (‘V’) for the speed of light in vacuum and the phase velocity of electromagnetic waves in vacuum according to the ‘Maxwell-Hertz equations’, linking thus special relativity with Maxwell’s theory, and at the same time linking the new time keeper (propagation of light) with the earlier ones (cf. [27]).”}. Two circumstances were decisive for Einstein’s argument: first, Maxwell’s equations can be simply made Lorentz covariant and thus consistent with the principle of relativity by appropriately defining the transformations for the electric and magnetic fields and charge and current densities (the fact already known (partly) to Lorentz [28] and (fully) to Poincaré [29], but also (partly) to Larmor [30], cf. [18]), and, second, empirical evidence for Maxwell’s theory in pseudo-inertial frame tied to the earth.

The above discussion reveals that Einstein’s second postulate conceptually precedes the principle of relativity. The rich content of the second postulate is understated in the laconic Einstein’s footnote [12] quoted above.

3. Relativity Without the Light Postulate

The traditional way of deriving the Lorentz transformations, starting from Einstein [11], is basically based on the following assumptions:

• There exists a frame of reference in which spatial coordinates x, y and z are Cartesian coordinates (Euclidean geometry applies), and time coordinate t is defined through ‘light–time’ (light signals in vacuum propagate rectilinearly and uniformly and the one-way two-clock speed of light is by definition the known constant, $c\equiv {\left({ϵ}_0{\mu }_0\right)}^{-1/2}$); we will call a frame of reference with such properties of space and time the ‘primitive frame.’
• Each observer in uniform motion with respect to the primitive frame (‘primitive observer’) has their own primitive frame so the speed of light is the universal constant, c, the same for all primitive observers.
• The principle of relativity applies to primitive frames.

These assumptions entail uniformity of space and time {In the view of the present author, “uniformity” is a better choice than the standard locution “homogeneity,” while the latter is preferred in the physics community and by modern translators of Newton. According to the first Scholium of Principia, absolute space always remains “similare & immobile.” The Medieval Latin adjective similaris means similar or uniform, and the key meaning of uniform is basically “having a consistent nature throughout,” that is, it describes an object that does not change its property in different parts. On the other hand, “homogeneous” implies a substance made of parts that are the same, whereas “uniform” implies a nature that is the same. Newton was not saying space is made of “stuff” that is mixed well, he was saying the very nature (geometry) of space is identical everywhere.} and isotropy of space, in all primitive frames.

The above assumptions are sufficient for deriving the Lorentz transformations. Since light–time is consistent with inertia–time, by adding Galileo’s principle of inertia each primitive frame is also inertial frame {Throughout the present paper, the principle of inertia is understood to apply strictly only to massless (geometrical) points. Specifically, according to general relativity, the presence of a mass point of finite mass would destroy the Euclidean nature of the space in which it resides. Thus, from the standpoint of general relativity, “inertial reference frame” is seen to be a misnomer in special relativity.}, but the venerable Galileo’s principle, involving as it does material points, is not necessary for the derivation of the Lorentz transformations.

One may ask at which coordinate transformations one arrives with the principle of relativity but without light and the second postulate, replacing in the above assumptions primitive frame with inertial frame, employing inertia–time based on Galileo’s principle of inertia. As is well known, we owe the query and the correct answer to it to Vladimir Ignatowski [14,15]. In what follows we give a simple derivation of Ignatowski’s main results.

The principle of relativity entails, fundamentally, the equivalence of all reference frames for the determination of space and time. This equivalence implies that these frames share one and the same 3-dimensional, uniform and isotropic space—devoid of torsion and curvature, and thus Euclidean—and furthermore, that time is uniform in all these frames (There is also, of course, a physical content to these properties: the repeatability of physical experiments.). Now, since the principle of relativity entails the principle of inertia, coordinate transformations between two inertial frames should map uniform motion to uniform motion. This condition implies that the required coordinate transformations must be affine ([31]; for a rigorous argument cf. [32,33,34]).

Consider two inertial frames S and $S^{\prime }$ in the standard configuration ($S^{\prime }$ is in uniform motion with respect to S along the common positive x-$x^{\prime }$ axes at speed v, y-and z-axis parallel to the $y^{\prime }$- and $z^{\prime }$-axis, respectively; the origins coincide at $t=t^{\prime }=0$). From the preceding discussion it follows that the required coordinate transformations between S and $S^{\prime }$ must be linear; also, isotropy of space implies that $x^{\prime }$ is independent of y and z. Consequently,

$$x^{\prime }=x/F_v+t/G_v,$$

(1)

where $F_v$ and $G_v$ are parameters dependent solely on v, as yet unknown. Setting for simplicity $y^{\prime }=y$ and $z^{\prime }=z$ (for a convincing deduction see [11]), our considerations will be restricted to x and t only. Employing the principle of relativity and velocity reciprocity {Velocity reciprocity means that the velocity of the $S^{\prime }$ frame with respect to the S frame is the opposite of the velocity of S with respect to $S^{\prime }$. Many authors, including Ignatowski [14] and even Einstein ([23], p. 130), take velocity reciprocity as an immediate and self-evident consequence of the principle of relativity alone. However, to derive reciprocity, the principle of relativity, i.e., its consequences, must be combined with the requirements of causality and temporal isotropy, as is pointed out by Berzi and Gorini [35] and by Lévy-Leblond [36]; a thorough discussion of the issue was recently published by Moylan [5]. A more sophisticated derivation that obviates these difficulties was given by S. Kichenassamy [32] and, recently, from a wider perspective, by his son Satyanad Kichenassamy [33]. Incidentally, Einstein’s first derivation of the Lorentz transformations (given in Section 3 of [11]) based on the postulated, finite and known, universal speed V (Einstein’s symbol for ‘c’), deduces velocity reciprocity. It is perhaps amusing to note that therein he denotes by $\phi \left(v\right)$ both his a and $a\beta$, where $\beta \equiv {\displaystyle \frac{1}{\sqrt{1-{(v/V)}^2}}}$. Moreover, he employs three reference frames (which he calls ‘coordinate systems’), ‘resting frame’ K with coordinates $x,y,z$ and t, ‘moving frame’ k with coordinates $\xi ,\eta ,\zeta$ and $\tau$, and a third frame $K^{\prime }$ with coordinates $x^{\prime },y^{\prime },z^{\prime }$ and $t^{\prime }$, moving relative to k. In the same Section 3 of [11] he uses the symbol $x^{\prime }$ also for $x-vt$, so $x^{\prime },y,z$ and t refer to coordinate systems introduced in the reference frame K. All that of course does not add to lucidity; reading Einstein, like reading Maxwell, is always an adventure par excellence [37,38].}, one has

$$x=x^{\prime }/F_{-v}+t^{\prime }/G_{-v}$$

(2)

Considering trajectories of the origins and lengths of moving unit rods, one obtains

$$1/G_v=-v/F_v,1/G_{-v}=v/F_{-v},F_v=F_{-v}.$$

(3)

The third Equation (3) is derived as follows. Equation (1) and the well-known Einstein’s operational definition of “the length of a moving rod” (cf. Section 2 of [11] and Section 3 of [26]) imply that a unit rod at rest in $S^{\prime }$, lying along the $x^{\prime }$-axis, has the length $F_v$ as measured in S (from Equation (1), since $\Delta t=0$ and $\Delta x^{\prime }=1$). Mutatis mutandis, a unit rod at rest in S, has the length $F_{-v}$ as measured in $S^{\prime }$. According to the principle of relativity and using the isotropy of space, these two lengths must be equal, taking into account what Born calls “the principle of the physical identity of the units of measure” (cf. footnote 12 in [26]).

Equations (1)–(3) give [39]

$$x^{\prime }={\displaystyle \frac{x-vt}{F_v}},\mathrm{and}{t}^{\prime }={\displaystyle \frac{t-{\kappa }_{v}x}{F_v}},$$

(4)

where

$${\kappa }_v\equiv {\displaystyle \frac{1-F_v^2}{v}}.$$

(5)

It is convenient to write Equations (4) in matrix form [40,41]

$$\left(\begin{array}{c}{t}^{\prime }\\ x^{\prime }\end{array}\right)={\displaystyle \frac{1}{F_v}}\left(\begin{array}{cc}1& -{\kappa }_v\\ -v& 1\end{array}\right)\left(\begin{array}{c}t\\ x\end{array}\right)\equiv {\Lambda }_v\left(\begin{array}{c}t\\ x\end{array}\right)$$

(6)

Introduce now a third inertial frame $S^{\prime \prime }$, in the standard configuration with $S^{\prime }$, moving at the speed u relative to $S^{\prime }$. One has

$$\left(\begin{array}{c}{t}^{\prime \prime }\\ x^{\prime \prime }\end{array}\right)={\displaystyle \frac{1}{F_u}}\left(\begin{array}{cc}1& -{\kappa }_u\\ -u& 1\end{array}\right)\left(\begin{array}{c}{t}^{\prime }\\ x^{\prime }\end{array}\right)\equiv {\Lambda }_u\left(\begin{array}{c}{t}^{\prime }\\ x^{\prime }\end{array}\right)$$

(7)

where ${\kappa }_u\equiv (1-F_u^2)/u$. Equations (6) and (7) obviously give

$$\left(\begin{array}{c}{t}^{\prime \prime }\\ x^{\prime \prime }\end{array}\right)={\displaystyle \frac{1}{F_u{F}_v}}\left(\begin{array}{cc}1+{\kappa }_{u}v& -{\kappa }_u-{\kappa }_v\\ -u-v& 1+{\kappa }_{v}u\end{array}\right)\left(\begin{array}{c}t\\ x\end{array}\right)\equiv {\Lambda }_u{\Lambda }_v\left(\begin{array}{c}t\\ x\end{array}\right)$$

(8)

It is natural to require that coordinate transformations between two inertial frames satisfy closure property, i.e., the successive application of two such transformations yields a third such transformation of the same form and content {This may seem to be an additional postulate, but this is not so: the principle of relativity implies the physical equivalence of all inertial frames—which necessitates transitivity—and this equivalence, in turn, implies a group structure for the set of all coordinate transformations between these frames [36]. Incidentally, an interesting generalization of Ignatowski’s original result, within a group-theoretical framework, is established in [42].}. Comparing Equations (6) and (8), the requirement can be satisfied if and only if

$${\kappa }_{u}v={\kappa }_{v}u,$$

(9)

and

$$1+{\kappa }_{u}v\ne 0.$$

(10)

Condition (9) is tantamount to

$${\displaystyle \frac{{\kappa }_u}{u}}={\displaystyle \frac{{\kappa }_v}{v}}\equiv \Omega .$$

(11)

Since u and v are arbitrary, $\Omega$ must be a universal constant, the same for all inertial observers. Conditions (10) and (11) give

$$1+\Omega uv\ne 0,$$

(12)

which excludes the possibility of negative $\Omega$ (otherwise, one would have $1+\Omega uv=0$ for $uv=1/|\Omega |$). Thus the universal constant $\Omega$ must be nonnegative

$$\Omega \ge 0.$$

(13)

Expressed in terms of $\Omega$, employing condition (12), matrices ${\Lambda }_v$ and ${\Lambda }_u{\Lambda }_v$ become

$${\Lambda }_v\equiv {\displaystyle \frac{1}{F_v}}\left(\begin{array}{cc}1& -v\Omega \\ -v& 1\end{array}\right),$$

(14)

$${\Lambda }_u{\Lambda }_v\equiv {\displaystyle \frac{1}{F_w}}\left(\begin{array}{cc}1& -w\Omega \\ -w& 1\end{array}\right),$$

(15)

where

$$w\equiv {\displaystyle \frac{v+u}{1+\Omega vu}},$$

(16)

and

$${\displaystyle \frac{1}{F_w}}\equiv {\displaystyle \frac{1+\Omega uv}{F_u{F}_v}}.$$

(17)

Equations (5) and (11) imply

$$F_v=\sqrt{1-\Omega v^2},$$

(18)

since only the positive value of $F_v$ makes sense. Thus also $F_u=\sqrt{1-\Omega u^2}$ and $F_w$ from Equation (17) must be equal to $\sqrt{1-\Omega w^2}$ with w given by Equation (16). One can verify that this is indeed so and consequently w is the speed of the frame $S^{\prime \prime }$ with respect to the S frame.

There are two possible choices for $\Omega$, zero or a positive value. $\Omega =0$ yields the Galilean transformations

$$x^{\prime }=x-vt,y^{\prime }=y,z^{\prime }=z,t^{\prime }=t,$$

(19)

which imply that there is one time for all inertial observers (identical with Newton’s absolute time), that a standard of length always has the same length independent of its velocity, and that v can be arbitrarily large. While in perfect agreement with our instinctive Galilean mentality, the Galilean transformations cannot be made compatible with a consistent interpretation of experience {Newton’s absolute space (that “by its own nature without relation to any thing external remains always uniform and unchangeable”) and absolute time (that “by its own nature without relation to any thing external flows equably”) appear to be the natural habitat for the Galilean transformations. As the parts of absolute space “do by no means come under the observation of our senses” (“non incurrunt in sensus”), and the same applies to absolute time, it appears that those absolute concepts remain transcendental, in the Kantian sense (compare Maxwell’s poetic discussion in his Matter and Motion [43]; by the way, what Maxwell calls “the doctrine of relativity of all physical phenomena” is the root of Poincaré–Einstein’s principle of relativity). But, keeping in mind that “everything which is not forbidden is allowed,” one should not dispense with a possible interpretation of the Galilean transformations. It should be stressed that in special relativity all inertial frames share one and the same space, cf. Section 3 of [11], which is analogous to Newton’s absolute space, without of course involving the idea of absolute rest.}.

The choice $\Omega >0$ yields the well known Ignatowski transformations [14]

$$x^{\prime }={\displaystyle \frac{1}{\sqrt{1-\Omega v^2}}}(x-vt),y^{\prime }=y,z^{\prime }=z,t^{\prime }={\displaystyle \frac{1}{\sqrt{1-\Omega v^2}}}(t-\Omega vx).$$

(20)

Putting

$$\overline{c}\equiv {\displaystyle \frac{1}{\sqrt{\Omega }}},$$

(21)

and employing Equation (18), one obtains that

$$v<\overline{c}.$$

(22)

Thus, $\overline{c}$ is the universal limit speed for massive particles, as yet unknown. Putting also

$${\overline{\gamma }}_v\equiv {\displaystyle \frac{1}{\sqrt{1-v^2/{\overline{c}}^2}}},$$

(23)

Equations (16), (17) and (20) become

$$x^{\prime }={\overline{\gamma }}_v(x-vt),y^{\prime }=y,z^{\prime }=z,t^{\prime }={\overline{\gamma }}_v(t-vx/{\overline{c}}^2),$$

(24)

$$w\equiv {\displaystyle \frac{v+u}{1+vu/{\overline{c}}^2}},$$

(25)

$${\overline{\gamma }}_w\equiv {\overline{\gamma }}_u{\overline{\gamma }}_v(1+vu/{\overline{c}}^2).$$

(26)

Since we concluded above that w is the speed of $S^{\prime \prime }$ relative to S, in Equation (25) one recognizes the familiar relativistic composition of velocities, and in Equation (26) the familiar relativistic transformation for gamma factors (cf., e.g., [31,44]). It can be shown employing Equation (24) and invariance of causality, that the speed of propagation of interaction must be less than or equal to $\overline{c}$, but the theory does not specify whether $\overline{c}$ is a physically attainable speed or not [31].

Thus, the Ignatowski transformations (20) are Lorentz-like transformations. To obtain the Lorentz transformations, i.e., to ascertain the value of the universal constant $\overline{c}$, one must turn to phenomena, without invoking properties of light or Maxwell’s electromagnetic theory.

4. Concluding Comments

Turning to phenomena, however, is tricky considering that inertial observers inhabit “ideal infinitely extended gravity-free inertial frames,” whereas in the real world their habitat is restricted to “the freely falling nonrotating local frames” [31] {This fact was basically well known to Newton, see corollaries V and VI of the Laws of Motion [17] and also [45].}. We quoted above Einstein’s clear-cut statement that “the Lorentz transformation, the real basis of the special relativity theory, in itself has nothing to do with the Maxwell theory […],” which is widely endorsed in the literature (“Special relativity would exist even if light and electromagnetism were somehow eliminated from nature” [31]). While various scenarios that have been proposed to ascertain the value of $\overline{c}$, without invoking properties of light or Maxwell’s electromagnetic theory, appear to be free from logical errors (cf., e.g., [6,39,46]), none of them can be actually implemented.

On the other hand, Einstein’s second postulate combined with the principle of relativity essentially states: first, there is a universal finite speed, c, the same for all primitive observers, and light propagates in vacuum at the universal speed; second, the universal speed c is a primitive quantity, and ‘time’ (‘light–time’, ‘duration’) in any primitive frame is a derived quantity through time interval $\stackrel{d}{=}$ light path/c; third, the magnitude of c is equal to the speed of propagation of electromagnetic waves in vacuum, ${\left({ϵ}_0{\mu }_0\right)}^{-1/2}$ as given by Maxwell’s electromagnetic theory, and also it is equal to the one-clock two-way speed of light as ascertained by terrestrial measurements and consistent with astronomical observations.

Thus, whereas in ‘relativity without light and electromagnetism’ $\overline{c}$ is the universal limit speed determinable only in principle through Gedankenexperiments, in the second postulate approach c is the speed which pertains to a real phenomenon (the essential one for our experience of the world), “concerning which we know something certain […] in a higher degree than for any other process which could be considered, thanks to the investigations of Maxwell and H. A. Lorentz” [47] {It should be stressed that relativistic length contraction and clock retardation cannot be verified directly. As Jefimenko [48] notes, Einstein’s method for measuring the length of a moving rod proposed in [11], “was, of course, merely a ‘Gedankenexperiment,’ that is, an imaginary procedure, a verbalization of [equation $x^{\prime }=\gamma (x-vt)$], that cannot be actually implemented.” Also, some familiar relativistic generalizations concerning clock retardation have to be amended [49]. The analysis of Jefimenko’s non-point clock in [49] explicity demonstrates that without the foundational tacit assumption of the point clock, measured proper time loses its standard correspondence with the uniform flow of the absolute Einstenian time.}. The second postulate combined with the principle of relativity, while counterintuitive, gives a simple path to the concepts of time and primitive frame, and to the Lorentz transformations. The second postulate is more fundamental than the principle of relativity because a definition of time that is as simple as possible must conceptually precede any discussion about “the laws according to which the states of physical systems change.”

Is it indeed a surprise that one can deduce the Lorentz-like transformations from the principle of relativity, employing Galileo’s principle of inertia, uniformity of space and time, spatial and temporal isotropy, and causality, without postulating a universal finite speed, as Einstein did? With hindsight, post festum ex post facto, it appears that the Ignatowski’s result could have been anticipated by the following simple argument. If there is the universal (signal) speed that is infinite, then $t^{\prime }=t$, according to clock synchronization. Applying modus tollens, if $t^{\prime }\ne t$, then either the universal speed is finite, or there is no universal limit speed, the same for all inertial observers (which possibility contradicts the principle of relativity). Thus, Ignatowski’s result is latent in the seemingly innocent starting assumption $t^{\prime }\ne t$. And what could be a more natural candidate for the universal limit speed in the world of phenomena than the speed of light?

The “mere” introduction of $t^{\prime }$ represents a fundamental physical shift. It acknowledges the possibility that interactions do not propagate at an infinite speed. Since the speed of propagation could therefore be finite in one frame, the principle of relativity necessitates that this finite speed be universal across all frames. In this view, each reference frame possesses its own absolute Einsteinian time. Velocity reciprocity is not required to derive the Lorentz-like transformations following Einstein’s 1905 path.

To summarize, Einstein’s definition of the “time” [11] is basically based on a tacit assumption that there is a universal finite speed of propagation of interaction; the limit speed is a primitive quantity equal to the speed of light from Maxwell’s electromagnetic theory, c, and “time interval” is a derived quantity. On the other hand, in Ignatowski’s approach, the universal limit speed of propagation of interaction (whether attainable or not), $\overline{c}$, arises as a consequence of the principle of relativity and his starting assumption, $t^{\prime }\ne t$. The value of the limit speed $\overline{c}$ remains unknown, perhaps transcendent, independent of any physical theory. In this sense Ignatowski’s relativity is more general than Eistein’s relativity [50]; true, the meaning of Ignatowski’s “t” is somewhat blurred. Obviously, if $\overline{c}\ne c$, then Maxwell’s equations are not Lorentz-like covariant, and thus do not conform to Ignatowski’s relativity.

Finally, a historical remark on the peculiar relationship between the principle of relativity and Maxwell’s electromagnetic theory at the end of the nineteenth century. In their recent papers, Browne [7] and Moylan [5] discuss briefly the evolution of the principle of relativity, recalling contributions of the saints and martyrs of the philosophical calendar, pride of place probably belonging to Galileo [5,7,50]. Illustrating the crucial role played by Poincaré in the vigorous defense of the relativity principle, Moylan notes that “at the end of the nineteenth century, physics was in a terrible state of confusion. Maxwell’s equations were not preserved under the Galilean transformation […], the followers of Maxwell’s electrodynamics were ready to uproot the relativity principle and reinstate a new form of geocentrism, where the relativity principle no longer held true.” However, there is a little-known episode that testifies that the situation was more complex. A topic of discussion among physicists in the late nineteenth century was the electrodynamic interaction between a charge and a current-carrying loop at relative rest that are moving uniformly with respect to the ether [51,52,53,54], cf. also [55,56]. The basic feeling of the ‘old’ physicists was simple: it is highly improbable that anything depends on the motion with respect to the ether; physical effects depend only on the relative motion between ponderable bodies and on their mutual relative position. Since in the problem considered Maxwell’s theory classically interpreted predicts a nonzero force [57], depending on unobservable speed v of the system with respect to the ether, Budde [51,52], FitzGerald [53] and Lorentz [54] postulated that charges were induced on the current loop in exactly that amount required to cancel the electrodynamic force due to the motion with respect to the ether. Their solution to the problem, reached through a rather contrived Budde’s ‘principle of neutralizing charge’ (cf. [55]), up to the second-order terms in $v/c$ coincides with what we think today to be the correct solution. Thus, some of the ‘old’ physicists were more ready to introduce an ad hoc hypothesis than to sacrifice the principle of relativity. Poincaré’s melancholy phrase, “les hypothèses, c’est le fonds qui manque le moins,” {Probably echoing the first two verses of the fable by Jean de La Fontaine, Le Laboureur et ses enfants: Travaillez, prenez de la peine: C’est le fonds qui manque le moins.} expresses his longing for a principle, instead of piling up hypotheses. The longed-for principle, while counterintuitive and far from being clearly stated, was the second postulate.