Relativity With or Without Light and Maxwell

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. It is a common experience of high school physics teachers that even the students with an aversion to physics lose their apathy and listen with ears wide open when confronted with the theory of relativity.

The historical path to special relativity stricto sensu began with the second postulate introduced by Einstein in 1905 [10]. Immediately after the publication of reference [10], he noted that the second postulate is contained in Maxwell’s equations [11], suggesting thus the approach recently elaborated by Aguirregabiria et al [1,12].

In the present paper, which was inspired by references [1,2,3,4,5,6,7,8,9,10,11,12], we venture to illuminate some dark corners and to simplify some involved points in relativistic arguments. We believe that the present attempt could save some time and effort to the teachers and students of special relativity at undergraduate level, and provides some fresh insights. In Section 2 we discuss the complex relationship between Maxwell’s equations and the second postulate. Section 3 is an attempt to deduce the main results of the ‘relativity without light’ approach in a simple way. 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.

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

Starting from the postulated validity of Maxwell’s equations in any given inertial frame involves that the meaning of ‘time’ has already been settled in that frame, 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 any one inertial frame S imply that one can also introduce ‘light-time’ in that frame, based on rectilinear propagation of electromagnetic waves1 in vacuum at the same speed $c\equiv {({ϵ}_0{\mu }_0)}^{-1/2}$ 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.2 Assumed uniformity of space and time implies that the transformations must be linear; employing also invariance of the speed of light c and assuming isotropy of space, one obtains the Lorentz transformations.3

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 [10]] is of course contained in Maxwell’s equations.” [11] 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 [10]. 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.” [21].

However, as noted in [22], 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” [10]), 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 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 [10]), appearing in the introduction and in paragraph 2 of [10], 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.4 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 [25] and (fully) to Poincaré [26], but also (partly) to Larmor [27], cf [16]), 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 quoted above [11].

3. Relativity Without the Light Postulate

The traditional way of deriving the Lorentz transformations, starting from Einstein [10], is basically based on the following assumptions:

1. There is 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’ (Einstein’s second postulate applies so the one–way two–clock speed of light is by definition the known constant, $c\equiv {({ϵ}_0{\mu }_0)}^{-1/2}$); we will call a frame of reference with such properties of space and time the ‘primitive frame.’

2. Each observer in uniform motion with respect to the primitive frame (‘primitive observer’) has her/his own primitive frame so the speed of light is the universal constant, c, the same for all primitive observers.

3. The principle of relativity applies to primitive frames.

4. Uniformity of space and time and isotropy of space.

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, but Galileo’s principle is not necessary for deriving 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 [28]. In what follows we give a simple derivation of Ignatowski’s main results.

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 $t=t^{\prime }=0$), Uniformity of space and time implies that required coordinate transformations 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 [10]), our considerations will be restricted to x and t only. Employing the principle of relativity, assuming velocity reciprocity5, 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)

Equations (1)–(3) give [33]

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

(4)

where

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

(5)

It is convenient to write eqs. (4) in matrix form [34,35]

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

(6)

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

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

(7)

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

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

(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. Comparing eqs. (8) and (6), 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 eq. (17) must be equal to $\sqrt{1-\Omega w^2}$ with w given by eq. (16). One can verify that this is indeed so and consequently w is the speed of the frame $S^{″}$ 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.6

The choice $\Omega >0$ yields the well-known Ignatowski transformations [28]

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

(20)

Putting

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

(21)

and employing eq. (18), one obtains that

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

(22)

Thus $\overline{c}$ is the universal limit speed for massive particles (a supremum of possible particle speeds), as yet unknown, for the case $\Omega >0$. Putting also

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

(23)

eqs. (20), (16) and (17) 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 $S^{″}$ relative to S, in eq. (25) one recognizes the familiar relativistic composition of velocities, and in eq. (26) the familiar relativistic transformation for gamma factors (cf, e. g., [37,38]).

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” [37].7 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.” [37]) 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., [6,33,40]), none of them can be actually implemented.

On the other hand, Einstein’s second postulate combined with the principle of relativity states, basically: 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, ${({ϵ}_0{\mu }_0)}^{-1/2}$ 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” [41].8 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 and Galileo’s principle of inertia, assuming 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 the 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?

To summarize, Einstein’s definition of the “time” [10] 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, a universal finite speed of propagation of interaction, $\overline{c}$, arises as a consequence of his starting assumptions; 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 [43], 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 [5,7,43]. 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 [44,45,46], cf also [47,48]. 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 [49], depending on unobservable speed v of the system with respect to the ether, Budde [44], FitzGerald [45] and Lorentz [46] 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 [47]), 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,”9 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.