The Speed of Light - A Limit on Principle ?

A physicist's view on an old controversy

Last update 16th June '97
by Laro Schatzer <>
(comments and criticism welcome)

"Easy" Treatise

Contemporary physics states that no object should be able to travel faster than the speed of light
c = 299'792'458 metres per second. 
Although the value of c appears enormous when compared with conventional traveling speeds, it suggests a limit which renders a practical realization of interstellar travel improbable. Whereas another planet in our solar system is reachable within minutes or at least hours at the speed of light, a journey to the nearest star system Alpha Centauri would already demand a traveling time of several years. Surely, the question remains: Are faster-than-light speeds possible? At the present time most scientists believe that the correct answer should be "no". However, it has to be emphasized that there is no definite proof for this claim. Actually, whether superluminal speeds are possible in principle depends on the real structure of the space-time continuum. Basically, one can distinguish two distinct notions of space-time in physics, which - according to present-day knowledge - both represent a valid possibility: Whereas Galilean space-time allows the realization of faster-than-light speeds, at least in principle, Minkowski space-time does not. What is the reason for this difference ? The key point is the different conception of global time, ie. what one accepts as a physical definition of simultaneity. Actually, what do we mean when we say two separated events to be simultaneous? What we need is a clear physical notion of past, present and future.

It is important to note that without a definition of global time the physical quantity speed (and thus light-speed) has no definite meaning anyway. Why? Consider an example: Imagine a train moving from point A to point B. Its speed v is given by

The start time t(A,start) and the finish time t(B,finish) are read off from two distant clocks. One is located at point A and the other one at point B. Now, the difference of the two times in the denominator t(B,finish) - t(A,start) is an indefinite expression, unless there is a rule how to synchronize both clocks, because clock B ignores the "current" time at clock A at first. In fact, the decision in favour of a particular synchronization rule is pure convention, because it seems impossible to send an "instantenous" (infinitely fast) message from A to B like "initialize the clocks now!". Thus, the actual quantity of speed is conventional too, depending on the particular choice of the simultaneity condition.

The question concerning global time is also important in the context of different reference frames. What is a reference frame? A reference frame (let us label it R from nowon) is simply a coordinate system of an observer. (For instance, let us imagine a physicist experimenting in his laboratory.) The observer can attach to all physical events coordinates, ie. space coordinates x, y, z (where?) and a time coordinate t (when?). Another observer in his personal reference frame R' (let us imagine another physicist sitting in a train moving with constant velocity v with respect to R) attaches to all physical events another (not necessarily equal) set of coordinates x', y', z' and t'. While two events may appear simultaneous in reference frame R (happening at equal time t), does this still hold in reference frame R'? And while the physical laws assume a particular form in frame R, do we obtain the same formulas in frame R' also? The answer is given by a theory which relates the new coordinates x', y', z', t' to the old ones x, y, z, t. Essentially, this is what the problem of relativity is all about.

Galilean Space-Time

In Galilean Space-Time the physical existence of an absolute time is assumed. The pioneer of physics Isaac Newton defined it in the following way [1]:
"Absolute, true and mathematical time, in itself, and from its own nature, flows equally, without relation to any thing external; and by other name called Duration. Relative, apparent, and vulgar time, is some sensible and external measure of duration by motion, whether accurate or unequable, which is commonly used instead of true time; as an hour, a day, a month, a year. It may be, that there is no equable motion, whereby time may be accurately measured. All motions may be accelerated and retarded, but the flowing of absolute time is liable to no change."
Because of this absolute time the notion of past, present and future is the same in all reference frames. If two events are simultaneous in one particular reference frame, this means that they are simultaneous in all reference frames. Thus, there is a unique separation between past and future events - the line of present in the space-time diagram (see below). Within Galilean Space-Time, faster-than-light speeds are possible in principle. However, electromagnetical waves are limited not to exceed the speed of light c. The latter is only a constant in the absolute space-time frame, which is also called the Newtonian rest frame.
Galilean Space-Time

There has been a variety of theories to describe electromagnetical waves (light) as excitations of some medium, quite in analogy to sonic waves which propagate in the medium air. This hypothetical medium was called the ether and it was supposed to be in rest in the absolute space-time frame. That is why this frame is also called ether frame sometimes. Since the establishment of the theory of special relativity it has become extremely unpopular among scientists to speak about an "ether". However, we know today that electromagnetical waves are indeed excitations of some "medium". However, this medium is not a solid or a liquid in the classical sense, but it is governed by the laws of quantum mechanics. Quantum field theoretists found the name vacuum for it. Some people interprete the vacuum as space-time itself, but this does not cover the fact that its true nature still remains a mystery. Anyhow, the term quantum ether might be used to indicate a possible modern synthesis of both concepts.

Minkowski Space-Time

Minkowski Space-Time does not know any absolute time. It was Albert Einstein who gave it a physical definition of simultaneity. Especially, because all experimental tests to determine the motion of an observer relative to some absolute space-time frame had failed, he decided to abandon the notion of absolute time at all. In his famous theory of relativity he postulated two principles which should hold for all physics: While the first postulate seems well established by observation and experiments, the second one is simply an assumption. It implies, in contrast to Galilean Space-Time, that simultaneity is not a true absolute physical quality, but a relative one, depending on the motion of the actual observer. (Again, the existence of absolute time could not be proved by experiments, but the theory of special relativity does not disprove it either.)

Now, how does the theory of relativity compare the time of two events at distinct locations? How can clocks be synchronized at different places? The answer Einstein gave, which is totally equivalent to his second postulate, is the following:

Set up a reference frame R. Put time measuring devices (clocks) at two locations. Let's label the clocks 1 and 2. To synchronize them place a mirror at clock 2 and emit a light signal from clock 1 at space-time point A to clock 2. The light signal arrives at clock 2 at space-time point B, is reflected in the opposite direction and arrives at clock 1 at space-time point C. As the speed of light is per definition constant and the light signal travels the same distance in both directions, the instant t(B) when the signal is reflected equals exactly t(P), which is in the mean-time of A and C. Or, more formally, t(B) = t(P) = (t(A)+t(C))/2.
With this definition of simulaneity, simultaneous events in one particular reference frame need not to be simultaneous in another frame. This can be checked by following the same procedure in a frame R' where all clocks are moving with relative speed v with respect to R.

Remark: For a better understanding of these reflections it is very fruitful to study a geometrical representation of space-time, the so-called space-time diagram (see below). Such a diagram is a reduced model of space and time from four to two dimensions. Instead of three space x, y, z and one time coordinate t, one uses only one space and a time coordinate, x and t, respectively. (It is much more easier to draw and think in two dimensions than in four dimensions.) For reasons of convenience the units are chosen such that the speed of light equals unity c=1. Hence, a light ray is described by x=+t or x=-t, appearing as a line at 45° or 135° in the (x,t)-plane, respectively.

The reader is strongly encouraged to reconstruct the important ideas by studying the space-time diagram.. Remember that the x-axis is the line of simultaneity (ie. with constant time t=0), and that the t-axis is the line of constant position (x=0).

Clock Synchronization by Light Signals

Now, as Newtonian time and thus the absolute space-time frame have disappeared in the theory of special relativity, all reference frames are equivalent. This implies that two superluminally separated events in space-time are instantenous in a particular reference frame. Present is no more a simple line in the space-time diagram, but equals the whole region of faster-than-light processes. As there is no Newtonian rest frame anymore which separates past and future superluminal events, faster-than-light motion would imply the possibility of time travel. Therefore, because this leads to the well known severe paradoxes of time travel, the theory of special relativity excludes faster-than-light speeds a priori.

Minkowski Space-Time


The question whether the speed of light is a true physical limit has no definite answer yet. It depends on the real structure of space-time. If there is an absolute time preserving causality (by preventing time-travel paradoxes), then faster-than-light speeds - and even faster-than-light travel - are possible, at least in principle. On the other hand, if superluminal processes are to be discovered, then absolute time will probably have to be reintroduced in physics. Although the theory of special relativity states against absolute time and superluminal phenomena, it does it not by proof, but only by assumption.

Are there indications that absolute time and faster-than-light processes exist ? The opinion of the author is "yes" ! It is the task of the next section to present some physical evidence.

Physical Treatise

For the description of physical phenomena it is sufficient to use only the first of Einstein's postulates [2]. Without loss of generality we may choose a reference frame R (with coordinates x, y, z, t) where the speed of light c is constant in all directions. The general coordinate transformation from this particular reference frame R to a general one R' (primed coordinates) reads
Equation 1

where the relative speed v of R' with respect to R is chosen to be parallel to the x-axis. The transformation properly expresses the apparent contraction of moving rods (Lorentz contraction) and the slowing of moving clocks (time dilation). The function S(x') determines the notion of simultaneity in frame R'. Generally, this may be an arbitrary function, but it is convenient to impose S(0) = 0 for that the clocks of the two reference frames R and R' become synchronized at the origin (x,t) = (0,0) = (x',t'). Furthermore, in order to avoid acceleratory effects, one usually imposes that S(x') is linear in x', ie. S(x') = s x'.

Minkowski Space-Time

It can be shown that Einstein's second postulate is equivalent to setting S(x') = - v/c^2 x', so that we obtain the well known Lorentz transformations
Equation 2

with the speed of light c' = dr'/dt'(r=ct) = c constant in all frames. Thus, from the viewpoint of relativity, all reference frames are completely equivalent.

The first postulate only ensures that physical phenomena have (locally) the same appearance in all reference frames, in the sense that one obtains the same result for all measurable quantities, which are but mean round-trip quantities (eg. the mean two-way speed of light). The second postulate states that there is no preferred reference frame and thus the expression of the global physical laws in mathematical formulas appears equally in all reference frames. The space-time coordinates (local Lorentz coordinates) are defined in such a way that the one-way speed of light is constant.

The success of the theory of relativity can be understood from the fact that the possibility to formulate all physical laws covariantly, ie. in a relativistically invariant manner, appears most tempting. One cannot deny that the involved mathematics is highly attractive from an esthetical point of view. For more information on special relativity and the principle of covariance one may consult eg. [3][4].

Galilean Space-Time

Another possibility is to set S(x') = 0. The reference frame R now plays a special role. It can be interpreted as the Newtonian frame of absolute time and space. The coordinate transformation looks very much like the old Galilean transformation x' = x - vt, t' = t, except that there appear additional time dilation and length contraction factors. This expresses the well known Lorentz-FitzGerald contraction hypothesis.

Although the one-way speed of light is not constant within this framework, the mean-speed c of a round-trip is again constant [3], as this would require the possibility of synchronizing clocks by some other means than finite-speed signals. Thus, some "experimental proof" of the constancy of the one-way speed of light has not been given so far.

Remark: It has to be emphasized that H. A. Lorentz version of the ether theory (which is set in such a Newtonian framework), ie. Lorentz relativity, is still a valid alternative to special relativity. It suffices to introduce the hypothesis that moving particles are contracted by some interaction with the ether (Lorentz-FitzGerald contraction), and that internal time is dilated by the same factor. Some physical justification was given by Lorentz in a paper (1904), where he showed how the physical length contraction can be explained by electro-magnetical effects.

Towards a Decision

Which conception of space-time structure is correct ? Obviously, the covariant framework is the most attractive one to describe matter in electromagnetical and gravitational fields. However, it is still possible that there exists an underlying absolute time preserving causality for superluminal phenomena. The theory of relativity offers no adequate framework for superluminal processes, a Galilean theory does. As is pointed out in the following part, several arguments indicate the non-generality of covariance and the existence of superluminal processes. The resurrection of absolute time is therefore possible, if not even necessary.

The Non-Generality of Covariance

Besides the principle of relativity, quantum mechanics is a cornerstone of modern physics. No physical theory evades relativity and quantum mechanics, but do these cornerstones actually fit together?

Let us repeat what is the time evolution of a physical state |s> in quantum mechanics (according to the Copenhagen interpretation). There are two steps:

The unitary time evolution is represented covariantly in a natural way, for instance, it leads to the Klein-Gordon or Dirac equation in the case of a relativistic particle. On the other hand - and what is less well known - there exists no covariant representation of the state reduction postulate [5]. If a physical reality is attached to the wave function, then the theory of relativity fails bitterly. In this context also belong EPR-like effects [6], which imply miraculous non-local (superluminal) correlations of measured quantities. Albert Einstein and other physicists could not believe in the validity of quantum mechanics because of such effects, which are apparently in conflict with the theory of relativity. One example is the violation of the Bell inequalities [7], which has been confirmed experimentally [8]. Thus, quantum mechanics has proven to be correct (see [9] for an overview). Although non-local effects are a constituent of quantum mechanics, most physicists still believe in the validity of special relativity, because EPR-like effects cannot be used to transmit information at superluminal speeds. Yet, EPR-correlations remain a mystery if special relativity and local realism are assumed to be valid.

While time and space are somehow "on equal rights" in the Lorentz transformations, time in quantum mechanics is completely different to space. In the field equations time appears as an exterior parameter only, whereas the position of a particle is described by some operator (a hermitian operator). But it is impossible to construct a valid time operator, and there are no time eigenstates. Thus, there exists no covariant 4-position operator in quantum mechanics. This is one of the main reasons why it has not yet been possible to construct a reasonable quantum field theory of gravitation. Thus, the usual theory of relativity and quantum mechanics appear to be incompatible.

Some Arguments in Favour of Absolute Time

One possible solution to the problem of time in quantum mechanics (and thus in quantum gravity) would be the reintroduction of a background Newtonian time. There are serious attempts to quantize gravitation in such a framework, eg. Post-Relativistic Gravity. This solution is also considered in more advanced research programs, eg. Canonical Quantum Gravity (see section "Further reading").

Moreover, there are some heuristic arguments which might further motivate the reintroduction of absolute time:

First, if there is a physical absolute time, then the number of fundamental constants reduces by one (the speed of light is not a constant any longer). This leads to a simplification and a new interpretation of the physical quantities and constants [2].

Second, it is well known that one can define a universal time, which appears in cosmological models. For instance, general relativity leads us to the Robertson-Walker metric [10], which describes the long-range structure of our universe:

Equation 3

Here, the time parameter t defines an universal time, the cosmological time. If there was an absolute beginning (with the big bang), it can be identified with the age of the universe. Anyhow, adopting absolute time would give it a further physical meaning. And, of course, there is a measurable preferred reference frame, which can be determined, for instance, from the absolute motion towards the uniform cosmic background radiation.

Interestingly, recent investigations of electromagnetic radiation propagating over cosmological distances seem to reveal a true anisotropy in the structure of our universe, suggesting that the speed of light might be not a true constant, but dependent on polarization. These results might possibly represent a further indication in favour of the existence of an absolute reference frame [11].


What is the true space-time structure ? Both Galilean space-time and Minkowski space-time have appeared to be valid physical concepts. However, the absolute generality of relativistic invariance (covariance) is set into doubt by the following arguments: It has been argued that a solution to these incompatibilities is the reintroduction of absolute time to physics. Further arguments in favour of the existence of a Newtonian absolute time have been given. Thus, the concept of Galilean Space-Time might be the correct one after all. Incidentally, there are active research groups trying to experimentally detect the existence of a preferred reference frame in this context.

Conclusion: If our universe has a Newtonian background, ie. if there is an absolute time in the space-time continuum, then there is no threat on causality by superluminal processes, because time travel and its paradoxes are excluded a priori. And thus, within this framework, faster-than-light travel is possible, at least in principle.

Remark: It may now come as a surprise to many physicists that even within the framework of general relativity faster-than-light speed is allowed, provided that the space-time metric of the universe is globally hyperbolic [12]. This condition simply implies that closed time-like paths in space-time (and thus time-travel) are not possible, therefore causality is again preserved. (Again, the time parameter can be interpreted as an absolute time of the universe.) However, in order to construct a propulsion mechanism for faster-than-light travel, exotic matter (with imaginary mass) would probably be needed in order to produce negative energy densities in space. Unfortunately, exotic matter is not known to exist, although negative energy densities have been shown to appear in quantum field theory. But, of course, such a hypothetical propulsion mechanism just provokes to be given the familiar name of the warp drive.


Further Reading (Scientific Papers)

Related Pages on the Web

Special Relativity:

The Ether Concept:

Alternative Gravity Theories:

Grand Unified Theories:


Interstellar Travel:

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