The Speed of Light - A Limit on Principle?
A physicist's view on an old controversy
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 = 299'792'458.5 metres per second. Ê
Although the value of c appears to be 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, which contemporary physics
ignores, however. Basically, there exist two distinct notions of space-time in physics, both
of which represent a possibility:
Galilean Space-Time (GST)
Minkowski Space-Time (MST)
Briefly, 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?
In the next sections it is exposed that the key point is the conception of global time, ie. the
physical significance of the term simultaneity. In fact, what does it mean when we call
two spatially separated events "simultaneous", actually? What we need is a clear physical
notion of past, present and future, not only on a local but on a global level.
It is important to note that without some definition of global time the physical quantity
speed (and thus light-speed) has no definite meaning anyway. Why? Consider an example:
Imagine an object moving from position A to B. Its speed v is given by the formula
Here, the start time t(A,start) and the finish time t(B,finish) are read off from two
spatially separated clocks: one clock 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 exists a rule how to synchronize both clocks, because clock B
ignores the "current" time at clock A at first. But, 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 definition.
The question concerning global time is also important in the context of different reference
frames. What is a reference frame? A reference frame R is simply a coordinate system of
some observer. (For instance, let us imagine a physicist experimenting in his laboratory.)
The observer attaches to all physical events personal coordinates, ie. space coordinates x,
y, z (where?) and a time coordinate t (when?). Another observer in his personal reference
frame R' attaches to all physical events another (not necessarily equal) set of coordinates
x', y', z' and t'. (Let us here imagine another physicist who is working in a train moving with
constant velocity v with respect to the reference frame R.) While two events may appear
simultaneous in reference frame R (happening at equal time t), does this still hold in
reference frame R' (at equal time t')? And while the physical laws have a particular form in
frame R, does one 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 a theory of relativity is all about.
Remark: For a better understanding of the distinct space-time concepts it is fruitful to
study a geometrical representation of space-time, the space-time diagram (see below). In
this picture four-dimensional space-time is reduced to two dimensions. Instead of three
space x, y, z and one time coordinate t, one uses only one space and the time coordinate, x
and t, respectively. (Obviously, it is much more easier to draw and think in two 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, which is described by x=+ct or x=-ct, appears as a
straight line in the (x,t)-plane at 45° or 135°, respectively.
The reader is encouraged to reconstruct the arguments 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).
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 global 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 also 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 the framework of Galilean Space-Time, faster-than-light speeds are
possible in principle. However, electromagnetical waves are limited not to exceed the speed
of light c, which usually depends on the direction of the light signal the reference frame in
which it is measured. The speed of light is constant only in the absolute space-time frame,
which is also called the Newtonian rest frame.
There has been a variety of attempts to describe electromagnetical waves (light) as
excitations of some medium, quite in analogy to sonic waves which propagate in the medium
air. The hypothetical light medium was called the ether and it was supposed to be in rest in
the absolute space-time frame. (That is why the absolute 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". But it is well known that
electromagnetical waves can be indeed interpreted as excitations of some "medium".
However, this medium is not a solid or a liquid in the sense of classical physics, but it is
governed by the laws of quantum mechanics. In quantum field theory it is simply called
vacuum ("void"). Some physicists prefer to 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 thinkable modern synthesis of both concepts.
Minkowski Space-Time
Minkowski Space-Time does not know any absolute time which is physically meaningful. It
was the revolutionary idea of Albert Einstein to give the notion of simultaneity a new
definition. Especially, because all experimental tests to determine the motion with respect
to some absolute space-time frame had failed, he decided to abandon the notion of absolute
time at all. In the famous theory of relativity he postulated two principles which should
hold for all physics:
1) All physical laws appear according to the same laws in all reference frames.
2) The speed of light is constant in all reference frames.
Now, 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 an absolute physical quality, but a relative one, depending on the motion
of the observer (ie. the reference frame). However, it has to be emphasized that although
the existence of a physical absolute time (or, equivalently, a preferred reference frame)
could not be established by experiments, the theory of special relativity does not disprove
it either.
Now, how does the theory of relativity allow to compare the relative time of two events at
distant positions? How can one synchronize clocks being spatially separated? The
definition Albert Einstein offered, which is completely equivalent to his second postulate,
is the following:
Choose two clocks (let us label them 1 and 2) in some reference frame R. In
order to synchronize them place a mirror at position 2, then emit a light signal
from clock 1 at space-time point A. The light signal arrives at clock 2 at
space-time point B, it is reflected in the opposite direction and arrives at
clock 1 at space-time point C (see the space-time diagram below). Since the
speed of light is per definition constant and the light signal travels the same
distance in both directions, the instant t(B) of the reflection 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 global time, 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
the former reference frame R.
Now, because absolute time and thus the Newtonian reference frame have disappeared in the
theory of special relativity, all reference frames are completely equivalent. This implies
that two superluminally separated events in space-time can be made instantenous by
choosing a particular reference frame. Hence, present appears no more as a simple line in
the space-time diagram, but it equals the whole region of "faster-than-light" processes.
Furthermore, since there is no absolute reference frame separating the regions of
superluminal past and future, faster-than-light motion in Minkowski space-time implies
the possibility of time travel. Therefore, because this leads to the well known severe
logical paradoxes of time travel, the theory of special relativity excludes faster-than-light
speeds a priori.
Summary
The question whether the speed of light is a true physical limit has no definite answer yet.
It depends on the real structure of the space-time continuum, which is presently unknown.
If absolute time (and a preferred reference frame) exist, then faster-than-light speeds -
and even faster-than-light travel - are possible, at least in principle. Although the theory
of special relativity states against absolute time and superluminal phenomena, it does it
not by proof, but only by assumption. If superluminal signals are to be discovered in the
future, then the notion absolute time will surely have to be reintroduced to physics.
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 one 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
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-Fitzgerald contraction) and the slowing of moving clocks (time dilation). The
function S(x') simply determines the notion of simultaneity in frame R'. Generally, S(x') can
be an arbitrary function, but it is convenient to impose S(0) = 0 such that the clocks of the
reference frames R and R' are 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 one obtains the well known Lorentz transformation equations
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 ensures that physical phenomena have the same appearance in all
reference frames, in the sense that one obtains the same result for all measurable
quantities being 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 physical
laws (when expressed in formulas using global coordinates) appear equally in all reference
frames. The space-time coordinates (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 leading to the affine coordinate transformation
It has to be emphasized that these equations are not equivalent although similar to the
well-known transformation equations of Galilean relativity,
as the former equations contain additional time dilation and length contraction factors
expressing the Lorentz-Fitzgerald contraction hypothesis.
In the Galilean framework the reference frame R (with unprimed coordinates x, t) has a
special significance: It is the Newtonian frame of absolute time and space.
Although the one-way speed of light is not constant in general (ie. when expressed in an
arbitrary reference frame), the mean-speed c of a round-trip is again constant [2], what is
in accordance with all experiments (like Michelson-Morley a.s.o.). It should be emphasized
again that there has been no experiment which determined the one-way speed of light [3],
since this would require the possibility of synchronizing physical clocks by some other
means than finite-speed signals. Thus, in fact, some "experimental proof" of the constancy
of the one-way speed of light has not been given so far.
Remark: It has to be noted that H. A. Lorentz version of the ether theory (which is set in
such a Newtonian framework), ie. Lorentz relativity, is 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.
Towards a Decision
Which conception of space-time structure is the physically correct one? 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
does not offers an adequate framework for superluminal processes, at least not without
refering to logical paradoxes, but a Galilean theory does. As is pointed out in the following
section, several arguments can be found which indicate the non-generality of covariance
and the existence of superluminal processes. The resurrection of absolute time in physics 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:
1) The unitary time evolution |s(t)> = U(t) |s(0)>
2) The reduction of the state |s(t)> into an eigenstate of an observable P |s(t)> in case
of measurement by an observer, where P is a projection operator. This is the famous
"collapse of the wavefunction".
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. However -
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 have
not allowed to transmit information at superluminal speeds so far. Yet, EPR-correlations
remain a mystery if local realism is assumed to be valid. Therefore, the possibility of
superluminal communication (and thus FTL travel) has been acknowledged by various
authors, eg. [10].
While time and space appear somehow "on equal rights" in the Lorentz transformation
equations, this is not the case within the formalism of quantum mechanics. In the quantum
field equations the position of a particle is described by a linear operator (a hermitian
operator) in the Hilbert space of physical states, whereas the time coordinate appears as an
exterior parameter only. It is well known that it is impossible to construct a valid time
operator. There exist no time eigenstates, what is basically a consequence of Heisenberg's
uncertainity relation of energy-time. Therefore, 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, it is evident
that the standard theory of relativity and quantum mechanics are 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
reduced by one, since the (one-way) 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 one to the Robertson-Walker
metric [11], which describes the long-range structure of our universe:
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 exists 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 direction
and polarization. These results might possibly represent a further indication in favour of
the existence of an absolute reference frame [12].
Summary
Which is the real space-time structure? Both Galilean space-time and Minkowski
space-time have appeared to be valid physical concepts. However, the absolute generality of
relativistic covariance is set into doubt by the following arguments:
The time evolution of a quantum mechanical state has no covariant representation,
because the "measurement process" cannot be described in a relativistically invariant
manner.
EPR-like effects seem to indicate non-local (superluminal) processes.
It is impossible to construct a quantum time observable, so that no covariant
4-position operator exists.
From a cosmological perspective the existence of a preferred reference frame
appears to be natural.
It has been argued that a solution to these incompatibilities could be the reintroduction of
absolute time to physics. 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
underlying 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 be a surprise for 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 [13]. This condition simply implies that closed
time-like paths in space-time (and thus time-travel) are excluded, so that causality is
again preserved. (In this framework, the cosmological time parameter can be again
interpreted as the 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.
References
[1] I. Newton: "Mathematical Principles of natural philosophy", (London, Dawson, 1969)
[2] J. P. Hsu, L. Hsu: "A physical theory based solely on the first postulate of
relativity", Physics Letters A 196 (1994), pgs. 1-6; F. Selleri: "Theories equivalent to
special relativity", in Frontiers of Fundamental Physics, edited by M. Barone and F.
Selleri, (Plenum Press, New York, 1994)
[3] H. Reichenbach: "The philosophy of space and time", (Dover, New York, 1958)
[4] J. D. Jackson: "Classical electrodynamics", (Wiley, New York, 1975), chapter 11
[5] Y. Aharonov, D. Z. Albert: "Can we make sense of the measurement process in
relativistic quantum mechanics?", Physical Review D 24 (1981), pgs. 359-370; A.
Peres: "Relativistic Quantum Measurements", Annals of the New York Academy of
Sciences, Volume 755 (1995) ("Fundamental Problems in Quantum Theory"), pgs.
445-450
[6] A. Einstein, B. Podolsky, N. Rosen: "Can quantum-mechanical description of
physical reality be considered complete?", Physical Review 47 (1935), pp. 777
[7] J. S. Bell: "On the Einstein Podolsky Rosen paradox", Physics 1 (1964), No. 3, pp.
195
[8] A. Aspect et al.: "Experimental realization of Einstein-Podolsky-Rosen-Bohm
gedankenexperiment: A new violation of Bell's inequalities", Physical Review Letters
49 (1982), No. 2, p. 91; "Experimental test of Bell's inequalities using time-varying
analyzers", Physical Review Letters 49 (1982), No. 25, pp. 1804
[9] R. Y. Chiao, P. G. Kwiat, A. M. Steinberg: "Faster than light?", in Scientific American
(1993), August
[10] O. Steinmann: "The EPR Bingo", Helv. Phys. Acta, Vol. 69 (1996), pgs. 702-705
[11] S. Weinberg: "Gravitation and cosmology", (Wiley, New York, 1972), chapter 14
[12] B. Nodland, J. P. Ralston: "Indication of Anisotropy in Electromagnetic
Propagation over Cosmological Distances", Physical Review Letters 78 (1997), No. 16.
3043-3046; e-print:astro-ph/9704196; see also here
[13] M. Alcubierre: "The warp drive: hyper-fast travel within general relativity".
Classical and Quantum Gravity 11 (1994), pgs. L73-L77, see also here.
Further Reading (Scientific Papers)
C. J. Isham: "Prima Facie Questions in Quantum Gravity": Relativity, Classical and
Quantum, eds. J. Ehlers and H. Friedrich, Springer-Verlag, Berlin (1994),
e-print:gr-qc/9310031
G. K. Au: "The Quest for Quantum Gravity", e-print:gr-qc/9506001
Related Pages on the Web
Special Relativity:
Rob Salgado: "The Light Cone - An Illuminating Introduction to Relativity".
Alan Pendleton: "Was Einstein right?" offers another critical look at Einstein's theory
of special relativity.
On the Nature of Space-Time:
Amara Graps: "Ether: What is it?"
Albert Einstein: "Ether and the Theory of Relativity". It was only 11 years, from 1905
to 1916, that Albert Einstein did not believe in the existence of an ether. In 1920,
some years after the publication of his theory of general relativity, he expressed his
opinion in favour of an existing ether in a talk at the University of Leyden.
Sten Odenwald: "The physical vacuum of space".
Alternative Gravity Theories:
Yilmaz Theory of Gravity, a new gravity theory that seems to resolve the defects of
general relativity and that appears to be closer to some kind of "ether" interpretation
of the gravitational field.
Quantum Mechanics:
Anton Zeilinger: Interpretation and Philosophical Foundation of Quantum Mechanics:
An excellent summary of existing (meta-)physical interpretations of the
"measurement process".
"Grand Unified Theories":
Brian Greene: "Superstring Theory". Superstring theory appears to be a very promising
attempt to unite all fundamental forces including gravity, but it is also not able to
resolve the measurement problem. However, it resides on a fixed space-time
background, and it does allow the existence of a background time parameter.
Cosmology:
Borge Nodland: "A Peek into the Crystal Ball of an Anisotropic Universe": Recent
measurements on the propagation of radio waves over cosmological distances seem
to indicate that our universe possesses a preferred direction in space.
Interstellar Travel:
"Warp Drive When?": What NASA has to say about interstellar travel.
John G. Cramer: "Space Drives": A collection of articles published in Analog, amongst
a well-done discussion of Miguel Alcubierre's paper on the warp drive.