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starship-design: Fwd: The Speed of Light - A Limit on Principle ?
My apologies to those of you whose editors/readers don't support word
wrap. Mine does by default and the following copy of the original
document did also. It is too long to reformat by hand. Other than that,
it should make interesting reading for everybody in the group.
The Speed of Light - A Limit on Principle ?
A physicist's view on an old controversy
Last update 16th June '97
by Laro Schatzer <email@example.com>
(comments and criticism welcome)
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:
.Galilean Space-Time (GST) .Minkowski Space-Time (MST)
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
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
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
In Galilean Space-Time the physical existence of an absolute time is
assumed. The pioneer of physics Isaac Newton defined it in the following
"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.
<Picture: 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 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:
1) All physical laws appear according to the same laws in all reference
2) The speed of light is constant in all reference frames.
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
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
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).
<Picture: Clock Synchronization by Light Signals>
Now, as Newtonian time and thus the absolute space-time frame have disap
peared 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.
<Picture: 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 t
ravel - 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.
For the description of physical phenomena it is sufficient to use only the
first of Einstein's postulates . 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
<Picture: 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'.
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
<Picture: 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
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. , .
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 , 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
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. On the other hand - and what is less well known -
there exists no covariant representation of the state reduction postulate
. 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 , 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 , which has been confirmed
experimentally . Thus, quantum mechanics has proven to be correct (see
 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 .
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 , which describes the long-range
structure of our universe:
<Picture: 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 mea
surable 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 .
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:
.The quantum mechanical state evolution has no covariant representation.
.EPR-like effects seem to indicate non-local (superluminal) processes. .It
is impossible to construct a valid time observable, there exists no
relativistic 4-position operator. .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 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
. 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.
 I. Newton: "Mathematical Principles of natural philosophy", (London,
 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
 H. Reichenbach: "The philosophy of space and time", (Dover, New York,
 J. D. Jackson: "Classical electrodynamics", (Wiley, New York, 1975),
 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
 A. Einstein, B. Podolsky, N. Rosen: "Can quantum-mechanical description
of physical reality be considered complete?", Physical Review 47 (1935),
 J. S. Bell: "On the Einstein Podolsky Rosen paradox", Physics 1 (1964),
No. 3, pp. 195
 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
 R. Y. Chiao, P. G. Kwiat, A. M. Steinberg: "Faster than light?", in
Scientific American (1993), August
 S. Weinberg: "Gravitation and cosmology", (Wiley, New York, 1972),
 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
 M. Alcubierre: "The warp drive: hyper-fast travel within general
relativity". Classical and Quantum Gravity 11 (1994), pgs. L73-L77, see
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
Related Pages on the Web
.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.
The Ether Concept:
.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
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.
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 for the
existence of a background time parameter.
.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.
."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
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