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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 <schatzer@ubaclu.unibas.ch>
(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:

.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 
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

<Picture>

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.

<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

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 
frames.
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 
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).

<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>



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 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.

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

<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'.

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

<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 
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:

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 
[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:

<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 [11].

Summary

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 
[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.

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] S. Weinberg: "Gravitation and cosmology", (Wiley, New York, 1972), 
chapter 14
[11] 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
[12] M. Alcubierre: "The warp drive: hyper-fast travel within general 
relativity". Classical and Quantum Gravity 11 (1994), pgs. L73-L77, see 
also here.


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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


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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.



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 
Leyden.



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.



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.


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