gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (11/05/84)
One fairly simple way to see why c (speed of light) is an absolute
speed limit is to consider the Lorentz equations relating two Cartesian
co"ordinate systems that are in motion at a uniform velocity relative
to each other; without loss of generality orient their X axes parallel
to the direction of motion and keep their Y and Z axes parallel to the
other system. Call the relative speed "v". The transformation from
one system to the other is
x' = (x - (v/c) ct) / sqrt(1 - (v/c)^2)
y' = y
z' = z
ct' = (ct - (v/c) x) / sqrt(1 - (v/c)^2)
These were obtained by Einstein from a conceptual analysis of motion
coupled with the assumption that the speed of light is the same in
all uniformly-moving reference frames (also assumes that similar units
are used to measure time and distance). There are other ways to derive
these equations; the important point is that they embody the MEANING of
relative motion with uniform velocity.
Now, as v approaches c, the Lorentz transformation equations develop a
singularity (denominator approaches 0), so it is hopeless to try to reach
superluminal speeds by accelerating from below the speed of light (if one
calculates the energy required to do so, it would be infinite).
However, these days we are well aware of the possibility of creating new
particles in subatomic reactions, so that it is at least conceivable that
a particle could be created that has v > c. If one considers v > c,
then the primed co"ordinates x' and t' become pure imaginary
quantities. This is a tip-off that we have pushed the model beyond
the realm in which its quantities have a direct physical interpretation.
One could stop here and say that this shows that such a speed has no
meaning, but if we nonetheless persevere and try to wring some meaning
out of the imaginary co"ordinates, here is what happens:
In order to relate these quantities to intuitive co"ordinates, the
simplest approach is to examine the Minkowski metric tensor, which for
the above Lorentz transformations always takes the form
( -1 0 0 0 )
( 0 -1 0 0 )
( 0 0 -1 0 )
( 0 0 0 1 )
(for simplicity I take (x,y,z,ct) as the four coordinates, in that order).
It is the special nature of the [4,4]-component of this metric (which is
already diagonalized, saving us some work) that indicates that the fourth
co"ordinate has a different nature ("time-like") from the other three.
If you don't know about the metric, is it one particularly simple way of
describing invariant geometric structure on a manifold (there are others
that are more fundamental, but they are beyond the scope of a special-
relativistic discussion). Denoting the metric tensor as "g", it relates
the co"ordinates (call them "w" collectively, which is not the usual
notation but I want to avoid multiple use of symbols) to an invariant
(co"ordinate-independent) measure of interval "s" by
(ds)^2 = Sum over i (Sum over j (g[i,j] dw[i] dw[j]))
(The d's are differentials.) Because ds is invariant and dw[] is a
vector (rank (1+0) tensor), the symmetric part of g[,] is a rank (0+2)
tensor. (The antisymmetric part is irrelevant to this discussion.)
For more about metrics, consult a differential geometry text.
There are several possibilities for mapping the imaginary co"ordinates
obtained for v > c back into real measure-numbers so that we can puzzle
out their meaning. All reasonable schemes amount to the same thing, so
consider the simplest, which is to just multiply the relevant
co"ordinates by i (the complex unit, sqrt(-1)):
x'' = i x'
y'' = y'
z'' = z'
ct'' = i ct'
This transformation takes the components of the metric tensor to
( 1 0 0 0 )
( 0 -1 0 0 )
( 0 0 -1 0 )
( 0 0 0 -1 )
The new (x'') co"ordinate is now the time-like one. To make comparison
easier, one can optionally apply a further transformation
x''' = ct''
y''' = y''
z''' = z''
ct''' = x''
in order to put the time-like coordinate back in its accustomed place.
This final set of space-time coordinates (along with its "standard form"
Minkowski metric) is suitable for direct interpretation as Cartesian
spatial coordinates and time coordinate of a reference system moving
"in the X direction with speed v > c" with respect to the original
"stationary" coordinate system.
But look what has happened in the process of straightening out the
imaginary coordinates! The relation between the triple-prime set of
coordinates and the original unprimed set, when you work it out, is
precisely that of an equivalent Lorentz transformation with parameter
u < c, where
u = c^2 / v
One way of looking at the above is to say immediately upon production
of imaginary co"ordinates for superluminal v that that demonstrates
the unreality of motion faster than light (for, in order to describe
"motion", we must have a thing that "moves", and such a thing must
be capable of having a real-valued space-time reference frame
associated with it). If one wants to go farther and try to interpret
such co"ordinates as meaningful, by the time one has coerced them into
a form such that their meaning is clear, he has shown that such motion
is entirely equivalent to (i.e., IS really) ordinary subluminal motion.
This duality of velocity is interesting. Try playing with the dual
interpretations in a variety of relativistic formulae, such as that
for composition of relative speeds:
v[AC]/c = (v[AB]/c + v[BC]/c) / (1 + (v[AB]/c)(v[BC]/c))
The practical significance, apart from keeping one from having to
worry too much about superluminal speeds, is that if one DOES allow
v > c in some formula, e.g. particle path-integral summations, then
he will be overcounting by a factor of 2 (except possibly at v == c,
which clearly calls for special treatment).herbie@watdcsu.UUCP (Herb Chong, Computing Services) (11/07/84)
Weren't the Lorentz transformations derived after it was decided that
the speed of light was the same in all inertial frames? I know that
the mathematics yields negative energy and all kinds of things like
that for beta greater than 1 in the energy/momentum equations.
Herb Chong...
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