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ds2 = c2dt'2 - dx'2 - dy'2 - dz'2, (18b)
by the following diffeomorphism,
x' = x, y' = y, z' = z, and t' = t(/c. (19a)
Eq. (19a) implies, however, that the unit of t' is c/( (sec). The light speed in the x'-direction is
= c() = (((cm/sec). (19b)
Thus, the light speed remains (. If ( = 2c, Metric (19a) implies that the light speed would be 6 x 1010 cm/sec; and metric (19b) implies that the light speed is 3 x 1010 cm/half-sec.
In the literature, the units of the coordinates are usually not specified. Then, the distinct metric (18a) would be confused with a rescaling of the Minkowski metric. This creates a illusion that all constant metrics were equivalence in physics. This example illustrates also that it is invalid to efine" light speed in terms of local Minkowski spaces in a manifold.
Moreover, if the equivalence principle were valid, according to Einstein approach, one would obtain
c2dT2 = (2dt2 , and (dX2 + dY2 + dZ2) = (dx2 + dy2 + dz2), (20a)
for a resting observer at a point (x0, y0, z0, t0). Eq. (20a) and ds2 = 0 imply that the light speed is
= = 1 (20b)
Eq. (20b) implies, however, that the light speed is c in the local Minkowski coordinate, but is ( ((2c) in the (x, y, z, t) space. On the other hand, since there is no gravitational force for this case, we can have also
x = X, y = Y, and z = Z (21)
Eq. (20b) and eq. (21) absurdly mean that for the same frame of reference, we have different light speeds. This is in disagreement with the principle of uniqueness for a physical measurement.
Example 2, consider the Minkowski flat metric and the transformation, which is a diffeomorphism,
t = C[exp(T/C) - exp(-T/C)]/2, where C = constant. (22a)
Then
ds2 = [exp(T/C) + exp(-T/C)]2dT2 - dx2 - dy2 - dz2, (22b)
is the metric transformed from the Minkowski metric. If metric (22b) is realizable, according to ds2 = 0, the measured light speed would be [exp(T/C) + exp(-T/C)]/2.
From (22b), the Christoffel symbols ((,(( are zeros except (t,tt = (tgtt/2. Then, according to the geodesic equation, the equation of motion for a particle at (x, y, z, T) is
+ (t,tt= 0, and === 0 (23a)
where
(t,tt = {ln[exp(T/C) + exp(-T/C)]}.
It follows eq. (23a) that one obtains, for some constant k
dT/ds = k[exp(T/C) + exp(-T/C)]-1 and dx(/ds = Constant (23b)
Now, consider the case dx/dT = dy/dT = dz/dT = 0. For this case, one has dx/ds = dy/ds = dz/ds = 0 and dx2/dT2 = dy2/dT2 = dz2/dT2 = 0. Thus, in such a ree fall", there is no change in the spatial position or acceleration. Physically, this means that such an observer would have the same frame of reference, whether ree fall" or not. Thus, he would absurdly have two different light speeds from the same frame of reference. Accordingly, the equivalence principle is not satisfied and metric (22) is not realizable. Note that, from metric (22), there is no acceleration for a static particle.
Nevertheless, some theorists would disregard all these inconsistency because they believe that space-time coordinates have no physical meaning. Therefore, they also disagree with Einstein [2,3] and regard that coordinate light speeds as meaningless. |
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