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Nevertheless, some discussions may be helpful in clarifying coordinate relativistic causality. To discuss the Friedmann model, one must first accept essentially by faith that the mass distribution of the whole universe is homogeneous and isotropic. One must decide also modeling a galaxy as a particle is consistent with the normal understanding of Einstein's equivalence principle. Then, in Cartesian coordinates,
ds2 = d(2 - 2((){dx2 + dy2 + dz2}, (A1)
the Robertson-Walker geometry, is believed to be appropriate. Then, the Einstein equation (1) with source energy tensor T(( = u(u(+ P(u(u( - g(() leads to the following general evolution equations [17]:
3 = -4((( + 3P) (A2)
and
32/2 = 8(( - 3k/2, (A3)
where ( is the mass density, and P is the pressure. For different values of k, there are different types of solutions: k = +1 for the 3-sphere, k = 0 for the flat space, and k = -1 for the hyperboloid. For k = -1, 2(() is unbounded [17] and is therefore incompatible with the equivalence principle [11].
The rate of change of R (the distance between two isotropic observers at time () is
v= HR , (A4)
where H(() =/is identified with Hubble's constant. This means, however, the constant is time-dependent. Note, however, the observed red shifts may not be due to the Doppler effect alone [11,54,56].
However, within the above constraint, a model-independent feature of (() is
(()(( ( ( = 0; (A5a)
and
((()n(() = constant, where n ( 3 (A5b)
On the other hand, ds2 = 0 could imply that the light speed in the x-direction would be
(A6)
Thus, (A5a) and (A6) lead to a result that the light speed could be larger than c. Thus, it seems, either that coordinate relativistic causality could be violated or metric (A1) would be invalid.
Nevertheless, one must be careful because things are not that simple. For ds2 = 0 leads to a light speed in vacuum. However, in the Friedmann model, when a(() is very small, according to (A5b), not only there is no vacuum but the mass density ((() would be too large for the light to go through. Thus, the argument that leads to (A6) breaks down. Moreover, to justify the Robertson-Walker geometry, the effects of gravitational radiation should have been shown to be negligible at least for the assumed early universe. The existence of gravitational radiation, as pointed out by Lorentz and Wheeler [1], is due to the theory of relativity. Thus, it is also not clear that Friedmann's solution must be deduced from general relativity.
In reality, a galaxy is not a particle, the mass distribution is not homogeneous, and a light speed has nothing to do with Friedmann's modeling. Thus, it is clearly unsuitable for a discussion on fundamental questions. Now, it should be clear also that the Big Bang theory, though can be related to (1), depends on too many dubious assumptions (see also [32,54]) for the claim of being a consequence of general relativity. (Also, in view of the idealizations, the possibility of deriving eqs. (A2) and (A3) from another equation cannot be rule out.) Nevertheless, this discussion illustrates also the importance of the equivalence principle. |
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