Subject: Spherically Symmetric Spacetime
Date: Wed, 30 Apr 2008 07:53:58 EDT
One of the most general line elements of the spherically symmetric spacetime is given by Carroll in his eq. (7.13) of his 1997 online notes. This was evaluated in paper 93 of _www.aias.us_ (http://www.aias.us) . Carroll then automatically neglects torsion by computing the Christoffel symbols of this metric. Our Maxima code in paper 93 was checked in this way for correctness. Carroll then gives the non-zero elements of the Riemann tensor for this line element, and again our Maxima code reproduced Carroll’s results. Finally or code was checked to reproduce Carroll’s Ricci tensor. Carroll then uses the Ricci flat condition, in which all elements of Ricci are zero. Our code was again checked in this way. The result of the Ricci flat assumption for the spherically symmetric line element is the CORRECT Schwarzschild solution of 1916, in which Schwarzschild’s alpha is denoted mu by Carroll in his eq. (7.26). The reason for the structure of the Schwarzschild line element is that it is a line element of spherically symmetric spacetime for zero torsion in which the Ricci tensor is assumed to be zero. The Einstein Hilbert equation does not enter into the analysis.
Finally the mu of this pure geometrical solution is assumed to be – 2GM / c squared.
However, the Einstein Hilbert field equation is
R sub mu nu – (1/2) R g sub m nu = k T sub mu nu
and in a Ricci flat spacetime:
T sub mu nu = 0
The 00 element of this tensor is the mass density, so in a Ricci flat spacetime there is no mass density anywhere in the spacetime. Therefore the identification of mu with -2GM / c squared assumes the existence of a mass M where there there is no mass M. Such a procedure was not used by Schwarzschild.
In paper 93 we computed a particular curvature tensor which appears on the right hand side of
D sub mu T sup kappa mu nu = R sub mu sup kappa mu nu
This curvature tensor is non-zero in general for the Christoffel connection, whereas the torsion is always zero for the same Christoffel connection. So the use of this connection conflicts with the Hodge dual of the Bianchi identity in general.
In the weak field limit ( r goes to infinity) we have (T sub mu nu = 0, no EH equation used)
g00 = – grr = – (1 + mu / r)
This is identified with the weak field limit of EH (T sub nu nu NOT zero, EH equation used):
g00 = -grr = – (1 + 2 phi)
where phi is the Newtonian:
phi = – GM / r
It is obvious that this is self contradictory because in one case T sub mu nu is zero (no EH used) and in the other case it is not zero (EH used). So this procedure, on scholarly examination, makes no sense at all, and in papers 93 to 100 we set out to revise it completely to include torsion self-consistently.