Re: Question about analysis of Schwarzschild solution on Fo. of Phys. 1988, 18, 6

On Dec 31 2006, 8:33 am, Surfer <sur...@xxxxxxxxxxx> wrote:
On 31 Dec 2006 05:46:32 -0800, "Juan R."
<juanrgonzal...@xxxxxxxxxxxxxxxxxxxx> wrote:

"On the gravitational field of a mass point according to Einstein's
theory."http://arxiv.org/abs/physics/9905030

On the gravitational field of a sphere of incompressible fluid
according to Einstein's theoryhttp://arxiv.org/abs/physics/9912033

An exact solution appears as expression 14 on Page 6 of the first
paper. It includes the following term R:

R = (r^3 + alpha^3)^1/3

However when alpha << r, it is possible to make use of the
APPROXIMATION R = r.

When textbooks present the Schwartzschild metric, they generally
present an approximate metric based on R=r, rather than
Schwarzschild's original metric.

The second paper contains an interesting statement at the bottom of
page 8.

"...there is a limit to the concentration, above which a sphere of
incompressible fluid can not exist."

This seems suggestive of gravitational collapse (eg into a black hole)

This is interesting, because the foreword to the first paper says that
the exact metric "leaves no room for the science fiction of the black
holes".

However, the exact metric was derived by assuming that all the mass is
concentrated at a point. Since this assumption is not physically
realistic, the exact metric is exact in a mathematical sense only, and
not in the sense of exactly matching reality.

So the approximate metric might match reality just as well or better.

In late 1915, Schwarzschild received the field equations. To solve
these equations in which the solution is the metric, he had to wade
through the vastly complicated Ricci tensor to do so. However, if he
could transform the coordinate system into something that the
corresponding metric would yield a determinant of -1, the complexity of
the Ricci tensor would reduce by half. That is from this new
transformed coordinate system with the reduced Ricci tensor that easily
yielded for him his original solution in 1915. It actually looks like
the Schwarzschild metric as a function of R. However, transforming it
back to the polar coordinate system, that results in the
Schwarzschild's original metric and not the Schwarzschild metric.
Schwarzschild's original metric and others do not manifest black
holes. Schwarzschild was able to find a solution so soon because there
are actually an infinite number of solutions to the field equations.

The Schwarzschild metric itself did not come about until Hilbert
introduced it a year or two later. Thus, Einstein did not have any
approximate solution to Schwarzschild's original metric. In fact,
before the field equations, it is impossible even to approximate any
thing about the curvature in spacetime. Einstein's 1915 or
pre-field-equation solution to Mercury's orbital anomaly was a total
BS.

.

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