Crothers said:
As for emotional responses, I have not become emotional. I offer only citations of relevant papers and mathematical truths. You asked for the standard rebuttal of the black holers and I simply stated it. It is the black holers who respond with emotion by resorting to accusations of "crackpottery" instead of rigour, with few exceptions.
Calm, non-emotional responses are good. If you can avoid attributing any particular motivation whatsoever to "black-holers" and simply recognize that they (we?) totally disagree with you it will be a good start.
Calling "black-holers" nasty names is not going to advance calm argumentation.
It will also be necessary to adhere to the PF guidelines about sources and the other PF guidlenes as well. The Abrams paper appears to me to meet those guidelines, though I still think it is misguided.
I ask now for a mathematically rigorous justification of the arbitrary corruption of Schwarzschild's solution, by which the black hole is alleged, addressing the points I have made in my previous postings and repeated above.
I have noted the remarks concerning the Kruskal-Szekeres "extension". This extension is based upon the very same corruption of Schwarzschild's solution, and is therefore invalid.
Could you go over, in more detail, why you think the Kruskal-Szerkes extension is invalid?
It is a "simple" algebraic manipulaiton of Schwarzschild'd solution in terms different variables.
It is hardly controversial - it is used in many textbooks, including the one I have right in front of me by Wald, "General Relativity", one of the standard textbooks.
(I see you acknowedge this point, though I'm not sure who previously mentioned Wald in this thread).
Are you claiming that the particular variables used to express a line element have some physical significance?
Let us start with the initial Schwarzschild metric:
ds^2 = -(1-2M/r) dt^2 + 1/(1-2M/r) dr^2 + r^2 d\Omega^2
Do you agree that this is a valid vacuum solution of Einsteins' Field equations? (EFE).
Now make the following variable substitutions. (These are from Wald, not that it particularly matters BTW).
(r/2M - 1) exp(r/2M) = X^2 - T^2
(t/2M) = ln [(X+T)/(X-T)]
Note that these are of the form
(X+T)(X-T) = f(r)
(X+T)/(X-T) = g(t)
hence we can solve them for X+T = sqrt(f*g) and X-T = sqrt(f/g)
We can then write:
<br />
dr = 4\, \left( 2\,X{\it dX}-2\,T{\it dT} \right) {M}^{2} \left( {e^{{<br />
\frac {r}{2M}}}} \right) ^{-1}{r}^{-1}<br />
<br />
dt = 2\,M \left( 2\,X{\it dT}-2\,T{\it dX} \right) \left( {\frac {r}{<br />
2M}}-1 \right) ^{-1} \left( {e^{{\frac {r}{2M}}}} \right) ^{-1}<br />
Now we can re-write the Schwarzschild metric, with r(X,T) implicictly defined by
(r/2M - 1) exp(r/2M) = X^2 - T^2
as
<br />
\frac{32 M^3 e^{r/2M}}{r}(-dT^2 + dX^2) + r^2 d \Omega^2<br />
We see that the new expression is perfectly fininte in the new variables X,T at r=2M (which is at X=T), removing the coordinate singularity at the event horizon (r=2M, or X=T).
However, a singularity remains at r=0. We know we can't eliminate that because the curvature scalar diverges.
In short, a "simple" (it's simple with computer algebra, anyway) variable substitution eliminates the singularity at the event horizon.
I do not understand your objection to this procedure.
Who thinks the lesser universe contains enough information to sustain the reality of the event horizon, and why? Where is the relevant information (i.e. the properties of the singularity causing the event horizon) contained, for it then to be communicated in the physical events that will follow, within that lesser universe?
