Stringy interior of non-rotating black hole
On May 7, 2006, at 5:28 PM, Paul Zielinski wrote:
I'm not sure what you mean here. In the SSS spacetime, r > 2M is simply a scaled radial coordinate.
"r" is defined so that the area of spherical surface concentric with the event horizon surface is 4pir^2 when r > 2m. This meaning is lost when r < 2m. I don't know what "scaled radial coordinate" means?
This idea that the curvature singularity at Hilbert r = 0 somehow fills a "spacelike hypersurface" makes no sense to me. As far as I can see it is at most a 4D line.
It's actually effectively a 2D world sheet since the transverse polar and azimuthal directions close up analogous to a Cerenkov radiation cone as the speed of the charge --> infinity.
From Hawking & Ellis: Note that in Appendix B pp 371-2 the NAIVE interior vacuum solution r < 2m is Eq. (A8)
with t -> r and r -> t
ds^2 = - dr^2/(2m/r - 1) + (2m/r - 1)dt^2 + r^2(d@^2 + sin^2@d&^2)
So that in these NAIVE coordinates one "sees" what the spacelike singularity "means" when r -> 0
The hypersurface of constant r, i.e. dr = 0 has the SPACELIKE 3D metric
3^ds^2 = (2m/r - 1)dt^2 + r^2(d@^2 + sin^2@d&^2)
With the effective interior proper radial differential
dR = (2m/r - 1)^1/2dt ---> infinity as r ---> 0 singularity
If you integrate along dt
R(2) - R(1) = (2m/r - 1)^1/2(t2 - t1)
So this is a "radial" stretch or dilation as r -> 0. It's as if the two transverse directions are getting compactified into a mini Kaluza-Klein "hose pipe" (with S2 instead of S1).
That is, the effective interior RADIAL "space" is warped to infinity with infinite tidal stretch-squeezed ripping-apart curvature. Therefore, when Penrose uses "centre" he is speaking LOOSELY, i.e. the NAIVE "Euclidean" INFERENCE of the OUTSIDE r > 2m observer.
OK, as r --> 0, one can ignore the transverse angular @, & terms relative to the "radial" dt term. The celestial sphere gets swallowed up, i.e. the effective solid angle of the in-falling observer's future light cone shrinks to a point. Of course he is long dead before he can notice! ;-)
Therefore, in a sense, the interior space morphs to a string as the infinite curvature a = J/m = 0 space-like singularity is approached, i.e. a 2-D world sheet in the (r,t) plane
Jack Sarfatti wrote:
That is, the physical meaning of "r" as Area of spherical surface about spatial center of symmetry = 4pir^2 only is true for r > 2M.
On May 7, 2006, at 2:19 PM, Jack Sarfatti wrote:
Quick note to Zielinski
re: SSS black hole
Strictly speaking one should only use the Schwarzschild radial coordinate r outside the event horizon 2M (G = c = 1) and use Eddington-Finkelstein coordinates inside the event horizon. The overlap of the two patches is important at the event horizon. As stated before, if one insists on using r inside it is no longer a space coordinate, but a time coordinate, with the spacelike singularity of infinite curvature at r = 0 being an entire spacelike hypersurface.
String Theory Scandal?
In the course of citing Polchinski & Witten that superstring theory at ~ Planck scale is the only game in town to explain "quantum gravity" Penrose says that his reaction, and that of his "close colleagues" is "Very negative." He gives several excellent technical reasons. By analogy string theorists have to posit Ricci flatness Rab = 0 in 10 Dim. No wonder they cannot explain dark energy ~ /\zpf that would be more like
Rab - (1/2)gab/\zpfgab = 0 in 10 Dim
But that is the least of the problems. There are too many degrees of freedom and string theory is basically perturbative (violating the generalized no action without reaction) although the duality arguments partially fix that - except in the important middle.