Monday, May 08, 2006

Einstein's Geons 1936

Someone, maybe Abrams?, completely garbled the history. Einstein discussed it in 1936 in "Physics and Reality" Section 6

Einstein writes:

ds^2 = -(1 - 2m/r)^-1dr^2 - r^2(d@^2 + sin^2@d&^2) + (1 - 2m/r)dt^2

Then Einstein writes

rho^2 = r - 2m

Note it's "rho^2" on the LHS.

Nowhere is there r* = r - 2m written, though one could do it that way of course. Nowhere does Einstein use the term "masspunkt" here. That would be an entirely different physical problem in which one forces a singular source term Tuv =/= 0 absent in the discussion below by Einstein.

Then

ds^2 =
- 4(2m + rho^2)dpho^2 - (2m + rho^2)^2(d@^2 + sin^2@d&^2) + [rho^2/(2m + rho^2)]dt^2

"This solution behaves regularly for all values of rho. The vanishing of the coefficient of dt^2 i.e. g44 for rho = 0 results, it is true, in the consequence that the determinant g vanishes for this value; but, with the methods of writing the field equations actually adopted, this does not constitute a singularity.

If rho extends from -oo to +oo, then r runs from +oo to 2m and then back to +oo, while for such values of r as correspond to r < 2m there are no corresponding real values of rho. Hence the Schwarzschild solution becomes a regular solution by representation of the physical space as consisting of two identical 'shells' neighboring upon the hypersurface rho = 0, that is r = 2m, while for this hypersurface the determinant g vanishes. Let us call such a connection between the two (identical) shells a 'bridge.' Hence the existence of such a bridge between the two shells in a finite realm corresponds to the existence of a material neutral particle which is described in a manner free of singularities."

So this is the "Einstein-Rosen bridge" the "Mass without mass" "geon".

There is obviously an event horizon still at 2m where g = 0. There is NO SINGULAR MASSPUNKT here as in what Carlos Castro suggests. Einstein's whole purpose here is to avoid singularities in his "wormhole" "geon". He is only concerned with the view from outside, i.e. r > 2m and then in 1936 Einstein was not aware of the future work of Penrose and Hawking after his death.

Tiny geons using Newton's G are very massive and cannot be elementary particles and in fact we now know they are closely related to black holes. However, in principle they can form spontaneously without the collapse of a star of ordinary matter.

Suppose you want a geon of size 10^-13 cm.

Note that a mass 10^-5 gm has a gravity size of 10^-33 cm, therefore the mass of the 10^-13 cm geon would be huge, i.e. 10^15 gm ~ 10^40 electron masses. On the other hand, if short fermi scale gravity were 10^40 stronger then the mass would be of order of the electron mass.

There is also the issue of the Hawking radiation from small black holes.