Thursday, March 22, 2007

How to visualize the spacelike curvature singularity of a non-rotating black hole

Here we have the absolute light cone field of the black hole (a static spherically symmetric vacuum solution of Ruv = 0).

Note that the above solution in terms of local objective absolute invariant light cones is independent of the choice of local coordinate frames and patches. This is a picture like Egyptian hieroglyphs with the Cartan form "co-frame" tetrads and spin connections as the invariant algebra.

In special relativity without gravity the vertical axis is "time" in a given global frame as is actually depicted in the upper & lower left of the picture. The frame is global because all the invariant light cones are parallel to each other. Now the light cone's origins are "events", i.e. local coincidences like the collisions of two particles. The event is not a raw manifold point. You can get away with that in special relativity but not in general relativity (Einstein Hole Paradox of 1917). The infinitesimal world line segment forming the axis of symmetry of the local light cone, now tilted by curvature relative to its close neighbors, is timelike. We see at the lightlike event horizon that outward bound light rays are trapped on it and the inward bound light rays hit the vertical line at the center. However, that vertical line inside the horizon is not timelike, but has switched over to a spacelike line. Inside the horizon both outward and inward light rays hit the singularity.

We need to look at the math, in what would be static coordinates outside the horizon at r = 2M

ds^2 = -(1 - 2M/r)dt^2 + (1 - 2M/r)^-1dr^2 + transverse metric

for 2M/r < 1

but inside the horizon 2M/r > 1, therefore there is a signature switch of sign to

ds'^2 = -(2M/r - 1)^-1dr^2 + (2M/r - 1)dt^2 + transverse metric

where (2M/r - 1) is a positive

Therefore inside the horizon r is a timelike coordinate, the singularity at r = 0 is not located at a tiny point in 3D space, but is rather a "moment" over the entire 3D spacelike surface with spacelike coordinates t,theta, phi.

Note that

transverse metric = r^2(dtheta^2 + sin^2theta dphi^2)

so at the singularity r = 0 the transverse metric disappears.

Actually if the diagram was drawn more accurately the local light cones at the r = 0 spacelike singularity should be tipped over all the way to the horizontal on the vertical line. It then becomes intuitively obvious that r = 0 is not a locally timelike line inside the local light cone, but is a spacelike line outside that local light cone, which means that r = 0 is the FINITE moment of ultimate destruction of any observer who has fallen through the horizon.

Of course George Chapline has a new idea with Robert Laughlin that there is a dark zero point energy core of negative pressure inside the horizon.

Note that finite electron shells of charge have positive pressure zero point energy inside them. They appear as point particles in large momentum transfer probes because of the enormous space curvature induced by the positive pressure "dark matter" zero point energy cores.

That is, for a simple non-rotating shell of charge toy model "electron", the effective surface area of the spinless "electron" in a scattering with momentum transfer p from probe to target is

Effective area of "electron" Bohm hidden variable ~ (h/mc)^2(1 - p/mc) -> 0 at the Compton radius.

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