Schwarzschild Interior

[WrathfulSpatula]

There is a loose argument attributed to John Wheeler, that the likelihood of the measured physical constants being what they are might be (at least partially) attributable to the relative tendency of any universe with those constants to produce black holes. Wheeler thought that black holes, as we see them from outside, might be connected to small "Big Bang" cosmogonies. Wheeler's argument is speculative, and so is our simulation here, but it is not ad hoc.

In the Schwarzschild coordinates, behind the event horizon of a black hole, the radial and time coordinates exchange sign, in the sense of signature of the metric tensor. Generally, we are cautioned against attempting to interpret this coordinate system on the interior, if for this very reason. Flouting some of the conventional assumptions about crossing the event horizon, we read these coordinates as literally as possible, treating the interior radial Schwarzschild coordinate as time-like and the time coordinate as space-like. (We flip the relative direction of time as well, assuming that perceived time proceeds from lower entropy to higher entropy, but the equations are at least time-reversible.) The result is that, instead of traveling into or out of a singularity localized in space, at the same spatial location along all interior world-lines, we travel into or out of a singularity localized in time, at the same time coordinate along all interior world-lines. This singularity certainly looks like a "Big Bang." It is not identical to the FLRW metric, as far as I can tell. Regardless, it makes for interesting mechanics concepts for a video game, if one is a little creative.

I called the demo a "Big Crunch," in one of the commit messages, but I think these parameters actually rather lead to a "Big Rip." That is, the seeming re-collapse of the space might be due to the contraction of the Hubble sphere, not the contraction of the scale factor.

[WrathfulSpatula]

It's worth noting, as we have simulated this, it is not possible for the player to cross the event horizon from a first-perspective perspective, or to see anything else cross it, either direction, at least if the black hole evaporates in any finite time.

[WrathfulSpatula]

As far as Lemaître coordinates go, I'm arguing that (\rho - \tau) = 0 should be read as a singular hypersurface shrinking at the speed of light, from the exterior, which is the event horizon, limiting to point-like, with the interior open region of Schwarzschild coordinates identified with the sector opposite this boundary.

[WrathfulSpatula]

The part of the picture that I'm saying was missing was the inevitability of total evaporation of a quiescent black hole into Hawking radiation, if not also into other forms like my hypothetical monopole term. Hence, from the exterior, an idealized, isolated, black hole must reach 0 mass and Schwarzschild radius of 0, therefore the appearance of a spatially point-like singularity at the trans-Planckian limit.

[WrathfulSpatula]

By the way, speaking of the trans-Planckian problem, I've already mentioned that the gravity lens treatment lacks a necessary time delay between multiple images. If we take Hawking's approach, and assume that the black hole formed a finite time in the past, then the limit point of greatest image delay at the visual center of the black hole originates at the (space-)time coordinate of formation of the hole, as per Hawking's treatment. (Consider that "skybox rays" are assumed to be in flight, at this and all time coordinates.)