29 - Saturday

There is a way to make our modified form of Kruskal coordinates well behaved, but it just gives you the standard Kruskal coordinates, or it rotates them by 90 degrees.

 

The motivation for this coordinate modification is to allow trajectories that hit the origin (in the evaporating case this is possible) to be continuous. Ordinarily, the trajectories will hit the origin, and then sharply continue upwards, enforcing this condition leads to them continuing smoothly into region III instead. 

The problem is that the way this is enforced is by mapping region II onto region III and IV onto region I. Finding the transform to Schwarzschild coordinates reveals that the interior regions (II and IV) are disallowed, as you end up with:

\[t=\text{arctanh}\frac{T}{X}\]

 Which is singular for any \(|T|\geq|X|\). One can even draw the \(r=0\) singularity on the modified plot, and will find that it occurs in regions I and III.  (The standard Kruskal coordinates are defined to allow X/T as the argument in the interior regions)

Consequently, the trajectories that allow for in falling radiation to escape are simply trapped trajectories plotted incorrectly.

 

A give away is that the upper right half and lower left half of the Kruskal diagram correspond to entirely separate sets of Schwarzschild coordinates, so trajectories that connect these sets don't make sense. (This is ignoring that there is the view of the lower left half being the antipode of the upper right)