Auto coordinate transformations

[fjebaker]

For some metric, we can calculate the horizon $g_{rr} = 0$, and therefore we can move to a set of coordinates where this horizon is removed.

E.g. for Kerr, move to a set of coordinates where the one-form filed $l_\mu = (- \Delta, - \Sigma, 0, \Delta a \sin^2 \theta)$ associated with the ingoing principle null congruences of the spacetime becomes $l_{\tilde{\mu}} = (-1, -1, 0, a \sin^2 \tilde{\theta})$.

The PNCs can be derived from the Weyl tensor acting on null vectors of the spacetime.

We can then build a Jacobian that maps from the Boyer Lindquist coordinates to the ingoing Kerr-Schild coordinates, giving an horizon-penetrating form of the metric.

I reckon we can probably come up with a way of numerically solving for this transformation, and then transforming the metric on the fly as we do ray tracing. This would be particularly useful for studying weird spacetimes.

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https://arxiv.org/abs/2408.09893