Comments (1)
The Kerr metric has two Killing fields aligned in the t and ϕ direction. Therefore, the Kerr metric is expected to obey the symmetry
( t, ϕ) -> (-t, -ϕ),
which implies that under time-reversal, the t and ϕ components of the velocity remain unchanged (else would have to be identically zero).
Consequently, when we "reverse" our integration picture, we need only reverse the radial component of the velocity, which is equivalent to the above (there's a modulo sign convention floating around here, but the general distinction is that the sign of the radial component is different from t and ϕ).
Sound about right @phajy ? I will maybe write a blog post with the full mathematics in the future, but I think this would explain the seeming contradiction, at least heuristically.
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