Greg Egan on Nostr: Without knowing the specific value of α, we can plug this into our formula for d and ...
Without knowing the specific value of α, we can plug this into our formula for d and take the limit as ε→0. This gives us:
d/M at horizon = 4 (α–1)/α √[1–2M/r₀]
But it turns out α isn’t hard to find. If we plug those same values into the equation that matches the time that elapses from the first astronaut falling, and take the limit again, we find:
τ = 4 √[1–2M/r₀] log(α)
If we solve that for α, we end up with the simple formula I gave earlier.
d/M at horizon = 4 (α–1)/α √[1–2M/r₀]
But it turns out α isn’t hard to find. If we plug those same values into the equation that matches the time that elapses from the first astronaut falling, and take the limit again, we find:
τ = 4 √[1–2M/r₀] log(α)
If we solve that for α, we end up with the simple formula I gave earlier.