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@Grinkle recently asked a question about detecting crossing the event horizon, which got me thinking. I think that, at least in principle, I can deduce when I cross an event horizon with a "closed box" experiment, basically by measuring tidal forces. I'm planning to see if the maths works, but there's enough numerical work in doing so that I'd like to check I've not gone off the deep end before I try it.

To make things a little more tractable I'm going to assume a simple case - I was released from rest at some finite distance from a Schwarzschild black hole. My release radius was ##R_u## (u for unknown) and I was released at my proper time ##-\tau_u## (another unknown). It's messy, but I can actually write down a closed form expression for ##\tau(r,\tau_u,R_u,R_S)## and ##t(r,R_u,R_S)## (there's no ##t_u## because I can set the zero of ##t## at any time I like) in this case of purely radial motion. Sadly, they cannot be inverted to obtain ##r## as a function of ##t## or ##\tau##, nor an explicit relationship between ##t## and ##\tau##.

Inside my box, I have a free-floating radar set and a rocket. I release the radar set as near the center of mass of the box as I can, and the rocket a short distance away in the direction of the hole (which I can determine by releasing a cloud of test particles and watching for elongation of the cloud). Now I program the rocket to fire in order to maintain constant radar distance from the radar set, and measure the necessary thrust.

I think I can calculate (numerically) the worldline followed by the rocket. It'll be parameterised by the radar set's proper time ##\tau##, using simultaneity established by radar. Given that, I can determine the proper acceleration of the rocket as a function of ##\tau##, ##\tau_u##, ##R_u## and ##R_S##. So if I make three measurements of the proper acceleration at known proper times, I can get the black hole parameter and my "orbit" parameters. Barring degeneracy in the maths, I can then tell you the proper time corresponding to me crossing the horizon.

Am I nuts?

To make things a little more tractable I'm going to assume a simple case - I was released from rest at some finite distance from a Schwarzschild black hole. My release radius was ##R_u## (u for unknown) and I was released at my proper time ##-\tau_u## (another unknown). It's messy, but I can actually write down a closed form expression for ##\tau(r,\tau_u,R_u,R_S)## and ##t(r,R_u,R_S)## (there's no ##t_u## because I can set the zero of ##t## at any time I like) in this case of purely radial motion. Sadly, they cannot be inverted to obtain ##r## as a function of ##t## or ##\tau##, nor an explicit relationship between ##t## and ##\tau##.

Inside my box, I have a free-floating radar set and a rocket. I release the radar set as near the center of mass of the box as I can, and the rocket a short distance away in the direction of the hole (which I can determine by releasing a cloud of test particles and watching for elongation of the cloud). Now I program the rocket to fire in order to maintain constant radar distance from the radar set, and measure the necessary thrust.

I think I can calculate (numerically) the worldline followed by the rocket. It'll be parameterised by the radar set's proper time ##\tau##, using simultaneity established by radar. Given that, I can determine the proper acceleration of the rocket as a function of ##\tau##, ##\tau_u##, ##R_u## and ##R_S##. So if I make three measurements of the proper acceleration at known proper times, I can get the black hole parameter and my "orbit" parameters. Barring degeneracy in the maths, I can then tell you the proper time corresponding to me crossing the horizon.

Am I nuts?

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