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I've been studying General Relativity on a friend's notes. There's an exercise I can't quite grasp.

An observer is located in a closed, windowless box. The box is falling radially towards a black hole. The observer has two identical metal spheres, a measuring tape, and a watch.

  1. Can he determine he is free falling, in a gravitational field?
  2. Can he determine, at any instant, the black hole's mass and his distance from the black hole? (Use explicit calculations to show if it's possible, or why it isn't.)
  3. Can he, at a given moment, know his exact postion with respect to the black hole?

The first one is easy, I think. The observer should drop both spheres at the same time and see that their trajectories aren't parallel, and that they tend to get closer. So he should assume he is in a radial gravitational field.

For the second one my answer would be no, because if he used his clock to measure time, or his tape to measure lenght, he would have nothing to compare the measurements to, so he wouldn't be able to find time dilation or lenght contraction. A more mathematical way to prove that it can't be done would be to find a relation that includes the black hole's mass and the observer's position, but I don't know how to go on about that.

The third question is directly related to the second one. Any input would be greatly appreciated.

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  • $\begingroup$ The field will be stronger at the side closer to the black hole and weaker at the other side so the spheres will drift apart. You can determine the gradient of the field by measuring the time and distance of the drift. Check the gradient again at some later time and you may be able to figure out 2 and 3. $\endgroup$ – 511mev Jun 3 at 1:46
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I think the first answer should also give the answer to 2nd question. The ball will have geodesic declination and you can calculate Riemann tension from that so you can get $T_{\mu\nu}$ And her the mass. For third one you can either use the 2nd method as you suggested or if it is a rotating blackhole then the guy from rocket may try to measure ISCO innermost stable circular orbir and there is a direct relation to isco and radious of black hole.

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