What is the Gravitational Acceleration Beyond the Event Horizon of a Black Hole?

[kuahji]

I couldn't find any information on this subject. I'm curious to find out how fast scientists believe objects accelerate &/or at which speed objects move beyond the event horizon of a black hole. For example, the gravitational acceleration on Earth is roughly 9.8 m/s. So, beyond the event horizon what is the gravitational acceleration? Any links/books would also be of help. Thanks.

 

[pervect]

Outside the event horizon, one can find the acceleration required to "hold station", measure it with an accelerometer in the ship or object that is "holding station", and call that the gravitational acceleration of the black hole.

Inside the event horizon, no object can "hold station". Therefore there isn't any simple coordinate independent answer for what the velocity or acceleration of an object is.

One can find solutions to the geodesic equation for the Schwarzschild r coordinate and the Schwarzschild t coordinate as a function of proper time "tau". These solutions are ill-behaved at the event horion itself (they are ill-behaved in Schwarzschild coordinates, anyway, they can be well-behaved in other coordinate) but work elsewhere.

If you think such a Schwarzschild coordinate solution for the trajetory of an infalling object would answer your question, I can probably provide one and/or a reference. If you want to deal with Friedman or Kruskal coordinates (which get around the coordinate singularity issue at the horizon) I can probably give you some information on those, too.

Note that inside the event horizon, the Schwarzschild 'r' coordinate acts like a time coordinate, and 't' acts like a space coordinate. This makes physical interpretation of these mathematical solutions a bit tricky. This is also why the coordinates are ill-behaved right at the horizon - the 'r' coordinate is in the process of switching over to measure time, and the 't' coordinate is in the process of switching over to measure space.

Note that these solutions will be solutions for an idealized "Schwarzschild" black hole. Real non-rotating black holes are expected to have a chaotic interior solution known as a BKL singularity - the nice regular Schwarzschild solution, which works well for the exterior region, is expected to be unstable in a real black hole. Thus the Schwarzschild solutions I can quote will be illustrative of what happens in an idealized case, but will probably not represent what would physically happen in an actual black hole.

 

[kuahji]

Thanks much for the info. It was a big help.

 

[pervect]

Anyone dropping into a black hole is not going to feel any gravity - like anyone in free-fall, they will be essentially weightless.

What someone can feel as they fall into a black hole is not the gravity, but the tidal force, which results from the difference in gravity between points closer to the center of the black hole and further away.This tidal force acts in such a manner as to turn people or objects into "spaghetti".

See for instance

http://www.museum.vic.gov.au/planetarium/stars/blackholes.html