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Hello! I have recently tried to calculate the gravity of a black hole at the event horizon. In other words, I want to calculate the surface gravity of an object if the escape velocity to leave that object is the speed of light. But I dont think I did the math right. I believe that the required gravity is less than infinity because there is only infinite gravity at the singularity. How can I determine the gravity of an object when the escape velocity is the speed of light (or more)? Please show how to calculate it. Thank you so much for your help!
Answer 1:

I'm sure you're using Newton's law of gravitation, F = GmM/r2, for doing this. That formula is incorrect when it comes to dealing with extremely powerful gravitational fields, such as those that create the event horizon of a black hole. In order to calculate the gravitational interactions of an object interacting with a black hole at or near the event horizon, you will need Einstein's general relativistic equations, which require more math than I have the capacity to explain. To try to explain it from what I do understand, Einstein's equations expand gravitational acceleration into an expansion series, i.e. F = n1 + n2 + n3 + n4 + ... on to infinity, of which n1, n2, n3, etc. are terms. The n1 term is Newton's GmM/r2 term, and all of the other terms are essentially zero for the purposes of calculating gravitational interactions between most objects (planets, stars, asteroids, spaceships, etc., even black holes if you're far enough from the event horizon). As a result, Newton's formula makes an outstandingly good approximation of gravity for most purposes - really, the rate of precession of the orbit of Mercury is the only thing in ordinary celestial mechanics that would tip you off that Newton's theory isn't quite right. But Newton's theory isn't right, and black holes are where Newton's theory breaks down.

Now, ironically, if you calculate the distance from which an object the escape velocity equals that of the speed of light, without using *any* relativity calculations (not even special relativity, which you've been using), it turns out that this magical distance happens to be the same as the event horizon you would derive using the full general relativistic calculations. This radius scales linearly with mass (as the radius of any escape velocity does), and is about 1.8 km per solar mass. However, this is something of a coincidence: you would not predict the existence of black holes, or any of their properties, from special relativity. You need general relativity.



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