Do black holes exert infinite forces?

Question

Black holes have infinite mass so according to Newton's law of universal gravitation should exert and infinite amount of gravity force.

Answer 1

There are important points necessary to answer the question:

1. black holes do not have infinite mass;
2. black holes are not well described by Newtonian gravity (unless at a really large distance).

The first point pretty much solves the issue. Black holes have finite masses, but having a lot of mass concentrated in a small region leads to powerful gravitational effects, including the formation of an event horizon. This does not need infinite mass, but rather just a good enough concentration of mass. In particular, a typical process for the formation of black holes is the death of very massive stars, which still have finite mass.

As for the second point, it is important to remember that Newtonian theory is not sufficient to describe black holes. From a large distance away, the Newtonian law will provide a good description, but as you get closer you'll need to employ General Relativity to understand what is going on. In case it interests you, I discuss a bit of the differences between Newtonian "black stars" and relativistic black holes in this answer.

What about forces?

While gravity is not a force in General Relativity, there is a different interpretation of the question that I had not considered at first and was pointed by . Namely, the one pointed out by safesphere in the comments, which I'll phrase as "Is the force needed to keep one from falling into a black hole infinite?". For the purposes of answering this, I'll consider massive particles for simplicity, since they are also the particles we deal with in Newtonian gravity.

At a distance from the event horizon, the answer is no. As I mentioned, if you're sufficiently far away, Newtonian gravity will provide a good description and you can just use the inverse square law to compute the forces at a reasonable precision (depending on your particular experimental interests, of course). However, as you approach the event horizon, General Relativity starts to kick in. As you get closer and closer, you would indeed need larger and larger forces to keep one from falling into the black hole. These forces do diverge at the event horizon, which is a way of saying that no amount of force could pull you out of a black hole. In this specific sense, black holes do exert infinite forces at the event horizon.