system
Given the following post, respond as if you're commenting on Reddit:
human
No information can travel faster than c, so no event inside an event horizon can have any causal effect on the external universe. How then do external objects "know" the mass of a black hole? Is that information somehow accumulated at/outside the event horizon over the life of the black hole?
gpt
The information is encoded in the gravitational field which extends all the way through infinity.

I'll explain it with charge because it's more immediate, but it's roughly the same with mass.

Say you have an electron outside and far from the black hole. This electron has a charge and field lines extending out to infinity. We can imagine a very large sphere surrounding both the black hole and the electron. Computing the total flux of the electric field through the sphere ("counting field lines that reach infinity") gives the total charge inside; this is Gauss' theorem. This charge must always, always be conserved. Nothing in the Universe can change this.

We lower the electron in the black hole. The field lines can distort, but they must still reach to infinity. There's nothing we can do by moving our puny electron that can change this very global fact about the behaviour of the field at infinity. As the electron crosses the horizon, we don't know/care what has happened to it, but we are sure there still is the same number of field lines, now emanating from the horizon, extending all the way to infinity. The total charge has not changed, and the black hole is now charged.

There is no need for the charge information to get out. Sure, if the electron wiggles inside the horizon, the *radiation* it emits cannot get out. But the Coulomb field is a bit different from radiation, and retains itself the information about the total charge. Charge can actually be seen as a geometrical property or even a topological invariant of the electromagnetic field itself - it has a sort of nonlocal nature. You can measure it even though you're far away and have no causal contact with the stuff that has the charge, you just need to scan a surface enclosing it.

In suitable situation and with smart definitions, mass in general relativity works in a very similar way.

An aside: obviously you cannot communicate from inside to outside the horizon using the information of the total mass/charge... because those things cannot be changed, since they are strictly conserved (superconserved is the actual term). Everything fits together.