I get email: Why doesn't a black hole get infinitely massive?

Greetings, relativists. I got an email from a former high school science teacher named Jeffrey that I thought would be fun to do here, rather than in the column. This is a fair bit more mathematical than even my normal “technical”, so the math-phobics should really, really consider skipping this. The math-philics, on the other hand, will love it.

Jeffrey asks:

Start at some distance from a non-rotating black hole (I can only handle simple things) and throw (or drop) something towards the black hole. The instant before it reaches the event horizon [probably relevant question: is it still matter? Or what form of energy?, or what?] is its speed c? Well, just shy of c. If so, has it’s relativistic mass [or is it momentum, I get confused] up to infinity? If so, what does that does that do to the mass of the black hole when an infinite (?) mass gets added to it?

I’ll get into the detailed answer in a moment, but the basic premise is quite good. Particles gain energy as they approach black holes, and they lose energy as they get further and further away. I’ve talked about this before in our discussion about how time runs slower near massive bodies than far away.

Particles, even photons, gain energy as they fall toward a black hole. To illustrate this point, I can’t show you Robo-Jeff and Dave shooting lasers often enough:

But for now, let’s just talk about dropping in a lump of clay. For convenience, whenever I plug in numbers, I’m going to assume that the black hole is 1 M_odot. I drop the clay very, very far from the black hole.

Far away, the energy of the clay will be given by good ol’


  E=mc^2

But as the clay drops, it will gain kinetic energy. Those of you who’ve taken college level classical mechanics may remember an equation along the lines of:


  Delta K=-Delta U

and, if you made it that far, you may also remember that the Newtonian potential energy for a particle near a massive body is:

  U=-frac{GMm}{r}

where r is the distance from the center, M is the mass of the black hole, m is the mass of the particle, and of course, G is Newton’s constant.

Infinitely far away, there’s zero potential energy. Closer in, energy conservation gives you:


  K=frac{GMm}{r}

In relativity, we always have to worry about the perspectives of different observers. So let’s imagine that we suspend a bunch of observers at different distances from the center of the black hole. It can be shown (in an even more technical discussion than this one), that the observers will measure a particle energy of:


  E_{obs}=frac{E_infty}{sqrt{1-frac{R_s}{r}}}

where R_s is the so-called “Schwarzschild radius” and has a value of:

  R_s=frac{2GM}{c^2}

For the sun, the Schwarzschild radius is about 3km.

This form looks extremely complicated, but it turns out to produce some nice results. Without crunching through too many numbers, you may immediately notice that things get crazy when rsimeq R_s. What happens, as Jeffrey points out, is that the observed energy of the incoming particle literally becomes infinite.

You can show that the speed of the incoming particle would, in principle, reach the speed of light (which for a massive particle, takes an infinite energy) to an observer who was literally exactly on the event horizon. The relation, in case you’re curious, is exactly the same as you would have gotten if you did the calculation in your classical physics course:


  v=c=sqrt{frac{r_s}{r}}=sqrt{frac{2GM}{r}}

which becomes the speed of light at the event horizon. Of course, since no one can actually stand “at” the even horizon, this tends not to be an issue.

But the basic idea remains, if I dump 1kg worth of stuff into the black hole, there are observers very close in who will measure 10kg (or arbitrarily more) worth of E=mc^2 energy falling inward. Who’s right?

What you need to keep in mind is that there are many perspectives (that’s why it’s called “relativity”). Even people near the black hole won’t measure the same energy, since some might be infalling and others are outgoing, and to them, the infalling material will be red- or blue-shifted respectively.

Whose opinion matters? The person infinitely far away.

Energy in relativity becomes very confusing as I hope I’ve just illustrated, but when we talk about the “mass” of something, what we really mean is the energy observed by a distant orbiting satellite or observer. When I say something like, “A 1 solar mass black hole,” I mean that the dynamics far away are the same as a 1 solar mass star.

Or, if you like thinking about things classically, it doesn’t matter that an infalling particle gains so much kinetic energy. It loses potential energy, which means that the total energy, by any reasonable standard, will remain the same.

Ahh… it’s fun to use equations every now and again.

-Dave

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