Black hole question: apparent rate of growth as seen by external observers

[JesseM]

Suppose we have an observer at a large distance from the event horizon of a black hole, pouring matter towards the BH at some constant rate of m per second. Will he see the BH's radius increasing at a rate of 2Gm/c^2 per second, or does gravitational time dilation change the apparent growth rate somehow?

 

[Physics Monkey]

This is a tricky question because in the Schwarzschild solution, for example, any infalling mass will take infinite coordinate time, as judged by an external observer, to reach the horizon. So how can the hole grow if matter can never enter it? The same thing is true for two black holes that merge: it seems that their event horizons have to actually touch for all time. Of course, no one knows an exact solution in this case so maybe this reasoning is flawed. You can get energy out of Kerr type hole with something like a Penrose process, but I don't see how this helps understand how black holes can grow. In my personal opinion, I think it may be that we need to understand black holes that don't require future and past infinity in the spacetime.

However, I may be way off base here, advanced black hole theory isn't exactly my specialty. Perhaps the problem is alleviated in quantum gravity, that would certainly be interesting, though I see no reason to suspect a classical GR explanation isn't possible.

 

[pervect]

An interesting question. I don't see how the external observer is going to measure the location of the event horizon - for instance, he won't get a radar echo from it...

Suppose the observer instead decides to find the mass of the black hole to compute the location of the horizon. He could observe orbital parameters of an object orbiting the hole. He may be far away from the hole, but he could observe the orbital parameters of an object "close" to the event horizon, I suppose. Close would have to be within the area that stable orbits existed - definitely outside the photon sphere, for instance - so there wouldn't be any huge gravitational time dilation effects.

So I think he's going to get the mass of the black hole plus the mass of some of the stuff outside the hole, I don't think he's going to get a pure measurement of the mass of the black hole. At least that's my first reaction.

 

[JesseM]

An interesting question. I don't see how the external observer is going to measure the location of the event horizon - for instance, he won't get a radar echo from it...

Is there any particularly "natural" coordinate system to use in the case of a growing black hole, perhaps one that would reduce to Schwarzschild coordinates for brief time intervals when the size wasn't changing appreciably? If so, the question could be rephrased as one about how the event horizon is growing in such a coordinate system, instead of one about what the observer will actually see.

Incidentally, the question arose in the context of a discussion on another board of what would happen if a small black hole fell to the center of the earth--someone suggested that perhaps we wouldn't have to worry about it gobbling up the entire planet, since from our external POV matter would never reach the event horizon so the black hole would never grow in size. This is definitely incorrect, no? There must be some meaningful sense in which we can observe the size of a black hole to change over time, otherwise the concept of black hole evaporation via Hawking radiation wouldn't make sense.

 

[Physics Monkey]

For a true classical Schwarzschild black hole, infalling matter will never reach the event horizon as judged by an external observer, it just won't. But then significant infalling matter isn't going to be described by a Schwarzschild solution anyway. If we take as given that "real" black holes can grow by absorbing material, then these holes must have a rather different topological structure than the usual Schwarzschild hole, right?

 

[JesseM]

For a true classical Schwarzschild black hole, infalling matter will never reach the event horizon as judged by an external observer, it just won't.

I realize an external observer can never see anything reach the horizon, but can't the size of the horizon itself change, carrying the near-frozen images of objects near the surface (assuming you could still see them at arbitrary redshifts) with it? Something like this seems to be what's described in http://cosmology.berkeley.edu/Education/BHfaq.html#q9, where they say an external observer would see an infalling object keep hovering closer and closer to the horizon as the horizon itself shrinks, finally reaching the horizon at the precise moment the black hole shrinks to nothing.

 

[JesseM]

I had a memory that Kip Thorne's book Black Holes & Time Warps had a discussion of what happens when a black hole swallows something and the size of its event horizon changes, so I went back and checked, it's discussed in chapter 12, which starts on p. 413:

Before November 1970, most physicists, following Roger Penrose's lead, had thought of a hole's horizon as "the outermost location where photons trying to escape the hole get pulled inward by gravity." This old definition of the horizon was an intellectual blind alley, Hawking realized in the ensuing months, and to brand it as such he gave it a new, slightly contemptuous name, a name that would stick. He called it the apparent horizon.

Hawking's contempt had several roots. First the apparent horizon is a relative concept, not an absolute one. Its location depends on the observer's reference frame; observers falling into the hole might see it at a different location from observers at rest outside the hole. Second, when matter falls into the hole, the apparent horizon can jump suddenly, without warning, from one location to another--a rather bizarre behavior, one not conducive to easy insights. Third and most important, the apparent horizon had no connection at all to the flash of congealing mental pictures and diagrams that had produced Hawking's New Idea.

Hawking's new definition of the horizon, by contrast, was absolute (the same in all reference frames), not relative, so he called it the absolute horizon. This absolute horizon is beautiful, Hawking thought. It has a beautiful definition: It is "the boundary in spacetime between events (outside the horizon) that can send signals to the distant universe and those (inside the horizon) that cannot." And it has a beautiful evolution: When a hole eats matter or collides with another hole or does anything at all, its absolute horizon changes shape and size in a smooth, continuous way, instead of a sudden, jumping way (Box 12.1)

...

Why were Penrose and Israel so wedded to the apparent horizon? Because it had already played a central role in an amazing discovery: Penrose's 1964 discovery that the laws of general relativity force every black hole to have a singularity at its center. I shall describe Penrose's discovery and the nature of singularities in the next chapter. For now, the main point is that the apparent horizon had proved its power, and Penrose and Israel, blinded by that power, could not conceive of jettisoning the apparent horizon as the definition of a black hole's surface.

They especially could not conceive of jettisoning it in favor of the absolute horizon. Why? Because the absolute horizon--paradoxically, it might seem--violates our cherished notion that an effect should not precede its cause. When matter falls into a black hole, the absolute horizon starts to grow ("effect") before the matter reaches it ("cause"). The horizon grows in anticipation that the matter will soon be swallowed and will increase the hole's gravitational pull (Box 12.2).

Penrose and Israel knew the origin of this seeming paradox. The very definition of the absolute horizon depends on what will happen in the future: on whether or not signals will ultimately escape to the distant Universe. In the terminology of philosophers, it is a teleological definition (a definition that relies on "final causes"), and it forces the horizon's evolution to be teleological. Since teleological viewpoints have rarely if ever been useful in modern physics, Penrose and Israel were dubious about the merits of the absolute horizon.

Hawking is a bold thinker. He is far more willing than most physicists to take off in radical new directions, if those directions "smell" right. The absolute horizon smelled right to him, so despite its radical nature, he embraced it, and his embrace paid off. Within a few months, hawking and James Hartle were able to derive, from Einstein's general relativity laws, a set of elegant equations that describe how the absolute horizon continuously and smoothly expands and changes its shape, in anticipation of swallowing infalling debris or gravitational waves, or in anticipation of being pulled on by the gravity of other bodies.

And Box 12.2 provides an example of how the apparent and absolute horizons change in response to matter falling into the black hole:

The spacetime diagram below illustrates the jerky evolution of the apparent horizon and the teleological evolution of the absolute horizon. At some initial moment of time (on a horizontal slice near the bottom of the diagram), and old, nonspinning black hole is surrounded by a thin, spherical shell of matter. The shell is like the rubber of a balloon, and the hole is like a pit at the balloon's center. The hole's gravity pulls on the shell (the balloon's rubber), forcing it to shrink and ultimately be swallowed by the hole (the pit). The apparent horizon (the outermost location at which outgoing light rays--shown dotted--are being pulled inward) jumps outward suddenly, and discontinuously, at the moment when the shrinking shell reaches the location of the final hole's critical circumference. The absolute horizon (the boundary between events that can and cannot send light rays to the distant Universe) starts to expand before the hole swallows the shell. It expands in anticipation of swallowing, and then, just as the hole swallows, it comes to rest at the same location as the jumping apparent horizon.

So, is anyone familiar with these notions of the apparent and absolute horizon of a black hole? When matter is being fed into the hole continuously as in my original question, I'd think the apparent horizon would move outward continuously rather than jumping discontinously--I wonder if it would simply coincide with the absolute horizon in this case, or if they would both move outward continuously but one would lag behind the other. And, as in my original question, if the mass were dropping in at a rate of M per second (as seen by a distant observer), I wonder if either one or both of the apparent and absolute horizons would grow at a rate of 2GM/c^2 per second.