Welcome to this week’s installment of the From Eternity to Here book club. Part Four opens with Chapter Twelve, “Black Holes: The Ends of Time.”
Excerpt:
Unlike boxes full of atoms, we can’t make black holes with the same size but different masses. The size of a black hole is characterized by the “Schwarzschild radius,” which is precisely proportional to its mass. If you know the mass, you know the size; contrariwise, if you have a box of fixed size, there is a maximum mass black hole you can possibly fit into it. But if the entropy of the black hole is proportional to the area of its event horizon, that means there is a maximum amount of entropy you can possibly fit into a region of some fixed size, which is achieved by a black hole of that size.
That’s a remarkable fact. It represents a dramatic difference in the behavior of entropy once gravity becomes important. In a hypothetical world in which there was no such thing as gravity, we could squeeze as much entropy as we wanted into any given region; but gravity stops us from doing that.
It’s not surprising to find a chapter about black holes in a book that talks about relativity and cosmology and all that. But the point here is obviously a slightly different one than usual: we care about the entropy of the black hole, not the gruesome story of what happens if you fall into the singularity.
Black holes are important to our story for a couple of reasons. One is that gravity is certainly important to our story, because we care about the entropy of the universe and gravity plays a crucial role in how the universe evolves. But that raises a problem that people love to bring up: because we don’t understand quantum gravity (and in particular we don’t have a complete understanding of the space of microstates), we’re not really able to calculate the entropy of a system when gravity is important. The one shining counterexample to this is when the system is a black hole; Bekenstein and Hawking gave us a formula that allows us to calculate the entropy with confidence. It’s a slightly weird situation — we know how to calculate the entropy of a system when gravity is completely irrelevant, and we also know how to calculate the entropy when gravity is completely dominant and you have a black hole. It’s only the messy in-between situations that give us trouble.
The other reason black holes are important, of course, is that the answer that Bekenstein and Hawking derive is somewhat surprising, and ultimately game-changing. The entropy is not proportional to the volume inside the black hole (whatever that might have meant, anyway) — it’s proportional to the area of the event horizon. That’s the origin of the holographic principle, which is perhaps the most intriguing result yet to come out of the thought-experiment-driven world of quantum gravity.
The holographic principle is undoubtedly going to have important consequences for our ultimate understanding of spacetime and entropy, but how it will all play out is somewhat unclear right now. I felt it was important to cover this stuff in the book, although it doesn’t really lead to any neat resolutions of the problems we are tackling. Still, hopefully it was somewhat comprehensible.
Dr. Carroll, I just gotta say that this chapter alone is worth a lot more to me than Hawking’s entire A Brief History of Time, a book I never really found that interesting or insightful. Dr. Hawking of course should be lauded for his accomplishments in the face of physical adversity, but as an outsider I think I can be forgiven for saying that your explanations in this chapter are superb. The holographic principle now makes sense to me.
Question: What is the “volume” contained with the event horizon of a black hole?
Black holes have come a log way. Or our concept of them has evolved. They exist at the center of galaxies, as solar massed size and recently confirmed at the center of a globular cluster. Their role in the universe is probably much greater than we currently understand. Is the concept of a singularity purely a mathematical concept? Or does it exist in a black hole in terms of occupying space and location? Is the black hole, mass(from stars and interstellar gas, etc) consumed in an unknown state of being, an event horizon and a singularity?
Found this chapter fascinating, really wonderful stuff.
A part that was rather unclear to me though.
When in frustration, I throw my laptop into a black hole, you said that the Hawking radiation will already be carrying the information of the laptop I threw in it, before the laptop even fell in, creating a paradox. Why does this have to be so? Why can’t the radiation which is carrying the information of the laptop only arise AFTER the laptop has completely fallen in? Is this somehow related to the slowing down of time and the laptop never really actually falls in, until we wait an infinite amount of time? (something which I’ve yet to really understand as well…) If that is the issue here, you didn’t state it explicitly and I’m still not sure I understand it. If this isn’t the issue here, then I don’t understand the paradox… Why can’t the radiation start only after the laptop falls in?
Clifford– I don’t think there is any good answer to that question. You can define the Schwarzschild radius by taking the square root of the horizon area, but you can’t associate a unique volume with the interior.
Lawrence– As I discuss in the book, singularities are probably an artifact of classical general relativity.
Oded– When your laptop crosses the event horizon, nothing special happens to it. It’s only lost for good once it hits the singularity. But the flexibility of general relativity allows me to define “time” in such a way that an outside observer can receive the information in the outgoing radiation before the laptop reaches the singularity. That’s the paradox.
Two remarks:
1) The thermodynamics of black holes is not ordinary thermodynamics but a formal analogy built by direct comparison of some well-known formulae in thermodynamics with some other formally similar, not so well-understood, black hole formulae. This is the reason which we usually denote the latter discipline by the term “black hole thermodynamics”.
2) You wrote,
There is nothing surprising here! Entropy in thermodynamics is not always proportional to volume. Why would it be? Take for instance the thermodynamics of surfaces routinely used in physical chemistry and biophysics: the entropy of a surface is proportional to the area of the surface
dS = gamma dA
where gamma is the well-known surface tension.
Bekenstein and Hawking really got the entropy of the black hole horizon. It is not strange that the temperature they got is proportional to the surface gravity (kappa) of the horizon
T_H = kappa / (2 pi)
In fact, Carlip has suggested that the Black hole entropy counts the number of horizon gravitational states.
Shameless plug/question: I have a Javascript tool on my website here for calculating various attributes of black hole, which I wrote a number of years ago. Do the equations look right? I’m not using natural units, so they’re a little more verbose than I gather is usual in common practice.
Suppose I fly past a box at some large speed, and the box has a surface area containing some amount of entropy? What happens if I go past the box even closer to the speed of light, thus reducing the surface area of the box? What does the holographic principle have to say about relativistic length contraction? Does time dilation precisely reduce the apparent entropy in the box, by “freezing out” the motions of the particles in the box? But then that would make measurements of entropy dependent on the observer. Is there some way to formulate a relativistically invariant measure of entropy?
What say ye o’ great oracle?
p.s. with thanks to Poincaré’s Theorem on the stability of differential equations in phase space, we have that in classical General Relativity, except for the trivial paths that were perfectly aimed at the singularity, no path will ever hit the singularity, but it may a-periodically orbit inside of the event horizon.
Aaron said: ” except for the trivial paths that were perfectly aimed at the singularity, no path will ever hit the singularity, but it may a-periodically orbit inside of the event horizon.”
This is utterly wrong, but it’s not your fault: plenty of professional physicists picture the interior of a black hole in this way too. There ought to be a law mandating that all authors writing popular books should devote a chapter to explaining this.
Aaron: see the Kruskal diagram of a black hole. The singularity is 3-dimensional and spacelike.
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Yes you can choose to cut arbitrary boundaries into a smooth manifold, and yes Poincaré’s Theorem will no longer apply, because flows can hit the boundaries. But unfortunately this also means your system is no longer Hamiltonian, or even measuring preserving, which kind of defeats the purpose doesn’t it? The only way to keep the system measure preserving is if the metric avoids the boundary or singularity, for example the boundary of the unit disc in the Poincaré Metric is infinitely far away, i.e. it takes an infinite number of recursions of the flow map to reach the boundary.
So one can’t have their cake and eat it to. Either the system is measure preserving, or it has paths that reach boundaries in a finite number of iterations of the flow. I don’t know which option is weirder, having metrics that place all boundaries and singularities infinitely far away, or having metrics that don’t preserve measure.
“But if the entropy of the black hole is proportional to the area of its event horizon, that means there is a maximum amount of entropy you can possibly fit into a region of some fixed size, which is achieved by a black hole of that size.”
Would you care to explain this in a little more detail?
Thanks.
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Aaron– In GR, singularities are boundaries to spacetime, so the evolution of test particles ceases to be Hamiltonian there. Furthermore, they are boundaries in time, not in space, just like the Big Bang; there’s no way to avoid them.
Will– I’m not sure which part you want explained; the chapter goes into details.
Ceasing to be measure preserving would strike me as being a major obstacle to both formulating a sensible idea of entropy (remember that when flows no longer preserve measure, then things like counting are no longer well defined) and square integrable representations of quantum states.
So that begs the question, what is the stress-energy of a metric that becomes like the Poincare Metric near a boundary? So that the flow avoids the boundary.
Could you comment on Richard’s question in chapter 9. To me it seems fundamental. I am wonder what you think about this. Is gravity just an artifact of mass seeking to reduce the rate of time? why does mass seek to reduce the rate of time?
Thanks for this forum.
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Question: what do you make of Gubser’s comment in “The Little Book of String Theory” that “Time running at different rates at different places is gravity. In fact, that’s all that gravity is….Things fall from places where time runs faster to where time runs slower. That downward pull you feel, and which we call gravity, is just the different rate of time between high places and low places.”
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I remember reading an old John Baez post in which he argued that gravitational does not “break” the second law of thermodynamics, and that entropy still increases in them. Unfortunately his argument wasn’t very convincing at he spent virtually all of it showing how the entropy in a closed gravitational system goes down!
Personally, it seems to me that the second law is to be interpreted in a statistical sense, not in an absolute one. I find it difficult to see how billions of bright hot stars can form from cool gas in a universe where entropy must always increase. Then again, I probably just don’t understand entropy, along with everyone else.
Sean,
While we’re discussing black holes, here’s a question that might be related but I’ve yet to get a good answer to from experts (I asked Mark this once in person, but he couldn’t answer it off the top):
In the excellent popular book “Black Holes & Time Warps” by Kip Thorne, he discusses the Hoop Conjecture, which — very roughly — says that if the energy contained within some finite volume of space exceeds the mass of a black hole which would fill that volume, then the region must contain a black hole, ie it must have a singularity in it and presumably an associated horizon. My question is, does this rule apply to an infinite continuum density of matter? ie like that spread through any smooth universe? The presence of a non-zero cosmological constant may change the answer on large scales, but we can ask the question for zero c.c.: if we rotate a large enough hoop through an infinite universe filled with a uniform density of ordinary matter, then at some point it will enclose more mass-energy than a black hole of the same size — does the Hoop Conjecture apply at that point? if not, why not?
Aaron– You just don’t like classical general relativity. Which is fine; it’s not the final answer.
Waveforms– Sorry I missed that question earlier. There’s a sense in which that’s true, but it’s a (completely understandable) oversimplification. More generally, both time and space are distorted by both mass and energy — i.e., gravity is the curvature of spacetime.
OMF– There are things we do understand about entropy; this includes the fact that the entropy goes up as a cloud of gas collapses into a star.
Paul– It does depend on the type of energy you have lying around; a cosmological constant wouldn’t count. But with matter, the answer is certainly yes; but that’s just the well-known fact that the universe either starts in a Big Bang, or ends in a Big Crunch, or both.
Sorry for the short answers, there’s a plane I need to catch.
I find the geometry of GR quite elegant actually, and I rather fancy it when used to analyze things like the CMB isotropy, or the accelerations of distant galaxies. But what Poincaré clearly shows is that if a path can reach a boundary in some form of finite evolution, then that evolution cannot be unitary, nor in fact even isometric in the direction of evolution towards the boundary.
It seems there is an obvious choice that must be made in these space-time models between preserving unitary evolution (and the ability to formulate actions) and postulating the existence of singularities and boundaries that are finitely accessible. The only wiggle room one has in Poincaré’s theorem is if the sigma algebra of the measure does not correspond with the usual topology of the manifold. But that is even stranger territory, where there is no relationship between distances and the measures of volumes, areas, and hyper-surfaces.
I just finished reading Sean’s book this afternoon…enjoyed it very much. Sean was very objective and quite detailed…a great conceptual introduction, in my opinion anyway…fun and easy to read. The progression of ideas was excellent! As I mentioned, the fact that Sean tied his dialogue to important events in the history of science was enlightening. Those areas he felt could be productive, but which he was not prepared to evaluate in detail, he at least mentioned and described.
Aaron said: “But what Poincaré clearly shows is that if a path can reach a boundary…”
No path inside a black hole “reaches” a boundary in the sense you mean. Instead it *ages* into the boundary, which is to the future of all events. Just as Sean famously said, “You can’t take a wrong turn into yesterday”, in exactly the same way “you can’t take evasive action to avoid tomorrow”.
Sean, last year I read “Reinventing Gravity” by John Moffet and found his alternate approach to gravity and black holes fairly convincing. Do you have any thoughts on his Modified Gravity theory and how that fits into your discussion about entropy and time?
Corey– I don’t find Moffat’s version of modified gravity especially compelling, myself. Alternative models are very useful as foils for GR, but none of them is at all competitive at the moment.
Hi Sean (replying to #18) —
Thanks for the quick reply, but I don’t think you’ve gotten to the essence of the question here regarding the hoop conjecture. Consider these two points:
1. If I understand the hoop conjecture (and maybe I’ll do better once I can afford your GR book) then the statement is that if we define a space-like hypersurface, ie a “now” slice, and within this “now” we can flip the hoop and encompass more rest-mass energy than a BH of the same size as the hoop, then there must be a BH within the hoop _now_, not just in the distant future or the distant past. Is this not correct? If so, then in any infinite, uniform (ie FRW) matter-filled universe it will always be possible to get a large enough hoop within any “now” to activate the conjecture, and so there should be a singularity & horizon within any such “now”. But we know that that’s not true, since the FRW metric is a perfectly valid solution to the uniform matter-filled universe and it has no singularities at times later than the Big Bang (and before the Big Crunch, if it’s closed).
2. The only catch I can see here is that some assumption that led to the Hoop Conjecture might be invalid if the hoop needs to be bigger than the horizon scale at that time. But even if we think this is a catch, we can easily get around it by just considering a matter-filled, flat, and hence critically dense (or arbitrarily close to it) universe. The horizon size in such a universe grows faster than the scale factor (I think), and so it should always be possible to spin a hoop which is both smaller than the horizon and contains enough matter to activate the conjecture if we simply choose a late enough “now” hypersurface to examine.
So, no, I don’t see that the Hoop Conjecture can apply to a continuum of ordinary matter in an infinite universe. But, why not? Does the conjecture depend on an assumption that the matter density is localized, and goes to zero at large radii within the “now” hypersurface? Would it apply to overdensities above some background density? I think these questions are actually directly relevant to your arguments about time, and so should be worth thinking about carefully.
Paul– A black hole is not a region with a singularity “now”; it’s a region with a singularity in the future. In particular, a region inside which every forward-directed timelike trajectory necessarily hits the singularity, and therefore does not escape to infinity. (That’s a Schwarzschild black hole, anyway; with charged or rotating black holes, you can miss the singularity, but still can’t escape to infinity.) So a cosmological singularity fits the bill.