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.
OMF and Sean
I always assumed that gravitational collapse was entropically favorable for the same reason that nuclear fusion is: while the system itself decreases in entropy (4 free protons have more entropy than 1 free alpha particle, a million-mile-wide star has less entropy than a light-year-wide gas cloud,) both transformations are exothermic and the heat released into the environment carries away more entropy than is lost in the fusion reaction or the gravitational collapse.
Now I can imagine that in some very high mass situations, a lot of the energy would be carried away in gravitational waves, and calculating the entropy of gravitational waves could be problematic without a decent theory of quantum gravity. Aside from that, though, is there a problem with the above explanation?
Exactly right. When black holes coalesce, a lot of entropy is carried away in gravitational waves.
Hi Sean — Thanks for the clarification at #25. Your description reminds me of a diagram from Penrose’s “Emperor’s New Mind” which shows an expanding and re-collapsing universe (embedding diagram looks like a football) with several BH singularities shown as “tears” in the manifold all merging at the Big Crunch. Basically all forward-going paths end at some kind of “tear,” so there’s no distinction between the singularities that form “earlier” inside BH’s and the one at the Crunch.
I’ll have to study up on the Hoop Conjecture and its assumptions before arguing any further. But, at this level one question still lingers: what about a universe with zero cosmological constant which never re-collapses? Consider an expanding, uniform, ordinary-matter-filled flat universe which doesn’t have any singularities in the far future; does this still agree with the Hoop Conjecture? (Or is the surface at infinite time somehow a singularity? that would sound like cheating to me.)
Why do we assume that most of the Galaxy’s black hole mass is in the central hole. If the galaxy has, say 10^5 stars big enough to collapse into black holes, and the stars live for 10^7 years, then over the 10^10 year galactic lifetime we expect there to be 10^8 stellar black holes. At 10 solar masses each, they give 10^9 stellar masses of black holes, much larger than the central hole.
Or have I made a stupid assumption?
Paul– The hoop conjecture doesn’t tell you whether the singularity is in the future or the past, so the Big Bang counts. It’s like a white hole.
Lab Lemming — The entropy of a black hole goes as the mass squared. To equal the entropy of a single million-solar-mass black hole, you would need 10^12 individual solar-mass black holes.
If a black hole eats a magnetic monopole, can it still be described only by mass, charge, and angular momentum?
Also, is there any way to experimentally or observationally verify any of the numerous qualities assigned to black holes (Hawking radiation, entropy, space-time curvature, event horizon weirdness, etc)?
The “charge” that black holes can have refers to any long-range (1/r^2) type of field, so both electric and magnetic charges are possible.
Verifying Hawking radiation or entropy would be extremely hard unless you find a microscopic black hole, because the temperature goes down as the mass goes up, and all macroscopic black holes are extremely cold. You can imagine measuring the geometry near the event horizon through X-ray interferometry or gravitational-wave observations, and that’s a relatively realistic goal.
Sean,
I have a theory that i’d like to share with you. Just not in a public forum. How can i properly forward it to you? Probably less than a paragraph. I do not have the educational background to prove or disprove my idea. You may be able too. If you were able to prove it to be true, I believe that it would be a revelation of our planetary interactions.
thanks for your consideration,
Michael
Sorry, with regard to the black hole entropy question, what I was wondering is how one makes it clear that the maximum entropy a volume of space can contain is that given by the Bekenstein formula?
Has anyone figured out the percentage of mass of a singularity in a supermassive black hole compared to the percentages of Dark, Normal, and Black hole matter in a typical Galaxy or perhaps all singularity mass in the universe as a whole?
Is it still zero volume with infinate density?