The event horizon’s involved, but the singularity is committed

Lenny Susskind — the Stanford string theorist who Shtetl-Optimized readers will remember from this entry — is currently visiting Perimeter Institute to give a fascinating series of lectures on “Black Holes and Holography.”

After this morning’s lecture (yes, I’m actually getting up at 10am for them), the following question occurred to me: what’s the connection between a black hole having an event horizon and its having a singularity? In other words, once you’ve clumped enough stuff together that light can’t escape, why have you also clumped enough together to create a singularity? I know there’s a physics answer; what I’m looking for is a conceptual answer.

Of course, one direction of the correspondence — that you can’t have a singularity without also having an event horizon — is the famous Cosmic Censorship Hypothesis popularized by Hawking. But what about the other direction?

When I posed this question at lunch, Daniel Gottesman patiently explained to me that singularities and event horizons just sort of go together, “like bacon and eggs.” However, this answer was unsatisfying to me for several reasons — one of them being that, with my limited bacon experience, I don’t know why bacon and eggs go together. (I have eaten eggs with turkey bacon, but I wouldn’t describe their combined goodness as greater than the sum of their individual goodnesses.)

So then Daniel gave me a second answer, which, by the time it lodged in my brain, had morphed itself into the following. By definition, an event horizon is a surface that twists the causal structure in its interior, so that none of the geodesics (paths taken by light rays) lead outside the horizon. But geodesics can’t just stop: assuming there are no closed timelike curves, they have to either keep going forever or else terminate at a singularity. In particular, if you take a causal structure that “wants” to send geodesics off to infinity, and shoehorn it into a finite box (as you do when creating a black hole), the causal structure gets very, very angry — so much so that it has to “vent its anger” somewhere by forming a singularity!

Of course this can’t be the full explanation, since why can’t the geodesics just circle around forever? But if it’s even slightly correct, then it makes me much happier. The reason is that it reminds me of things I already know, like the hairy ball theorem (there must be a spot on the Earth’s surface where the wind isn’t blowing), or Cauchy’s integral theorem (if the integral around a closed curve in the complex plane is nonzero, then there must be a singularity in the middle), or even the Nash equilibrium theorem. In each of these cases, you take a geometric structure with some global property, and then deduce that having that property makes the structure “angry,” so that it needs a special point (a singularity, an equilibrium, or whatever) to blow off some steam.

So, question for the relativistas: is there a theorem in GR anything like my beautiful story, or am I just talking out of my ass as usual?

Update (3/22): Well, it turns out that I was ignorantly groping toward the famous Penrose-Hawking singularity theorems. Thanks to Dave Bacon, Sean Carroll, and ambitwistor for immediately pointing this out.

41 Responses to “The event horizon’s involved, but the singularity is committed”

  1. Ambitwistor Says:

    The Penrose-Hawking singularity theorems are what give conditions on the existence of singularities within black holes. The geometric structure they use is the existence of a “trapped null surface”, at which outward-directed light rays converge towards each other.

  2. Dave Bacon Says:

    What you’re looking for, I think, are the theorems of Hawking and Penrose proven in the sixties. I think Hawking and Ellis’s book contain lots of info on this.

  3. Dave Bacon Says:

    Doh. What ambitwistor said.

  4. Sean Carroll Says:

    Depends on whether or not you’re asking “What does GR predict?” or “What could conceivably happen?”

    If you’re sticking to GR, plus some reasonable conditions on the matter fields (no negative energies etc), then the aforementioned singularity theorems kick in. Once you squeeze enough stuff into a region to make an event horizon, it will collapse all the way to a singularity. Imagine a sphere in space where both inward-going and outward-going light rays actually converge on each other. That’s a “trapped surface,” and a good indication that you’ve crossed an event horizon. It’s telling you that space is getting smaller in every direction. No suprise that a singularity is the ultimate result.

    If you don’t restrict yourself to GR (or you allow negative energies), you can avoid the singularity. Your paths have to go somewhere, but you can imagine other possibilities — maybe you could slip through a wormhole to another asymptotic region, for example.

  5. Scott Says:

    Duhhhh, thanks everyone! I’d heard of the singularity theorems, but I stupidly failed to make the connection. I guess the key point I was missing is that the event horizon enters into the theorems as one of the assumptions. (Incidentally, it’s interesting to see yet another instance where focussing of light rays seems like the crucial thing about the event horizon…)

  6. Pascal Koiran Says:

    Thank you Scott for answering the question that always puzzled me about black holes.
    If I still can’t form a clear mental picture of a black hole, that must be due only to my utter ignorance of general relativity.
    Could someone knowledgeable tell us whether these singularity theorems have something to with the fact that space-time is 3+1 dimensional ? (I hope that no string theorist is reading this…)
    I ask because you mentioned the hairy ball theorem. There is definitely no hairy disk theorem…

  7. Pedro Pinheiro Says:

    What would the time dilation factor be for an observer approaching a singularity? This has been something that I’ve thought about for a while – would it approach stretching time “forever” from the observer’s viewpoint?

  8. Scott Says:

    Pascal: Last night I was reading this tutorial about the singularity theorems, which (since it defines everything mathematically) you and I might in principle fully understand. 🙂 According to the tutorial, the theorems work in any number of dimensions.

    Incidentally, while the statement of the result reminds me of the Hairy Ball Theorem, Cauchy Integral Theorem, Kakutani Fixed-Point Theorem, etc., the proof (insofar as I understand it) doesn’t really remind me of any of them.

  9. John Sidles Says:

    IMHO, Scott’s insight that “I know there’s a physics answer; what I’m looking for is a conceptual answer” is very useful, and perhaps can even be sharpened.

    As everyone knows, in the 20th Century mathematicians realized that every deep question can be answered in three ways: algebraically, differentially, and geometrically. There’s no sense asking which is most fundamental (unless one wants to provoke an enjoyable squabble over a dinner and/or glass of wine, which mathematicians often do!).

    Nowadays, information theory is “busting down the door” by demanding to be admitted to the holy triad of algebra, analysis, and geometry. To further this door-busting information-theory agenda, what Scott and many others are looking for (AFAICT) is an informatic derivation of the Penrose-Hawking singularity theorems — and they surely are right to think that this may be feasible.

    Tracking this exciting mathematical revolution (and borrowing mathematical ideas from it) is the main reason that I read this blog. Thanks, Scott! 🙂

  10. Greg Kuperberg Says:

    It’s important to understand that the mere existence of an event horizon does not imply that there is a singularity anywhere. Astronomers now think that we live in a de Sitter universe, which will develop a backdrop event horizon in all directions. An event horizon is no more and no less than a threshold of no return, which can arise from a non-zero cosmological constant. It is also observer-dependent. You have to select the observer that you can’t return to.

    I do not doubt the Penrose-Hawking theorems, but they would have to impose geometric conditions on the event horizon in order to derive a singularity.

    The most interesting question, in my view, is the converse, whether every singularity is always censored by an event horizon. As far as I know, this is still an open problem in GR.

  11. Sean Carroll Says:

    What would an information-theoretic viewpoint on the singularity theorems even mean? I’m not asking to be a smartass, I’m honestly curious. I know that black holes have an entropy, and entropy has an information-theoretic definition, so it’s not too hard to imagine a connection there, although I’m not sure how to extend it beyond the buzzword level. But a singularity is (nominally) way inside the event horizon, and I don’t see any obvious connection.

    It also might be worth pointing out that the singularity theorems are rigorous results about a theory that is not correct — classical general relativity — in a regime where we don’t expect that theory to be applicable. Either quantum gravity will be important (most likely), or some new classical theory. As far as the real world is concerned, they’re better thought of as GR-breaking-down theorems.

  12. Scott Says:

    Right, the singularity theorems are basically GR predicting its own breakdown! Regarding Sean’s question, I have no idea what an “information-theoretic viewpoint” would mean here. I asked for a conceptual answer, and a conceptual answer is what I’m getting! 🙂

    In particular, I think I now understand what was wrong with my previous intuition. I was operating under the impression that, once you pass through the event horizon, you can sort of float around freely for a while before finally landing at the singularity. The question then arises of why you ever have to land at a singularity, and why there has to be a singularity in the first place. What I wasn’t thinking through is just how severe the bending of spacetime is in this region — how it makes a spatial coordinate behave more like a time coordinate. In particular, if there’s only radial direction you can move in at the event horizon, then imagine how much worse it is inside the event horizon! When you think of it that way, it’s indeed not surprising (as Sean says) that a singularity will be the result — the singularity theorems are really just formalizing the intuition rather than telling us anything unexpected.

  13. John Sidles Says:

    Scott says: the singularity theorems are really just formalizing [geometric] intuition rather than telling us anything unexpected …

    … which is precisely why information theory is even more likely than geometry to tell us something really unexpected. 🙂

  14. Dave Bacon Says:

    One thing one has to be careful with here is the fact that there are many different types of event horizons. For example the cosmological event horizon which Greg discusses, I think, is different that the notion of a trapped surface used in the singularity theorem. Also there are event horizons which are observor dependent, the most famous being the one when you constantly accelerate the observor. Hmmm…I really need to brush up on my general relativity!

  15. Scott Says:

    Last night I was feeling mighty sheepish for wasting people’s time with such an elementary question, when I could’ve easily looked up the answer myself. But then, in Lenny’s lecture this morning, the mere fact of having blogged my doofus concerns seemed to dramatically improve my comprehension — so maybe I should try this more often!

    Anyway, the more of these lectures I go to, the more obvious it becomes why physicists like black holes. They’re like a blockbuster where all the audience favorites — GR, QM, QFT, thermodynamics, electromagnetism — reveal their true characters by being pushed to the most extreme situations.

    In fact, how does the following sound for a revised undergraduate physics curriculum?

    First year: Quantum computing and information
    Second year: Black holes
    Third and fourth years: Everything else

  16. Greg Kuperberg Says:

    Also there are event horizons which are observor dependent

    Which is to say, all of them.

  17. Pascal Koiran Says:

    Sean Carroll pointed out that black holes as described by general relativity probably do not exist in physical reality.
    Then what should we make of the claim that black holes have already been (indirectly) observed?
    If the singularity does not really exist, does the event horizon really exist?

  18. Greg Kuperberg Says:

    Pascal: The singularity in a black hole is what is unlikely to exist. Moreover, Hawking radiation is an indirect prediction that simply supposes that there is some consistent theory of quantum gravity. Despite these caveats, you do expect black holes in the universe that are virtually identical, from the point of view of practical observation, to black holes as predicted by Schwarzschild using classical GR.

  19. Greg Egan Says:

    Greg Kuperberg wrote:

    Also there are event horizons which are observor dependent

    Which is to say, all of them.

    Could you elaborate on what you mean by this in the context of a Schwarzschild black hole?

    Of course there are “phoney” horizons like the Rindler horizon, which exist for observers who choose to constantly accelerate in a fixed direction in flat spacetime, and are basically just an arbitrary choice of a set of flashes of light moving in parallel. You can point at this light and say: “Hey, if this light passes you, you’re never going to catch up with it!” (i.e. if you pass through this “horizon” in one direction, you’re never going to pass through it in the opposite direction). The choice of the set of light flashes is completely up to you, in as much as by choosing to constantly accelerate with a particular rate and direction, you determine the flashes of light that will never catch you; this is your Rindler horizon.

    Now the same Rindler horizon is actually shared by all constantly accelerating observers who choose the same direction to travel in, and set their rate of acceleration according to the formula a=c2/s, where s is their distance from the horizon. The world lines of all these observers form a set of concentric hyperbolas in spacetime. In 1+1 spacetime, the Rindler horizon is a line asymptotic to all the hyperbolas. If you start on one of the hyperbolas and then cross that line, then you can never get back to any of those hyperbolas via a timelike path.

    In the Schwarzschild black hole case, the analogue of that set of observers sharing a Rindler horizon is the set of observers who have fixed Schwarzschild r-coordinates. They are all constantly accelerating in such a way as to keep them at fixed distances from a common horizon, and if you visit either Mary or Bob, who are doing this at different r coordinates, then whether or not you can visit either of them again depends on whether or not you’re dumb enough to cross their common Schwarzschild horizon.

    However, if Jane is moving away from the black hole, and if she’s committed to undergoing constant proper acceleration, you might lose your chance to see her twice without ever crossing the Schwarzschild horizon, for the Rindler-ish reasons described above.

    And if Sam decides to accelerate towards the black hole (or not accelerate away from it fast enough to keep his r-coordinate constant), then once again the Schwarzschild horizon isn’t the right boundary to decide whether you can travel away from him and then meet him again. You and Sam can do a joint suicide dive whereby the two of you start together, you rush ahead and cross the Schwarzschild horizon first, then you fire your retro-rockets hard enough for Sam to catch up with you in the black hole’s interior.

    Is that what you’re getting at? I still think that Mary and Bob’s horizon is worthy of a special distinction: it’s the smallest horizon that doesn’t involve suicide.

  20. Greg Egan Says:

    In my preceding post, I mangled the link to a web page I wrote on the Rindler horizon.

  21. Greg Kuperberg Says:

    Could you elaborate on what you mean by this in the context of a Schwarzschild black hole?

    You are of course correct that the observer at infinity of a Schwarzschild black hole has special geometric properties. Any observer who can return to infinity may perceive a different event horizon for a while, but it will not be so different that it can’t be restored to the horizon that the infinite observer sees. All of that is true, and that is a kind of special status.

    Even so, the Schwarzschild event horizon is qualitatively equivalent to all other event horizons. That includes event horizons like the de Sitter example that do not have a distinguished observer and are fundamentally observer-dependent. So as I see it, the special status of the observer at infinity in the Schwarzschild spacetime is a geometric artifact. There is no physical membrane of any kind parked at a Schwarzschild event horizon (or any other event horizon). If you travelled to encounter it, you would no longer be the observer at infinity who is needed to define the horizon.

    Another way to express my view is that all event horizons are a type of mirage. A mirage is a physical effect, but it is not a physical object. It is an artifact of a physical object (such as a desert); it may also be true, depending on geometry, that a certain class of observers who see a mirage may be in a distinguished geometric position.

  22. Greg Kuperberg Says:

    Greg Egan: Also, in your geometric argument, you missed a case. It is possible to fall towards a black hole and see its event horizon shrink, then bail out and watch it grow again. So it is not quite true that the asymptotic event horizon is the smallest one that you can see without committing suicide. You can see a smaller one, temporarily, if you play a game of chicken.

  23. Greg Egan Says:

    Greg Kuperberg wrote:

    So it is not quite true that the asymptotic event horizon is the smallest one that you can see without committing suicide. You can see a smaller one, temporarily, if you play a game of chicken.

    Fair enough. I was thinking of classes of observers undergoing constant proper acceleration.

    Even so, the Schwarzschild event horizon is qualitatively equivalent to all other event horizons.

    Sure. At least locally, everything about a Schwarzschild horizon can be seen in a Rindler horizon. It all boils down to the fact that you can’t catch up with light, but you can stay ahead of it by constant acceleration.

    There is no physical membrane of any kind parked at a Schwarzschild event horizon (or any other event horizon).

    I’m 100% percent with you on that. This is why people who think classical GR breaks down in some observer-independent sense at event horizons drive me crazy.

  24. Greg Kuperberg Says:

    This is why people who think classical GR breaks down in some observer-independent sense at event horizons drive me crazy.

    I agree with the sentiment, but what is subtle is the GR probably does break down in an observer-dependent sense at event horizons. Otherwise there would be no Hawking radiation. People believe that the quantum information in the Hawking radiation is only the same information that fell in, scrambled. Let us suppose that it is so. Then if Alice sees Bob fall into a black hole, she sees his quantum state assimilated into Hawking radiation. (She may have a tough time decoding the Hawking radiation to reconstruct Bob, but in principle she can do it.) Bob himself lives a different future, incompatible with what Alice sees. There is no contradiction if Alice and Bob have parted for good. However, you can’t have a split story like this in classical GR.

    Because of observer dependence, I cannot say just from the abstract that gr-qc/0012094 is entirely wrong. It still looks wrong-minded to me, but I don’t really know.

  25. Scott Says:

    Another interesting question is whether it’d be possible to surround our solar system by an event horizon that prevented any outside light from getting in — a “quarantine,” so to speak…

    (Sorry, couldn’t resist. Welcome to the blog, Greg E.!)

  26. Greg Egan Says:

    Thanks for the welcome, Scott. It’s a great blog that I only discovered recently, after you got mentioned on the N-Category Café in this entry.

    I’m still trying to live down all the dumb quantum mechanics in Qu*r*ntine, 15 years after writing it. Every SF novel is entitled to one outrageous idea, but even given that licence I think I mangled the consequences. When I get the time, I’m planning to write a detailed mea culpa, cataloguing the havoc that decoherence would wreak on various aspects of the plot …

    Greg Kuperberg: FWIW, my suspicion is that most people who think that weird stuff happens locally at event horizons are picking unphysical observers who manage to stay fixed on those horizons. Riding on a lightbeam always tends to give you an odd view of the universe.

  27. Scott Says:

    Greg E.: I actually used a quote from Quarantine as the epigraph for my survey article “NP-complete problems and physical reality”. (See also this PowerPoint talk.)

    What struck me is that you were (in effect) asking some of the same questions theoretical computer scientists would ask about quantum computing, but a few years before they did. It took a theorem proved in 1994 — namely, the BBBV theorem — to prove that the plot of your novel is impossible in ordinary (linear) quantum mechanics!

    Incidentally, do you still collaborate with Dan Christensen? Years ago, I spent a several days working with him to try to define a model of computation based on spin foams. Alas, I eventually decided that it’s a premature question. (As I sometimes put it, “first tell me what time is in QG; then I can try to tell you what polynomial time is!”)

  28. Greg Egan Says:

    Scott: I looked at your survey article a couple of weeks ago, though I haven’t had a chance to study it properly. (I’d assumed you were citing Quarantine as an example of ignorant parallelism, to which I can only plead guilty, at least at the time I wrote it.) Your article’s packed with analyses of lots of questions that fascinate me, including the computational power of closed timelike curves (I once wrote a story in which someone who isn’t quite Alan Turing encounters a time traveller with an interest in that question).

    I’ll certainly include the BBBV theorem in my critique of Quarantine!

    I collaborated with Dan Christensen on a second paper, with John Baez, concerning the asymptotics of the gadgets Dan and I worked out how to compute efficiently. But from 2002-2006, I got called away from both SF and physics by other things, and I’m only just starting to resuscitate my SF career. It’d be fun to do something more on quantum gravity, but I think Dan and John have both advanced into the deepest wilds of category theory, and it will take a while (if ever) for me to catch up.

  29. Greg Egan Says:

    Pedro Pinheiro wrote:

    What would the time dilation factor be for an observer approaching a singularity? This has been something that I’ve thought about for a while – would it approach stretching time “forever” from the observer’s viewpoint?

    According to General Relativity, anyone who free-falls into a Schwarzschild black hole will experience a finite amount of proper time before they hit the singularity.

  30. Blake Stacey Says:

    My favorite line out of Quarantine was the one which went something like, “I have no idea what the Ensemble is, but it’s the most important thing in my life.” That had a certain resonance for me when I took Prof. Kardar’s stat-physics class. . . . Now, the only question is if Ensemble was canonical or grand canonical.

  31. Pedro Pinheiro Says:

    Greg, thank you for your answer. My question was badly formulated – what I meant is – would the situation I described would compress “forever” of the time of the universe outside the event horizon as it appeared to the observer within? How far into the future would that observer be able to see until the unpleasant end? (supposing a very sturdy solid state observer).

  32. Greg Egan Says:

    Pedro, that’s an interesting question!

    For an observer who free-falls into a black hole, she sees light that falls directly from the zenith to be red-shifted, but as she looks towards the edge of her view, the red-shift changes to a blue shift.

    Where she sees a red shift, she is seeing fewer periods of the light per unit of her proper time than if she stayed outside; where she sees a blue shift, she is seeing more periods of the light.

    The exact formula for the red/blue shift is a bit complicated, but I’ve worked this out in detail here.

    As the observer approaches the singularity at r=0, the blue shift at the edge of her view goes to infinity like 1/r. However, dr/dtau for her (where tau is her proper time), also goes to (minus) infinity, like -1/sqrt[r]. So if I’m analysing this correctly, the number of periods of blue-shifted radiation will be the integral of some constant times 1/sqrt[r], which is a finite integral. So even for the most blue-shifted radiation she sees, she only sees a finite time period of the history of the radiation source.

  33. Carl Says:

    Ah, as a kid, I often wondered if when you fall into a black hole, since to an outside observer it looks like you never cross the event horizon due to time dilation, you could look out your window and watch the universe end before you cross the threshold. On a related note, I suppose if you only get to see a finite amount of what’s happening outside your aft window, there’s also usually no hope for you that the black hole will decay due to Hawking radiation before you hit either (unless obviously, the black hole was just about to decay anyway)?

  34. Greg Egan Says:

    Carl, FWIW I’d be amazed if any object falling towards a black hole would get a reprieve due to Hawking evaporation, unless the hole’s final decay was already imminent. But don’t ask me to quantify “imminent”.

  35. Pedro Pinheiro Says:

    Greg, thank you again for your answer – although the math in your explanation is waaaaay over my head 🙂
    Another corollary to the already decreasing health of my very sturdy solid state observer (besides the gravity effects) would be the blue-shifting of all outside incoming radiation into high-energy gamma rays, correct? Which would make empirical sense as the same amount of energy of a longer period of time reaching the black hole being experienced in a relatively shorter timed frame of reference would need to have the overall energy conserved?

  36. Greg Egan Says:

    Pedro, sure the blue-shifted incoming radiation would be a big problem. Not just for the observer — it would also mean that most black holes (even non-spinning ones) wouldn’t really be the Schwarzschild black holes of this nice mathematical fantasy. I think the current belief among the GR cognoscenti is still that real black holes actually have something called Belinsky-Khalatnikov-Lifshitz geometry, which is horribly complicated and makes calculations near the singularity much more difficult (try Kip Thorne’s book Black Holes and Timewarps for more on that).

    As for the link between conservation of energy, red and blue shift, and the time you see pass in the outside universe, the reasoning goes like this: starting with the Schwarzschild geometry, you can make use of the fact that it has a “time translation symmetry” to get something like a conservation law along the world lines of the incoming photons. (In general, conservation of energy is a more complicated notion in curved spacetime than it is in special relativity or Newtonian physics.) That’s how you derive the red and blue shifts.

    Once you know those shifts, it’s just a simple counting argument to deduce the amount of time you see passing. If a one-hertz wave is emitted from a distant star, and what you observe due to blue shift is a ten-hertz wave, then for every second that passes for you, you must see 10 seconds pass for that star.

  37. Neratin Says:

    Greg, have you read Etesi&Nemeti paper? How realistic is scenario they describe? Can you use BLK black hole, instead of Kerr BH, to do the trick and observe the universe infinite history (let’s forget about quantum gravity, Hawking radiation, etc)?

  38. Pedro Pinheiro Says:

    Greg, I think the saying goes something like “In theory, practice and theory are the same, but in practice, theory and practice are different” 🙂 Thank you for your patience!

  39. Greg Egan Says:

    Neratin, it would probably take me a month to quantify anything about the BKL geometry, but I can offer you two hand-waving intuitive arguments against seeing the entire future of an external radiation source by jumping into a BKL hole.

    In Schwarzschild geometry, you suffer tidal stretching in the direction you’re falling, and (half as much) tidal squeezing in each direction perpendicular to it. The red shift you see aligns with the stretch, the blue shift with the squeeze.

    In BKL geometry, the same arrangement just oscillates: each of the 3 directions takes turns being the stretch while the other two are the squeeze.

    As you integrate down to the singularity, I think you’d just be averaging (and possibly diluting) the same net effect as you’d get from the Schwarzschild geometry. Basically, if you integrate an oscillating function (of constant peak-to-peak size) multiplied by some original function, you’re not going to get an infinite result when the integral of the original function was finite.

    My other argument is this: suppose Bob falls into the hole, and he gets to see every wavefront ever emitted by some radiation source. I think that would mean that the family of null geodesics describing those wavefronts would have to converge on a particular null curve (you could think of this limit curve as “the light from the infinitely far future”, though of course there is no actual light taking this path). That null curve would intersect the singularity at some distinguished event E. But the BKL geometry, although it’s not exactly time-invariant like the Schwarzschild geometry, is always the same on average; it’s just oscillating, it’s not evolving. So there is no “distinguished event E” coming from the geometry.

    Putting this another way, suppose the time for Bob when he observes wavefront N, tau(N), converges, as N goes to infinity, on the time when he hits the singularity. If Alice fell into the hole before Bob, I think it’s inevitable that she’d fail to see the entire history of the radiation. And if Carol fell into the hole after Bob, I think she’d see the entire history completed some finite time before she struck the singularity. But what’s so special about Bob? There’s nothing about his relationship to the black hole or the radiation source that can make him special like this.

    I haven’t read the paper you cited on Malament-Hogarth spacetimes, but it looks like fun. In Scott’s review article, I think the only real objection he raised to Malament-Hogarth spacetimes was quantum effects. FWIW I’m generally sceptical about these kinds of exotic solutions of classical GR (or at least I am when I’m not writing science fiction).

  40. Neratin Says:

    Thank you for your answer, Greg! I am no way General Relativity Guy, I just skimmed through some textbooks, but I think I can see your point…

    BTW, I always thought GR _is_ about exotic solutions (just look at the arXiv statistics), just like science fiction (speaking of which, your ‘Permutation City’ was – at last – published here in Poland four days ago 😉 )!

  41. Philosophia Naturalis #8 « {metadatta} Says:

    […] Penrose is an excellent segway into two posts by Scott Aaronson. The first poses the question: “what’s the connection between a black hole having an event horizon and its having a singularity? In other words, once you’ve clumped enough stuff together that light can’t escape, why have you also clumped enough together to create a singularity?” (This is related to the Penrose-Hawking theorems of general relativity). The second (or rather, the subsequent comments) deals with possible connections between the brain and quantum computers, something Roger Penrose has discussed in a good deal of depth. (Matt Leifer has a similar post, asking the question: “if quantum computers are more efficient than classical ones then why didn’t our brains evolve to take advantage of quantum information processing?“) […]