Sunday, May 19, 2019

10 things you should know about black holes

When I first learned about black holes, I was scared that one would fly through our solar system and eat us up. That was 30 years ago. I'm not afraid of black holes anymore but I am afraid that they have been misunderstood. So here are 10 things that you should know about black holes.

1. What is a black hole?

A black hole contains a region from which nothing ever can escape, because, to escape, you would have to move faster than the speed of light, which you can’t.  The boundary of the region from which you cannot escape is called the “horizon.” In the simplest case, the horizon has the form of a sphere. Its radius is known as the Schwarzschild radius, named after Karl Schwarzschild who first derived black holes as a solution to Einstein’s General Relativity.

2. How large are black holes?

The diameter of a black hole is directly proportional to the mass of the black hole. So the more mass falls into the black hole, the larger the black hole becomes. Compared to other stellar objects though, black holes are tiny because enormous gravitational pressure has compressed their mass into a very small volume. For example, the radius of a black hole with the approximate mass of planet Earth is only a few millimeters.

3. What happens at the horizon?

A black hole horizon does not have substance. Therefore, someone crossing the black hole horizon does not notice anything weird going on in their immediate surroundings. This follows from Einstein’s equivalence principle, which implies that in your immediate surrounding you cannot tell the difference between acceleration in flat space and curved space that gives rise to gravity.

However, an observer far away from a black hole who watches somebody fall in would notice that the infalling person seems to move slower and slower the closer they get to the horizon. It appears this way because time close by the black hole horizon runs much slower than far away from the horizon. 

That’s one of these odd consequences of the relativity of time that Einstein discovered. So, if you fall into a black hole, it only takes a finite amount of time to cross the horizon, but from the outside it looks like it take forever.

What you would experience at the horizon depends on the tidal force of the gravitational field. The tidal forces is loosely speaking the change of the gravitational force. It’s not the gravitational force itself, it’s the difference between the gravitational forces at two nearby places, say at your head and at your feet.

The tidal force at the horizon is inversely proportional to the square of the mass of the black hole. This means the larger and more massive the black hole, the smaller the tidal force at the horizon. Yes, you heard that right. The larger the black hole, the smaller the tidal force at the horizon.

Therefore, if the black hole is only massive enough, you can cross the horizon without noticing what just happened. And once you have crossed the horizon, there is no turning back. The stretching from the tidal force will become increasingly unpleasant as you approach the center of the black hole, and eventually rip everything apart.

In the early days of General Relativity many physicists believed that there is a singularity at the horizon, but this turned out to be a mathematical mistake.

4. What is inside a black hole?

Nobody really knows. General relativity predicts that inside the black hole is a singularity, that’s a place where the tidal forces become infinitely large. But we know that General Relativity does not work nearby the singularity because there, the quantum fluctuations of space and time become large. To be able to tell what is inside a black hole we would need a theory of quantum gravity – and we don’t have one. Most physicists believe that such a theory, if we had it, would replace the singularity with something else.

5. How do black holes form?

We presently know of four different ways that black holes may form. The best understood one is stellar collapse. A sufficiently large star will form a black hole after its nuclear fusion runs dry, which happens when the star has fused everything that could be fused. Now, when the pressure generated by the fusion stops, the matter starts falling towards its own gravitational center, and thereby it becomes increasingly dense. Eventually the matter is so dense that nothing can overcome the gravitational pull on the stars’ surface: That’s when a black hole has been created. These black holes are called ‘solar mass black holes’ and they are the most common ones.

The next common type of black holes are ‘supermassive black holes’ that can be found in the centers of many galaxies. Supermassive black holes have masses about a billion times that of solar mass black holes, and sometimes even more. Exactly how they form still is not entirely clear. Many astrophysicists think that supermassive black holes start out as solar mass black holes, and, because they sit in a densely populated galactic center, they swallow a lot of other stars and grow. However, it seems that the black holes grow faster than this simple idea suggests, and exactly how they manage this is not well understood.

A more controversial idea are primordial black holes. These are black holes that might have formed in the early universe by large density fluctuations in the plasma. So, they would have been there all along. Primordial black holes can in principle have any mass. While this is possible, it is difficult to find a model that produces primordial black holes without producing too many of them, which is in conflict with observation.

Finally, there is the very speculative idea that tiny black holes could form in particle colliders. This can only happen if our universe has additional dimensions of space. And so far, there has not been any observational evidence that this might be the case.

6. How do we know black holes exist?

We have a lot of observational evidence that speaks for very compact objects with large masses that do not emit light. These objects reveal themselves by their gravitational pull. They do this for example by influencing the motion of other stars or gas clouds around them, which we have observed.

We furthermore know that these objects do not have a surface. We know this because matter falling onto an object with a surface would cause more emission of particles than matter falling through a horizon and then just vanishing.

And since most recently, we have the observation from the “Event Horizon Telescope” which is an image of the black hole shadow. This is basically an extreme gravitational lensing event. All these observations are compatible with the explanation that they are caused by black holes, and no similarly good alternative explanation exists.

7. Why did Hawking once say that black holes don’t exist?

Hawking was using a very strict mathematical definition of black holes, and one that is rather uncommon among physicists. 

If the inside of the black hole horizon remains disconnected forever, we speak of an “event horizon”. If the inside is only disconnected temporarily, we speak of an “apparent horizon”. But since an apparent horizon could be present for a very long time, like, billions of billions of years, the two types of horizons cannot be told apart by observation. Therefore, physicists normally refer to both cases as “black holes.” The more mathematically-minded people, however, count only the first case, with an eternal event horizon, as black hole.

What Hawking meant is that black holes may not have an eternal event horizon but only a temporary apparent horizon. This is not a controversial position to hold, and one that is shared by many people in the field, including me. For all practical purposes though, the distinction Hawking drew is irrelevant.

8. How can black holes emit radiation?

Black hole can emit radiation because the dynamical space-time of the collapsing black hole changes the notion of what a particle is. This is another example of the “relativity” in Einstein’s theory. Just like time passes differently for different observers, depending on where they are and how they move, the notion of particles too depends on the observer, on where they are and how they move. 

Because of this, an observer who falls into a black hole thinks he is falling in vacuum, but an observer far away from the black hole thinks that it’s not vacuum but full of particles. And where do the particles come from? They come from the black hole.

This radiation that black holes emit is called “Hawking radiation” because Hawking was the first to derived that this should happen. This radiation has a temperature which is inversely proportional to the black hole’s mass: So, the smaller the black hole the hotter. For the stellar and supermassive black holes that we know of, the temperature is well below that of the Cosmic microwave background and cannot be observed.

9. What is the information loss paradox?

The information loss paradox is caused by the emission of Hawking radiation. This happens because the Hawking radiation is purely thermal which means it is random except for having a specific temperature. In particular, the radiation does not contain any information about what formed the black hole. 

But while the black hole emits radiation, it loses mass and shrinks. So, eventually, the black hole will be entirely converted into random radiation and the remaining radiation depends only on the mass of the black hole. It does not at all depend on the details of the matter that formed it, or whatever fell in later. Therefore, if one only knows the final state of the evaporation, one cannot tell what formed the black hole. 

Such a process is called “irreversible” — and the trouble is that there are no such processes in quantum mechanics. Black hole evaporation is therefore inconsistent with quantum theory as we know it and something has to give. Somehow this inconsistency has to be removed. Most physicists believe that the solution is that the Hawking radiation somehow must contain information after all.

10. So, will a black hole come and eat us up?

It’s not impossible, but very unlikely. 

Most stellar objects in galaxies orbit around the galactic center because of the way that galaxies form. It happens on occasion that two solar systems collide and a star or planet or black hole, is kicked onto a strange orbit, leaves one solar system and travels around until it gets caught up in the gravitational field of some other system. 

But the stellar objects in galaxies are generally far apart from each other, and we sit in an outer arm of a spiral galaxy where there isn’t all that much going on. So, it’s exceedingly improbable that a black hole would come by on just exactly the right curve to cause us trouble. We would also know of this long in advance because we would see the gravitational pull of the black hole acting on the outer planets.

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  1. "Compared to other stellar objects though, black holes are tiny because enormous gravitational pressure has compressed their mass into a very small volume"

    It could mentioned be that very large holes have the density of air.

    Also, the "black holes suck" misconception.

    1. Transparency,

      I find it misleading to speak of a "density" of black holes because that suggests that the mass is distributed in the inside.

    2. I think a layperson (your target audience?) would read that as "BH are tiny compared to stars". (I could not find a pertinent definition of "stellar object" - saying "other" seems to imply it includes BH.)

    3. Black holes are tiny compared to stars of the same mass.

  2. Since the radius is proportional to the mass, a black hole's mass should grow according to dm/dt = k m^2, provided the incident flux of matter is constant. Under those conditions a black hole would become infinitely large in a finite time. In reality, a black hole will grow until it has eaten up all the food available. So, doesn't this mean that the Earth, Sun or the future white dwarf the Sun will turn into will not escape being eaten by a black hole?

    1. Count,

      What you are saying is that a black hole surrounded by motionless dust would eventually eat up all that dust. That's correct, but not the situation we find ourselves in. Besides this, the same would be the case for any massive body.

      The gravitational pull of black holes doesn't work any different from the gravitational pull of other bodies. If you put something in an orbit around a black hole, it'll stay there, in principle forever. If something is moving away from a black hole faster than the necessary escape velocity, it'll not fall in. It's just that if you want to escape from the horizon itself, then the escape velocity (starting at this point) is faster than the speed of light.

      What will happen to celestial bodies in the long run depends on whether the cosmological constant is indeed constant or increases or decreases. So really one can't say much about it.

  3. 1. I've always disliked this explanation "to escape, you would have to move faster than the speed of light". Of course, a proper explanation would use the appropriate mathematical formalism, but I prefer "nothing can escape, because all directions point toward the interior". Is this correct? If it is, it seems preferable to me as it conveys the idea of space and time being strongly curved. With the "faster than light" explanation, it's like space isn't curved at all, the black hole just has a strong gravitational pull.

    One can show how this happens qualitatively with an umbrella (helped with strong wind). Initially, rain falls down on the umbrella and then on the ground. But if you turn the umbrella upside-down, and it becomes flat and then curved again but in the opposite direction, rain accumulates in the reservoir that has been created: it can't reach the ground anymore. No speed involved.

    3. "Therefore, someone crossing the black hole horizon does not notice anything weird going on in their immediate surroundings". This is the strict interpretation of Einstein's equivalence principle, but isn't there something that does go weird at this point, and can, in principle, indicate that one just crossed the horizon of a black hole? I'm not completely sure of this, but it seems to me that as one falls down, all light they see appears coming from behind them, with lower and lower energy, in a cone that becomes narrower and narrower. Right before they cross the horizon, this cone becomes a single ray of light coming from one direction. Then they can say "I must falling toward a direction opposite to that ray of light". And as they cross the horizon everything becomes totally black.

    1. dlb,

      There are "directions" in space-time that do not point towards the singularity. It's just that these "directions" are not directions you can take. You cannot take them because you cannot move faster than the speed of light.

      In other words, what you say is correct, but the underlying reason that you can't escape is the speed of light limit.

    2. dlb,

      There is a big difference between what someone freely falling into a black hole sees and what someone who is trying to hover sees. In the former case nothing unusual is detected at the horizon. But imagine orbiting far outside the horizon (at least 6 times the "radius", quotes because the visible radius is significantly larger, owing to the gravitational lensing effect), gently lowering a tethered camera toward the horizon and watching the feed.

      What you will see is like being at the center of a black bowl slowly getting deeper and deeper as the camera goes down, and eventually becoming a dark sphere with a bright open aperture right above you, getting smaller and smaller as the camera gets closer and closer to the horizon. Sort of like you describe. You won't actually see that "everything becomes totally black," because of the time dilation effect, and because the tether will break at some point, as the force required to keep something from falling through the horizon grows without bound.

      So, the only way to detect that you are near the horizon is to try to hover motionless, relative to far away beacons, lower and lower, and notice that it takes more and more force (or thrust) to do so.

      Unrelated, but worth mentioning: even though the tether at the lower end would experience forces beyond breaking point, the observer holding it would not be "sucked in", and would only need a limited effort to hold the tether. The apparent weight of the camera when it is close to the horizon, divided by the camera's mass, is known as the "surface gravity" of the black hole.

  4. (Continued)

    4. I think there is a flaw of logic in the generally accepted conclusion, that makes me believe there is indeed a singularity in the black hole. Bear with me, I'm no scientist, but here is the reasoning.

    According to General Relativity (GR) the interior of the black hole is causally disconnected from the exterior. There is nothing we can learn about the inside, even in principle. Now in any theory that replaces GR to include the quantum world that would still be true, for solar-mass black holes, because they are large enough to not impact this conclusion. At least until they evaporate but then all bets are off and you need this theory to draw any conclusion. So, since no experiment or observation of any kind can tell whether I'm right or wrong about the interior of the black hole, a statement about it is more a philosophical stance than a scientific one: it depends how well-motivated is your reasoning, and nobody can prove you wrong, they can only disagree with you.

    With this said, what do we have that can help us make a statement about the interior? We have GR, and its equations that show there is a singularity if you follow them to the end. In support to the fact that there is no singularity, we have, well, nothing. We don't know of a single theory that would help us make a claim, however tiny, about that question. None. There are several candidates, say for example String Theory, but none of them has the slightest beginning of a test that could help us decide if the theory is right, or is just an intellectual construction that has nothing to do with reality.

    Now for the flaw in the logic: you can say that equations from GR should not be followed to the end, that the theory breaks down at this scale. And I agree. But you cannot say that means there is no singularity. You can only say it if there is a small indication, however tiny, of the opposite. And there isn't, GR is our only guide, everything else is belief. Belief that a quantum theory of gravitation will ultimately show there is no singularity, when in fact we don't know.
    So my claim is that until we have the beginning of a theory, that has the beginning of an experimental or observational support, we should accept that GR tells us there is a singularity.

    You may say it's not "real", but then by that logic nothing inside a black hole is "real". And I bet a lot of scientists accept the rest of the description of the interior of a black hole, just not the singularity, because they dislike it. If you follow the logic of the theory that tells you what's inside the black hole, the singularity is as real as everything else.

    1. dlb,

      As Sabine said, we know that quantum effects must manifest strongly enough when the spacetime curvature gets very high. This is unrelated to it being inside or outside a horizon. For tiny enough black holes these effects, whatever they might be would be visible outside the horizon, assuming it could hold together long enough without evaporating. You can certainly believe anything you want, but the current best theories predict that classical General Relativity will break down at that point, so its prediction of a curvature singularity cannot be trusted.

  5. As regards (9) and the claim that there are no irreversible processes in QM: there are actually irreversible processes in quantum mechanics. They are known as measurements, and without them we wouldn't be able to use quantum mechanics. But a lot of physicists prefer to pretend that they are not needed.

    1. Sofie,

      That's right, the way that we presently understand measurement in quantum mechanics it's irreversible. Black hole evaporation introduces an irreversibility into the time-evolution prior to collapse, which is what causes the problem.

      There is a long debate in the literature about whether or not this new type of irreversibility can be made compatible with quantum mechanics or not. But in that case you have to change something about quantum mechanics. As I said, something has to give.

      I apologize for being sloppy on that point, but it would have taken too much time to go into this in the video.

    2. Quantum mechanics is about the evolution of complex valued amplitudes who's modulus squared gives probabilities for the state. The Born rule associates this with eigenvalues of operators. However, none of this tells us how a particular outcome of a measurement occurs. In that sense we might say the problem of measurement is outside of quantum mechanics. This irreversible aspect to what we observe phenomenologically may be a manifestation of our perceptions and understanding of the world, rather than as something intrinsic to the world.

  6. The problem with Hawking radiation has been that Tr(ρ) ≠ Tr(ρ^2), which for Tr(ρ) = 1 would mean that pure states evolve into mixed states. Here ρ = |ψ)(ψ| (for bra-ket notion) is the density matrix. This is not allowed in QM. Hence this is why Hawking radiation is said to not contain any information pertaining to the quantum states that composed the black hole. This is a main aspect of the quantum information problem.

    As I see it spacetime in the setting that Raamsdonk et al have established spacetime as an emergent phenomenon from large N entanglement. This is related in ways to the theory of laser coherent states. These states have classical-like properties of being in a subspace of the Hilbert space of states, but that has symplectic structure. These coherent states also exist in projective spaces, and for 3 dimensions this is CP^3 that is the projective twistor space. The distance between two spin coherent states, states that define point in CP^n or rays in C^{n+1}, are then equal to Riemannian distance between two points on this orbit space. The coherent states in CP^n define a minima in the Hilbert space, where this minima has classical-like properties. With the connection to spacetime this means that quantum gravitation coherent states are classical-like and that quantum gravitation is a theory involving deviations from these coherent states. This means the IR theory of quantum gravitation is a form of protected quantum gravity that Weinberg proposed.

    Quantum information is then carried off not by Hawking radiation, but in gravitons. The emission of Hawking radiation adjusts the size of a black hole as m → m - δm and this is modeled as metric back reaction. This is a semi-classical fix introduced “by hand.” I think however the quantum information contained in the black hole is emitted to I^{+∞} in gravitons as a set of charges or BMS symmetries.

    Apparent horizons are sort of odd. If you fall into a black hole the horizon does not suddenly disappear upon crossing it. It becomes an apparent horizon. The strange thing is the exact moment this transition happens can never be ascertained, for that would require a time piece able to measure Planck units of time and would be an appreciable mass contribution to the black hole. That perturbation would by itself mess up any attempt to determine when an invariant horizon transitioned into a more frame dependent apparent horizon. The existence of Hawking radiation means there is a quantum ambiguity in knowing if an event horizon is “true” or “apparent.”

    To be honest I would be thrilled if a black hole was found to be on track to come within say .1 light years of the solar system. We could send a probe out there to do all sorts of cool measurements. The only plausible problem is this might increase the likely hood of Oort cloud objects getting their orbits perturbed into the inner solar system.

    Crazy idea time: The thought occurred to me that maybe we could get a Bose-Einstein condensates to form black holes. A BEC is in effect a single wave function, so if we were to send one through an accelerator to 10TeV/nucleon, then a scattering of a BEC with 10^{15} atoms might then be where two wave functions at or beyond the Planck energy form a quantum black hole.

    Ok, yeah I can hear the howls of protest. Getting BEC with laser traps is tough as it is and how are we going to get them in a collider and …. . But the idea seems workable, and maybe with diligent work who knows?

  7. when the very last black hole emits the last bit of Hawking radiation, the universe will be very, very large and very, very cold ... and there will be no more matter ... right? can you comment on Roger Penrose's Conformal Cyclical Cosmology? If I understand his postulation correctly, at this stage the universe no longer has scale or length or time and "forgets" that it is big, essentially leading to another Big Bang. Is there any real physics or math that would explain or predict this outcome? Thanks!!

  8. How do physicists formally define "information" in the context of the information loss paradox?

  9. Jeff,

    You do not need to define "information" to come to the conclusion there is a problem. As you may have noticed, the explanation I have given does not rely on any notion of information. The problem is simply that you end up having an irreversible process where you expect a reversible one.

  10. Hi Sabine,
    What about the controversy on the 'firewall' issue? May I ask what your take is on that?

    1. John,

      The whole firewall issue is a based on a mathematical mistake. I demonstrated this explicitly here, though I pointed this out two years earlier here. The supposed proof in the paper is just technically wrong. And it's not a particularly deep mistake.

      To sum it up quickly, the firewall claim comes from showing that four assumptions are inconsistent with each other. But these assumptions are not inconsistent. What makes them appear inconsistent is a 5th assumption which you find hidden in the appendix of the paper. Drop this assumption and the contradiction vanishes. (The conclusion is then that the requirement of consistency of the first four assumptions implies that the 5th assumption is wrong. There is no firewall in that case.)

      I don't mention this in my video because the vast majority of physicists would use it to claim that I'm a nutcase. Certainly if there have been so many papers written about it (books been written, prizes handed out) there must be something to it.

      So I don't talk about it. But since you ask, that's the situation.

    2. Interesting, and surprising: from all the activity on the subject, I thought there was a genuine physical problem.

      Thanks for your explanation, and for the links to your articles (although they are a bit over my head, I am afraid).

    3. Sabine,

      "That the early radiation is entangled with the late radiation seems a natural assumption to make."
      No sign of being nuts when one is suspicious about words or sentences that contain "natural".

      (If one further drops the assumption of an exclusively unitary evolution then the black hole info loss problem does not get off the ground to begin with - all these are self-inflicted problems based on wrong assumptions, but better not talk about it ;-)

    4. I thought your paper on this was rather clever. You might not have taken it far enough. I think your disentanglement of Hawking radiation and the BH is really an entanglement swap, where Hawking radiation instead of being entangled with the BH is entangled with IR gravitons that escape to "infinity." This is a feeder for what I wrote above.

    5. It's an unjustified assumption, but I couldn't write that, so I wrote it's natural, which, for what I am concerned, means pretty much the same ;)

    6. Lawrence,

      It is an entanglement swap, yes. I do not know a relation to gravitons. I know that 't Hooft thinks that is so but it doesn't make a terrible lot of sense to me. And in any case, to tell you the truth, I have lost interest in this because it's all untestable anyway, so who really cares.

    7. t'Hooft is right, but I am not sure about how he goes about it.

      If there is this entanglement swap with gravitons it means there is quantum hair or fuzz on stretched horizons. A related physics would then result in signals in black hole coalescence associated with graviton production. This might be observable.

    8. Sabine: If the early radiation is not entangled with the late radiation, then not as much information can escape from the black hole as we think can escape, because we expect some information to be encoded in the entanglement between the early Hawking radiation and the late Hawking radiation.

      So unless the Beckenstein entropy formula is wrong, and black holes can store less information than we think, we now have the information loss paradox all over again, just coming from a different angle.

    9. Peter,

      If the early radiation is not entangled with the late radiation then you cannot encode information in this entanglement between the early and late radiation. That you (and other people) expect that to be the case does not mean it is necessary.

    10. We know exactly how much Hawking radiation a black hole emits. So you can compute how much information is emitted, assuming the early radiation and the late radiation are and are not entangled. Assuming there is entanglement, then the amount of information emitted is (up to a constant) is the Beckenstein bound - and I assume it gets the right constant; somebody should do the calculation. If the radiation is not entangled across large spans of time, then the amount of classical information that comes out doesn't change, but the amount of quantum information emitted is LESS than the Beckenstein bound, and you have something you have to explain.

    11. Peter,

      Entanglement between modes that are localized at different times is not the only way to encode information in a state. I am sure you know that, so I don't understand why we are having this conversation.

      Also, with all due respect, it is pretty clear you didn't read my paper because I stress explicitly in the paper that I am *not* talking about the information loss problem but about the firewall problem. Whether and how the information loss problem is solved is a different question entirely.

      Look, it is not a complicated argument that I am making, it is just drowning in a lot of math. The main point is really that you need to enforce stress-energy conservation at the horizon and not any "typical" state will allow that.

      I am quite frustrated about the whole story, because first no one wanted to listen to me because supposedly I didn't have enough calculations. Then I added all the calculations (though, frankly, they're obvious and you don't need them), then they complained that was too much math and I'd need a simple example. Then I added the simple example. Then they gave up and just outright ignored me and that's been the case since.

      You find the simple example in section 4.1. If you find anything wrong with that argument, please let me know. Srsly, I'd feel much better if I knew it's wrong, so I could stop bitching about people ignoring me, which makes me feel like a crank. But the status is, so far no one has found anything wrong with my paper, which makes me conclude that I have demonstrated the whole firewall "problem" is based on a mistake and no one cares.

    12. If you solve the firewall problem, but it's clear your solution leads to information loss, I don't see what you've gained. If you assume that there is information loss, there's no firewall problem in the first place.

    13. To avoid the problem Peter mentions with less quantum information than the Bekenstein bound is to swap the entanglement with gravitons. The gravitons are a quantum response change in the holographic screen or stretched horizon. Currently we just do a classical back reaction. This information then makes its way to I^{+∞}.

    14. To expand on my previous comment, is black holes destroy information, and the Hawking radiation is completely random, you don't need firewalls. So I don't see what the point of solving the firewall problem is if your solution implies that black holes destroy information anyway.

    15. Peter,

      I am merely pointing out that these are two separate problems. I did not assume that there is information loss, of course. If the Hawking radiation is not entangled across the horizon, you can trace out the inside all you want, it will not result in any information loss. I am merely saying that this doesn't explain how information from an infalling particle gets into the Hawking radiation.

    16. Peter,

      Regarding your second comment.

      The reason Hawking radiation causes a problem is not that the spectrum is thermal ("random") but that the particles are pairwise entangled across the horizon. What I am doing is simply swapping the entanglement from across the horizon (in-out) to in-in and out-out. The resulting radiation is *still* entangled, but it's not what AMPS called a "typical" state. The out-state is separately pure. There is no information loss if you trace out the inside qua construction.

  11. "The diameter of a black hole is directly proportional to the mass of the black hole".
    I would have thought that the mass scales as the volume, so the diameter is proportional to the cubic root of the mass. Am i missing smth?

    1. Opamanfred,

      Yes, you are missing something. The mass of a black hole does not scale with the volume. It scales with the diameter.

  12. Regarding (4), "What is inside of a black hole?", Nobel Laureate Gerard t'Hooft has over the past couple of years posted on arXiv an intriguing series of papers on the decades-old "antipodal" interpretation of where matter exits after entering a black hole.

    Oversimplified, antipodal just means that after an intriguing and typically very slow journey, such particles wind up on the far side of the same black hole.

    While not fully answering what is inside a black hole, t'Hooft provides a persuasive argument that for sufficiently small black holes, infall and Hawking radiation become time symmetric versions of each other. Nice! (I'd keep wanting to say "beautiful", but that word may be a bit of a no-no here... :)

    The papers are accessible to a broad audience. I recommend them highly to anyone interested in what happens in the long term to matter that falls into a black hole. Search for t'Hooft on arXiv to find them.

  13. "So, if you fall into a black hole, it only takes a finite amount of time to cross the horizon, but from the outside it looks like it take forever."

    How is that compatible with the observation of, say, supermassive black holes at the center of galaxies? We are told that these black holes got so huge by eating up the surrounding matter. So, from our point of view, the "digestion" of that matter has already been completed (because we are observing HUGE black holes). But according to the quoted sentence from your post, we should still be observing the surrounding matter falling forever into the black hole.

    One could ask the same question about the recently observed black hole mergers: why doesn't it take forever (from our point of view) for the two black holes to merge?

    1. Pascal,

      Stuff that gets close to the horizon becomes dim and frequencies are shifted into the red, so you can't see it anymore. This basically means fapp it looks the same as if something fell in and crossed the horizon.

      Black holes merging is a different story altogether because this changes the causal structure of space-time. There's no reason to expect this taking forever. Keep in mind that the statement that it takes an infinite amount of time for something to cross the horizon depends on a particular choice of time-coordinates. These coordinates look very different for a black hole merger.

    2. Pascal,

      "Is there going to be one last (very red) photon emitted by the infalling body?"

      This is a question for which you would have to know the future into all eternity, so I consider it unanswerable.

  14. You say that for all practical purposes, apparent horizons can be treated the same as true event horizons. But is this really true when we consider such foundational questions as the information paradox? I would appreciate some insight on the subtle distinctions here.

    I suppose apparent horizons also emit Hawking radiation, given that they look the same over our timescale of billions of years. But is that radiation truly thermal? Or do these proofs (such as the proof that Tr(rho) != Tr(rho^2)) assume that the horizon is apparent?

    For example, I could imagine a scenario where the Hawking radiation from an apparent horizon _looks_ thermal over normal timescales, but if we observe it over the entire lifespan of the black hole, the distinction between true and apparent horizons affects the radiation somehow. The information about the ingoing matter would be encoded in this difference.

    1. Audun,

      The horizon really has nothing to do with the information loss paradox. The horizon is merely the location where information gets lost for all practical purposes. What causes the ultimate loss is the singularity. If you remove the singularity, there is no problem. Yes, that the horizon is only apparent is one of the possible ways the scenario might pan out. Then again, this might not be the real answer. Who knows? No one knows.

    2. So can actual singularities form from a smooth spacetime within a finite time in a far-away reference frame? Even though an apparent horizon never develops?

    3. If you are asking whether naked singularities exists in nature, the answer is no one really knows.

    4. The really interesting black hole is the Kerr black hole, or similarly the Reissner-Nordstrom or Kerr-Newman black hole. The Kerr black hole is rotating and it has an outer region called the ergosphere, where the dragging of space with its rotation can't be resisted, or else you are must move faster than light and that is not allowed. Then there is the event horizon that is a boundary to a spacelike region inside that defines the radial direction as time and you are pulled this way inexorably. There is another event horizon called the inner horizon, which has some possible unpleasant features as you cross it. This leads into another timelike region that is most peculiar.

      This region permits closed timelike curves that orbit a ring singularity. There is an inner ergosphere here that contains the ring singularity. This inner region contains closed timelike geodesics and a subset is the inner ergosphere that is a region where geodesics are necessarily closed time loops. These closed geodesic have radii smaller than the ring singularity, which is a repelling region. In fact nothing actually reaches it, which gives some question on what is meant by it. This singularity is a pure timelike curve or ring with no spatial extent and with divergent curvature.

      The odd thing about this interior timelike region is that with closed timelike curves there is no proper Cauchy data one can specify on a spatial surface. A closed timelike curve means a timelike normal to a spatial surface, which is tangent to a timelike curve, loops around and this results in a degeneracy or mass-redundancy in possible spacelike Cauchy data for the foliation of spatial surfaces that define the spacetime. The means there is something similar to a residue evaluated by a loop integral around a pole. This is a monodromy which defines a type of quantum phase. This can have physical implications for the exterior world, for a spatial surface here can be equally one that extends outside this interior timelike region or one that is inside. This has some quantum phase or entanglement property associated with topology.

      A space or manifold can be partitioned into its orbit spaces. This interior region is one where if spacial surfaces have no unique Cauchy data, they then do not define curvatures according to connection terms. This is something in gauge theory that can happen with topological monopoles. The orbit space of spacetime then is partitioned, or stratified, into different classes, and the interior of a black hole is a class distinct from what we normally think of with spacetime.

      There is then the cosmic censorship conjecture which says these singularities can't exist in ordinary spacetime outside a black hole. These types of spacetimes are related to odd things such as wormholes. Now if a wormhole did exist it could be made into a time machine, where one could then throw a quantum state into it and at a previous time you received a copy of the quantum state you will throw. This duplicates quantum states and is a violation of the “no-cloning” theorem of quantum mechanics. So it seems unlikely the craziness in black hole interiors or with wormholes are really possible in the world outside of being wrapped in event horizons.

    5. Thank you, this is indeed crazy. We need the courage to go beyond mathematical formalism and dare to imagine a meaning to this. My intuition is that our existence and perception of space time is closely related to these loops. However, in order to make sense of it, we would need to create something like an emotional geometry, the ability to link our emotions and consciousness with the insights maths provide us.

    6. I am not sure about the idea of emotions or our psychological basis. The interior region with this ring singularity may be accessible to the intrepid soul who ventures in. You have to cross the inner horizon that has a set of null rays piled up to it from the outside. This is a Cauchy horizon that sometimes is called a mass inflation singularity. Some controversy exists over this. Some indications are that if you follow certain geodesic paths this pulse of radiation can be Doppler shifted sufficiently into the IR so as not to be too bad. The rule is mostly not to accelerate or fight the gravity flow inwards.

      If you get into the interior then so long as you avoid the inner ergosphere you can venture away. This region is a sort of different universe, and there is a boundary or I^{+∞} that once crossed can lead into another universe similar to this one. We may be reading tea leaves on that, or it might just mean this boundary is inaccessible

      A paper last year by Luk and Defermos goes into some of this. Warning, it is mathematically pretty dense.

    7. Thank you very much for the references as well as for the travel recommendations ;) I'll do my best when i'll get there.
      My reference to a "emotional geometry" in my previous post was rather poor, without giving a context. The thing that puzzles me the most is that any of this would make sense without us, making sense of it. When we talk about black holes, we go to the outer limits of our representation; perhaps the event horizon represents our own limits, today. It is perhaps obvious that the idea itself of entropy or information exists only from our perspective and the values we assign them. I am a big fan of black holes precisely because of this: I think they tell something about us as humans, being the place within us where "the most intrepid souls can venture". In a certain way, black holes exist in the way we represent them because we have the cognitive ability to do so ( it might seem obvious and tautologic, but it's also an important "feature" of the observable universe). Now I switch from science to a personal belief or intuition which is: what we learn about black holes, we lear about ourselves; the inner horizon you describe might not (only) be thousands light-years away, but perhaps finds itself in our own mental states, seen as non-local features. When you describe the Kerr black hole, I can almost feel it! That's what I meant by "emotional geometry"; I know that René Thom worked on such topological features.

  15. "Stuff that gets close to the horizon becomes dim and frequencies are shifted into the red, so you can't see it anymore. This basically means fapp it looks the same as if something fell in and crossed the horizon."

    Is there going to be one last (very red) photon emitted by the infalling body? If there is, the time of emission of that last photon would give the time when (from our point of view) the body has crossed the horizon.

  16. If it appears, to a distant observer, that it takes someone forever to fall into a black hole, then I'm guessing that for symmetry reasons an observer falling into the black hole would see the surrounding universe age infinitely long. Is that consistent with the black hole evaporating in finite time?

    1. Fulis,

      No one knows how long it takes a black hole to evaporate. How much an infalling observer gets to see of the history of the universe depends on exactly how they fall in, ie how long they manage to avoid hitting the singularity.

    2. In a funny sense an infalling mass or system of quantum harmonic oscillators is observed to be quantum evaporated as it crosses the horizon. The quantum modes are red shifted on the stretched horizon so the external observer can only witness Planck scale modes red shifted. These are Hawking modes, and as one observes the BH evaporate you in effect measure the demolition of this system just before crossing the horizon. Of course if this is an infalling observer nothing of the sort is observed, but the demolition happens internal to the BH with a singularity.

      There is no meaning to the locality of quantum events. The concept of there being a quantum amplitude "here" or "there" is not absolute.

    3. "for symmetry reasons an observer falling into the black hole would see the surrounding universe age infinitely long."

      Shouldn't that be "infinitely short"?

    4. In fact that does not happen. Light rays that leave a region close to the event horizon along ongoing null geodesics peal off from the horizon and are delayed, while null geodesics entering the BH enter directly. The second of the figures here illustrate Eddington-Finkelstein coordinates.

      The event horizon is the vertical red line.

      If you decided to put the brakes on and remain in a highly accelerated frame close to the black hole then you would observe this accelerated speed up of the exterior universe. In fact if you are close enough, and the limiting acceleration is about 10^{52}m/s^2 at a Planck limit the black hole would cease to exist in a very brief period as its Hawking radiation would come at you in a torrent or burst.

  17. What's this talk about 'information' - incorrectly taken as 'reversibility' in physics? And regardless, isn't 'reversibility' gone forever in Quantum Mechanics? Just take Young's experiment with one particle at a time: if you don't know where the particle ends up, you don't know where it came from - end of the 'reversibility' story. I do understand the claim that statistically the wave function distribution is deterministic, but to go from there to claims that "determinism" is still relevant in the QM universe, is pure philosophical desperation. Let's do the experiment: a photon is detected, can you or anyone else tell where it originated? If not, let's quit talking about "information" and "reversibility".

    1. Nonlin,

      No one says that "information is taken as reversibility in physics". That claim makes absolutely no sense.

    2. Hmm... Maybe not exactly equate them with one another, but you do link information to reversibility in item 9. and your reply to Jeff @11:58 PM, May 19, 2019.

      Also see

      "Calculations suggest that physical information could permanently disappear in a black hole, allowing many physical states to devolve into the same state. This is controversial because it violates a core precept of modern physics—that in principle the value of a wave function of a physical system at one point in time should determine its value at any other time."

      Why not address the main point of the comment, namely that "determinism" / "reversibility" / "information" are dead in physics post QM? Thanks.

  18. If we don't have a theory of quantum gravity yet, we don't have a serious theory of black holes, therefore we do not know what black holes actually are, or if they even exist at all.

    1. Albert Zotkin,

      This is wrong. A black hole's defining property is the event horizon. The effects of quantum gravity at the event horizon are negligibly small for black holes of astrophysical masses. This is a very basic estimate. Besides this, as I explain in the video we have experimental evidence for the existence of black holes, and no alternative explanation for the observations.

  19. AFAI (try to) understand General Relativity, an object falling towards a black hole will - for a distant observer! - never reach or even pass the event horizon. If this is true, how can we - distant observers that we are -external point of view, nothing in eternity has ever reached or passed the event hoizon of a black hole. Right or wrong - and please: why? :-|

    1. Auch der noch,

      Question doesn't parse, sorry, the grammar isn't working. I speak German, if that is easier.

  20. Sabine,
    This is a very nice summary. My only potential quibble is with the phrase:

    "So, eventually, the black hole will be entirely converted into random radiation"

    At present I do not find convincing the arguments that say that black hole remnants are impossible. Perhaps the endpoint is some form of a remnant plus radiation. To answer this would seem to need a full UV theory of quantum gravity, but it is assumed to be true in discussions of BH information. If you have comments to the contrary, I would be interested.

    (PS This comment will probably label me as "Jack" but you know me as John Donoghue.)

    1. Jack/John,

      You are right of course. Personally I think quasi-stable remnants are the most plausible solution. But really what I was trying to get across in this part of the video was merely that there is a problem which requires a solution.

  21. Time slows for an object that approaches an event horizon. For example, a radioactive isotope approaching an event horizon will decay at the “normal” rate relative to an observer at the event horizon, but for an observer at the far field, the decay rate will increase as the isotope move toward the horizon and viewed from the far field.

    Even the U238 isotope at the event horizon would stabilize almost instantaneously to an observer far from the event horizon without regard for its 4.5 billion year half-life.

    Time operates at a normal rate for an object near the event horizon but increases asymptotically toward infinity for an observer far outside the event horizon.

    For Hawking radiation emissions from near the event horizon, its rate of production would appear to be occurring at a nominal rate. But for an observer far outside the event horizon, the rate of Hawking radiation emissions would appear to be zero. To the far observer, these particles would appear to be frozen at the event horizon and would only be released to the far field when the event horizon terminated when the black hole was destroyed.

    Does this not mean that a black hole will never decay from erosive emissions of Hawking radiation?

  22. I’m curious about how we know there is mass at the center of black holes. I’ve thought a lot about time and how it mathematically can’t be one long, fluid instance. Mostly because it would require an infinite number of smaller increments to be exhausted for any practical increments of time to actually pass. If this is true, it means there are spaces between our finite planes of existence where everything we know ceases to exist before we reach the next plane of existence. If any of that logic is true, could it be possible that the mass that creates a black hole could be crushed down to a volume small enough to tear it’s way between planes of existence? Could it simply be too much mass for the so called fabric of space/time to support, essentially tearing an actual hole in the space and time that it used to occupy?

  23. Weird: just before I saw this blogpost, I woke up and recalled that I was dreaming I was in the Stanford (ugh!) Physics faculty and had stumbled into Susskind's office to ask him to explain the Black Hole's firewall. Unfortunately he was speaking nonsense...

  24. Is the strength of gravity calculated based on the radius of a mass or is radius irrelevant?

    1. RandomNewbie,

      If you are outside of the mass distribution, then the radius doesn't matter.

  25. The question as been raised how it would be to live inside a black hole.

    Johannes Koelman has written a nice blog on a thought experiment regarding this matter. It might not be as strange as you think (for a short while at least).

    How to stomach a black hole

  26. In the interest of things we should know about black holes; how can a singularity with a radius of zero rotate? Without a radius or circumference, how would it be possible to compute its speed of rotation as anything but infinite?

    It isn't a tiny sphere. Even if a mass was rotating as it collapsed, at some point, some outer point on the sphere would have to be moving faster than the speed of light to keep pace with an inner point, and since that's not allowed it can't happen; something else happens instead.

    But then, for any rotational speed that will end up true for any points on any radius > 0.

    How do physicists resolve this?

  27. How can we be sure that we are not inside a giant black hole, yet? Inside the black hole, would we not see the center in all directions? Is the background radiation perhaps the center of a black hole?

  28. Dr. A.M. Castaldo,

    Note that one gets this "paradox" even without any general relativity, and with very little quantum mechanics. We have known for some 100 years that electrons "rotate" yet they are, when probed, point particles. The naive paradox stems from associating rotation (spin) with a classical model of a rigid body. This is far from the first time in physics where our macroscopic classical intuition breaks down. As far as we know, spin (angular momentum) is a property of quantum objects unrelated to any physical rotation of anything. Similarly, a vacuum of space can also rotate, even if it ostensibly consists of nothing. This effect has been measured in space, see Gravity Probe B.

  29. One might mention an 11th, Spinal Tap fact - all this depends on the strict validity of general relativity as it stands, in particular, the decoupled status of the gravitational and electromagnetic fields. If say a strong gravitational field also produced an electromagnetic field, then the entire problem of horizon formation would be knocked in a cocked hat. Since it is as of now impossible to test GR in conditions of actual dense matter in which its second and higher order effects become apparent, well, doubts remain, and should remain. (Strictly speaking, the only real accurate test of GR is the weakest of first order effects - the Schwarzschild solution around a point mass.)


  30. Two well established aspects of Relativity Theory argue against the existence of black holes:

    1. According to GR, the speed of light is not constant in a gravitational field, rather it is inversely related to the strength of the gravitational field along its path (see>

    2. E/m=c^2 - the energy content of an object with rest mass is proportional to the speed of light squared.

    Combining those two characteristics leads to the conclusion that any gravitational collapse beyond a neutron star will be inherently self-limiting, as a steepening decline in the speed of light will result in a reduced mass content of any object involved in the collapse, the excess mass being radiated away.

    Black holes are no more a consequence of General Relativity than is the big bang model of the "universe". Both are dependent on the misapplication of GR to a conceptual model that is structurally inconsistent with GR. In the case of the big bang, the FLRW metric is essentially a universal reference frame, which is anathema to GR. The resulting big bang model is a qualitatively ludicrous description of physical reality.

    The Schwarzschild equation that describes black holes is wrong for a simple, straightforward reason - it does not account properly for the GR specified behavior of light and mass in a relativistically significant gravitational field.

    The foregoing considerations do not preclude the existence of compact objects of higher density than neutron stars. What is precluded is the runaway gravitational collapse ad absurdum of the black hole model.

    1. Bud,

      Consider actually learning the math of General Relativity before making misinformed categorical pronouncements. More specifically, you confuse the constant c with the apparent velocity measured by a distant observer.

    2. Sergei,

      Perhaps you should consider learning to think about physics rather than simply mathematical models. Einstein himself stated that in GR the speed of light in a gravitational field varies with position. This is not a statement regarding the point of view of a distant observer, it is the local condition where the gravitational field is relativistically significant. The failure to take this GR fact into account leads to the nonsensical black hole model.

      Math is not physics and conflating the two has produced models of physical reality that do not, in their particulars, resemble empirically observed physical reality. In some cases (black holes, the big bang), the models wind up in singularities which are scientifically and logically absurd, as well as mathematically meaningless. That such models are accepted within the scientific community despite their unavoidable divergence from any kind of realistic physics is a scientific disgrace.

  31. Bud rap, is it the speed of light that changes or the length of paths?
    The wikipedia article that you are pointing to seems to favor the latter explanation.

    1. Pascal, you raise a good point. That wikipedia entry is quite ambiguous.

      The article quotes Shapiro as saying "according to the general theory, the speed of a light wave depends on the strength of the gravitational potential along its path". The article then goes on to note:

      Throughout this article discussing the time delay, Shapiro uses c as the speed of light and calculated the time delay of the passage of light waves or rays over finite coordinate distance according to a Schwarzschild solution to the Einstein field equations.

      In the original paper ( things are a bit clearer:

      The right-hand side of Eq. (1) is due primarily to the variable speed of the light ray; the contribution from the change in path, being of second order in (r,/c), is negligible.

      So, the change in path length is not the primary cause of the time delay, it is the change in the speed of light that is the main causal factor.

  32. Not a scientist here, so please bear with me. I have two questions:

    1. Since the electromagnetic force is carried by photons, and since photons cannot cross the event horizon outwards, it seems to me to follow that when a body falls into a black hole the part of it that crosses the Schwarzschild radius first will disintegrate from the rest of the body as the EM attraction is no longer there to keep the parts together. Surely that cannot be the case if "someone crossing the black hole horizon does not notice anything weird going on". I, for one, would definitely notice if my foot disappears all of a sudden. Another issue along the same line of thought is what about charge conservation when a charged particle falls into a black hole - if the EM field cannot cross the event horizon the rest of the universe must not be neutral any more (if it was neutral to begin with).

    2. When discussing the arrow of time it is often said that the reason the early universe had a very low entropy is somewhat of mystery, while to me it seems a trivial corollary of Bekenstein-Hawking theory of black hole entropy since at the first split-seconds after the big bang the whole universe had a minuscule surface area.

  33. 1) The external observer sees objects slowing down and staying "forever" outside the horizon. How does SHE explain that the mass of the BH increases ? Some sort of Tunnel effect, with finite probability for the object to appear on the "inside" of the horizon.

    2) As the BH grows in mass - and therefore in diameter - does the external observer see the BH "swallow" the objects that are just outside the horizon ? Or do they appear to recede before it?

    1. jan:

      1) the external observer sees objects slowing down and getting redder, quickly disappearing from sight. She does not see them staying forever outside the horizon.

      2) There are no observers that can hover just outside the horizon, but you can see the horizon jumping outward in the youtube videos of simulated black hole mergers.

    2. I second Jan's question, and I am still confused after Sergei's reply. From the point of view of an outside observer, should we consider that the infalling matter hovers forever near the horizon or that it gets eaten up by the singularity? In the first case, most of the mass attributed to the BH could be hovering at the horizon with no increase over time in the singularity's mass.
      Imagine now a merger of two such black holes. From the point of view of an outside observer we would in effect see a collision between the two "shells" of matter hovering at the 2 BH horizons. If this picture is wrong, does it mean that the infinite slowing down of time is not "real" but is an illusion due to the redshifting?

    3. Pascal,

      Because the time flows at different rates for different observers, there is no uniform picture of what happens and when. If you are falling in, you cross the horizon and get torn to shreds very quickly. If you are watching from a safe distance, an object tiny enough to not significantly disturb the spacetime around the black hole would look like slowing down and redshifting away near the horizon. There is no contradiction. Further more, the mass is not in the singularity. A (hypothetical) wormhole lined inside with exotic matter does not have a singularity inside, yet it has the same mass as the black hole of the same radius. The observed mass (also known as the ADM mass) is the property of the spacetime itself.

      In the merger of two black holes the two apparent horizons "jump" and merge into a single one once the black holes are close enough. Anything that was seen moments before the merger disappears from the outside observer's view near instantly, the last image of what happened to be near the horizon of each one redshifting out of sight. From the point of view of whatever was hovering just outside, not that anything can hover for any length of time, they are now inside the newly formed black hole and are about to meet their demise after a few short moments.

      I hope one day we will get to see this event with the event (ho pun intended) horizon telescope, or its successor.

    4. Thank you Sergei for the explanations. I did not know about the mass not being in the singularity. This seems different from what one can read about black holes in the popular press.
      Do you literally mean that the singularity has mass 0, or is it that it contains only the mass of the collasping star when the BH was formed?

  34. Maybe, we don't know about black holes as much as some people think or pretend?

    But be sure: we have in spite of all that a "photography" of a maybe black hole.

    Who would have doubt about such "sure" things? We can see it, next we will touch it!

  35. If *nothing* -- from the external point of view -- can fall onto a black hole due to the time dilataion, how can another black hole fall onto it and still be observed *now* instead of infinitely far in the future? In other words, what has LIGO seen?

    1. Piotrw,

      The statement that nothing can fall into a black hole in finite time, as seen by an outside observer is a statement about the time-dilation nearby a single black hole. It simply does not apply for two black holes. If you have a second black hole, the causal structure is entirely different. The second black hole also bends time, if you wish. The two horizons can therefore merge in finite time (as seen from infinity).

  36. Sabine Hossenfelder12:19 AM, May 20, 2019
    " If you put something in an orbit around a black hole, it'll stay there, in principle forever. "

    This is wrong in my opinion. Or holds only for an else totally empty universe and without tidal gravitational forces. Else any motion slows down by deceleration forces, albeit over long time periods.

  37. Hi SABINE!
    Hello Friends.
    @ Bud Rap, you introduce a number of interesting and valid points.(would love to discuss sometime)
    I always get a kick out of people referring to the speed of light as a 'constant'.
    Really, maybe on this planet.
    The only thing GR (and SR) - and experiment on this planet actually show is that it has a limit.
    To then take that 'Universally' and suppose a 'constant' is a mistake.
    - for a number of reasons.
    ( will discuss anytime)
    It is, in my opinion, also a mistake to say that all stars
    are the same or that their
    phases are actually accurately
    ( Go ahead , .. one anomaly
    ... and you lose.)
    Science is an amazing thing.
    - we need to get back to it.

    All the best,

  38. Sabine, what do you make of the ECO model ("eternally collapsing object") as an alternative to the standard black hole model?
    I ask again because my previous comment did not get through. Slow moderation, or is it that you consider this model not worthy of consideration?

  39. Do we know anything about what happens to dark matter that falls into black hole? Does it radiate and became interacting (more) with normal matter? Or radiate differently? Or not radiate at all?

  40. Can black holes totally alter the wave functions of particles? I mean is there a possibility that a particle entering a black hole can get its wave function so mixed up that the highest probability of finding it is actually far far away from the black hole?


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