Are We Wrong About Black Holes?
quantamagazine.org
quantamagazine.org
No. It is posible to define the entropy of an object that has no internal indivisible parts, for example a black hole.
> Here’s another way in which the analogy falls flat. Black hole entropy is given by the black hole area. Well, area is length squared, volume is length cubed. So what do we make of all those thermodynamic relations that include volume, like Boyle’s law? Is volume, which is length times area, really length times entropy? That would ruin the analogy. So we have to say that volume is not the counterpart of volume, which is surprising.
This paragraph makes no sense at all. No sense at all!
First, Boyle's law is a law for ideal gases (that can be applied somewhat correctly to real gases). It can't be applied to solids, it can't be applied to liquids, it can't be applied to a photon gas, it can't be applied to black holes. Trying to apply it to black holes shows a completely lack of understanding of the subject.
In particular:
> Is volume, which is length times area, really length times entropy?
What? First the entropy of a black hole is not equal to the area, it's only proportional to the area, there are some constants. (I had to look up S = A * c^3 / 4Għ.) Second, you can't pick any three length [1] and multiply them and expect that the result that has the units of a volume can be used in any formula you want.
[1] For example, the height of the Eiffel tower, the atomic radio of an iron atom in the Eiffel tower and the width of the door of the elevator of the Eiffel tower. You can multiply them and get a volume, but this volume doesn't follow the Boyle's law when the atmospheric pressure changes in Paris.
How? If there is no system of parts, there is no entropy. I'm not sure why you have a "black hole" as an example, when you talk about the entropy of a black hole afterward and it's has not been concluded that black holes violate the second law. You have failed to give an example and I think it's important to at least clarify your statement.
> First, Boyle's law is a law for ideal gases (that can be applied somewhat correctly to real gases). It can't be applied to solids, it can't be applied to liquids, it can't be applied to a photon gas, it can't be applied to black holes.
He didn't say that Boyle's law applied to black holes. He was expounding on the initial point you objected on. If you have a system of parts, it has a volume.
> Trying to apply it to black holes shows a completely lack of understanding of the subject.
You're being disingenuous by not even attempting to understand the assertion and then attacking every logical conclusion as if that means something. He has an understanding, even if you think it's wrong.
> Is volume, which is length times area, really length times entropy?
To your first point, it's semantics when you drop the constants. This isn't some obscure way of speaking and he uses it generously.
To your second point, he didn't.
This was a generalization of the concept expanded to how you calculate the event horizon of a black hole. A sphere's surface area can be calculated from the length of the radius (ie A=4πr2).
(from google) S/V = 3/R 4πr2/Volume = 3/r or Area/3/radius = Volume
So the interpretation is volume = Area x 1/length or Area x Length (ie radius) when dealing with the variants.
A x c^3 / 4Għ = Entropy (Bekenstein–Hawking formula) and dropping all the invariants (A = E) we get Area = Entropy So the volume of a black hole (given it has entropy) could be the volume of the black hole (Area x Length) = Entropy and since we can get area from Length, we can drop the invariants and we get Length = Entropy with constants and invariant proportions.
Define entropy?
Wikipedia says this about entropy:
> In statistical mechanics, entropy is an extensive property of a thermodynamic system. It is closely related to the number Ω of microscopic configurations (known as microstates) that are consistent with the macroscopic quantities that characterize the system (such as its volume, pressure and temperature). Entropy expresses the number Ω of different configurations that a system defined by macroscopic variables could assume.
The number of possible configurations of a black hole is hilariously large and demands delving into up arrows and beyond. It includes collapsing stellar cores, primordial black holes, neutron stars accreting matter until collapse, merging black holes, kugelblitzes, etc., which all are bizarrely dislike microstatically. Yet the possible macroscopic variables are only three: Mass, angular momentum, and charge.
If you pretend quantum mechanics don't exist, sure, black holes don't have entropy. But we're pretty sure something that looks a lot like quantum mechanics does exist.
From there you deduct in the last paragraph that "Length = Entropy". I guess you want to use the radius or diameter of the black hole (and ignore a few more numerical constants). It doesn't make any sense, they aren't equal, they aren't proportional.
The first error is that at some point you use that the volume is "equal" (proportional?) to the entropy, but this relation is wrong.
The second error is that in "(Area x Length) = Entropy" you can replace Area by Length^2, and get Length^3 = Entropy. You say that "we can drop the invariants" and you drop the exponent 3, and you get "Length = Entropy", but now you are removing and exponent, not a multiplicative constant. So the "=" is not "equal", is not "proportional", it is only "somewhat related".
> It doesn't make any sense, they aren't equal, they aren't proportional.
That's not relevant. You increase one, you increase the other. [1]The equality symbol is not a literal equality. In a macro sense, it's a relationship that is directly correlated as opposed to equal in any sort of specific calculus.
> The first error is that at some point you use that the volume is "equal" (proportional?) to the entropy, but this relation is wrong.
You missed the point, which was initially:
> Here’s another way in which the analogy falls flat.
> So what do we make of all those thermodynamic relations that include volume, like Boyle’s law
Callendar was making his core argument that the entropy of a black hole is not expressed as a function of volume in one sense - because there's an assumption of quantum mechanics that take over, as you say it's "wrong" following the unproven "black hole thermodynamics" models ... but in another sense (Bekenstein–Hawking) there is a paper describing how surface area is a function of entropy. So this looks like a dichotomy since the surface area (which can tell you the entropy) can tell you the volume of a sphere, but volume of a black hole sphere isn't used to correlate to the entropy...so there's a dichotomy that is related to breaking laws (equivalence doesn't mean anything and it's all nonsense). Basically there's an assumption that beyond the event horizon we assume that the rules of physics break down into something new, although the black holes (theoretically per Bekenstein–Hawking) exhibit what a classical physicist would expect. Why not treat the black hole as if it has entropy as per the classical physics models, since it exhibits that quality already and see what insights that yields, rather than a whole new branch of thermodynamics that are necessarily exotic theory?
> now you are removing and exponent, not a multiplicative constant
See [1]
I have rephrased the position and I think maybe you should have a talk with the originator (ccallender@ucsd.edu) if you are still confused.
More generally, one cannot rely on intuitions from Euclidean relations in substantially non-Euclidean geometries -- and where there is Ricci curvature, we depart from Euclid.
Y C Ong has a perhaps slightly more accessible overview and further calculations at https://plus.maths.org/content/dont-judge-black-hole-its-are... under the subheading "Growing with time". Ong also writes, quite reasonably, that the spatial 3-volume of a black hole is not a well-defined notion. I would go further and say that applies to any region of any general curved spacetime in which there is "sufficient" Ricci curvature. Unfortunately, "sufficient" is hard to divorce from coordinate conditions, and I think that is the underlying source of the non-well-definedness.
I'd also like to emphasize that there is no reliable probe for the spatial 3-volume bounded by the horizon of a collapsed-matter BH given only its instantaneous location, linear & angular momentum, charge, and mass. We would also need to know the BH's age and initial conditions. Consequently, while one might talk about some sort of time-dependent black hole entropy related to its internal volume, I don't think it's likely to be useful at all, and at the very least it's an impractical measure in practically all astrophysical circumstances.
By comparison, given mass, charge and angular momentum (we can always fix linear momentum and position to zero by coordinate choice) we can calculate a Kerr-Newman BH's inner surface's area in arbitrary coordinates. Given area "A" and a choice of time coordinate, classically we say dA / dt >= 0, and A relates to entropy by Bekenstein 1973's (1/2 ln 2) / 4 pi relation. Hawking's 1974 modification given turns this into Entropy_{BH} := (boltzmannconstant A) / (4 plancklength^2) and breaks the monotonic area condition. The 1973 paper and the 1974 letter [ doi:10.1103/PhysRevD.7.2333 resp. doi:10.1038/248030a0] are fairly resp. very short; both are straightforward and can be found easily including at one's favarrrite source of old papers. I think if you read them you will change your mind that they form "a whole new branch of thermodynamics that are necessarily exotic theory". Indeed, it is because nobody seriously proposes rejecting standard thermodynamics that evaporating Schwarzschild black holes seem like weird objects.
Given the above,
> (from google) S/V = 3/R 4πr2/Volume = 3/r or Area/3/radius = Volume
and the consequences you draw from that in this thread are not generally correct.
I know a lot of cases where equivalence/comparison the relations are not linear. These are very powerful toys. In these cases people usually uses "equivalent" or a custom name for the equivalence. Using "equal" is too confusing unless it is very extremely super clear from the context.
I recommend to stick to proportional relations and to say that the things are "proportional", not "equal".
Quoting again the part we both quoted from the article:
> Here’s another way in which the analogy falls flat.
> So what do we make of all those thermodynamic relations that include volume, like Boyle’s law
The problem is that there is no analogy. There is no real thermodynamics for gases and a fake thermodynamic for black holes that is somewhat analogue to real thermodynamic for gases, and where each magnitude for gases is replaced for a magnitude for black holes that is somewhat related. So it is not necessary to copy the Boyle's laws for gases to a analogue law for black holes where each magnitude is replaced by an analogue magnitude.
The main error is that this is an extension of the theory, not an analogy.
Just imagine that someone has discovered thermodynamics for (ideal) gases, including entropy and Boyle's law. Now someone comes with an object that is a solid. What it the Boyle's law for solids? To extend thermodynamics to solids, there is no analogies, it is necessary to define some properties like entropy and internal energy. It is not necessary to copy the Boyle's law to solids, because Boyle's law is a law that is valid only for gases (and even only for classical ideal gases).
> Callendar was making his core argument that the entropy of a black hole is not expressed as a function of volume in one sense
As the sibling comment says (in a too technical way), it's difficult to define how much is volume is inside the sphere of the event horizon of the black hole. The space-time inside it is very distorted and you can no longer use the classical formulas. But it is clear how much surface are it has and is easy to calculate. So it's wise to write a theory that uses the surface area instead of the volume.
How one gets to an entropy is ultimately via no-hair. If gravitational multipoles "bald" away in about a horizon light-crossing time, then we have no chance of distinguishing between having formed the black hole from (classically) one collapsing homogeneous spherical shell of neutral dust of mass M (and no angular momentum), or two such shells each of mass M/2 and each with identical-but-differently-signed angular momentum, or three such shells whose mass sums to M and whose angular momenta sum to zero, and so on. The shell count and the details of their individual angular momenta are "lost" from the black hole.
The spacetime-filling metric tensor encodes all of this into something that far from the infalls of these shells everything looks like Schwarzschild far from the central mass. This is a bit frustrating if the shells of dust are eternally present, because it doesn't give a strong hint about an observer -- who is far from the central mass at all times -- should slice up spacetime so that the metric is time-dependent. (Breaking the maximal symmetries of the black hole at some point can, though, but we are avoiding that by using these classical shells: if we choose an observer who is sufficiently far away from the shells then at all times they look pointlike (or more precisely always like a central spherically symmetric uncharged mass that is the sum of the black hole and all the shells we throw in, and the observer struggles to determine whether all the mass is within the Schwarszchild radius)).
The frustration arises because we can construct the situation so that for a far future observer there is no way to observe how many of these shells were in that observer's distant past, nor what their individual masses, momenta, and charges were: only that all that remains is a nonzero mass, zero angular momentum, and zero charge occupying such a small solid angle of the observer's sky that the observer concludes it is a black hole.
The Boltzmannian macrostate is the black hole and its mass, angular momentum and spin; the microstates are the details of the shells, starting with their individual macrostates (each shell has some mass, angular momentum, and charge), and then considering their internal degrees of freedom (dust particles can be of arbitrarily small mass without breaking our spherical-and-homogeneous condition). Thus the "balded" state can have arbitrarily high entropy, because we could have thrown in any number of shells each containing any number of dust particles, because we are in a classical setting where infinitesimals are OK. If we massively increase the particle-mass and reduce the particle number of a shell (constant M>0, J=0, Q=0), we still get the same macrostate compared to arbitrarily lowering particle mass and increasing the particle number to keep the same M>0, J=0, Q=0.
If we were to be like Merlin or similar fictional characters and travel with a different arrow-of-time, starting with closer-in view of the black hole as the only thing in the test universe of pure Schwarzschild, we would not be able to predict whether in our personal "future" the black hole will spontaneously spit out one shell of mass M, or two shells each of mass M/2 with an angular momentum summing to 0, or three shells, etc etc. Not being able to predict successor time-indexed spacelike slices of spacetime given the entirety of data on one slice frustrates the initial values approach which works remarkably well in much of physics. (Singularities appear in other physical theories that are far from fundamental, such as Euler/Navier-Stokes fluid dynamics, and they too frustrate initial values studies of systems in which they can arise).
("Initial" is a free choice, and generally means where the physicist starts from, rather than initial as in the sense of in the ultra-distant past. It's fairly common to choose as an initial values surface some final state and work backwards from there.)
The quantum picture is a bit more complicated because of the vacuum state problem (I do not have time to get much into that right now[1]) but one can think of throwing in a bunch of neutral hydrogen atoms of mass M versus a bunch of helium atoms of mass M: after "balding", we are again left with only having mass M, angular momentum summing to zero, and no charge: all the quantum information is in the balded black hole. (There are issues about keeping the metric tensor indistinguishable from Schwarzschild in the quantum case; and about how one defines a quantum vacuum).
The relevant difference here is that we expect that "Merlin" in the quantum case must only see QFT observables in the stress-energy tensor as he watches the stable black hole spontaneously emit matter, and this is where \hbar comes in in the equation you found.
We still think about Boltzmann entropy in the same way though: all the quantum numbers and degrees of freedom in the atoms we throw in result in a no-hair black hole with some mass M, some angular momentum (here we carefully keep it at zero), and some other quantum numbers. The entropy of a black hole is thus extremely high.
Sorry that this comment is a bit messy due to lack of time. Also, sorry, I'm not sure when I'll be able to clarify or comment further if needed.
- -- [1] (Briefly, some people are unhappier in the quantum case where Merlin cannot predict what QFT observables would appear and when, than in the classical case when he predicts Schwarzschild black hole forever in one arrow-of-time direction, but can't predict shells and shell-components in the other. In particular, the quantum case introduces black hole evaporation, and what gets spit out "spontaneously" in the future does not seem to be what fell in. In the sketch quantum BH above Merlin sees a spitting out of hydrogen or helium atoms, whereas ordinary observer sees an evaporation into mostly just photons (with possibly neutrinos and other massive things in the very late stages of evaporation, but probably never neutral hydrogen or helium atoms (too heavy and too bound during early evaporation, too easy to ionize or transmute during late evaporation)).
He has only half the "no drama" conjecture here. Given a sufficiently massive Schwarzschild BH the steam engine would work for a while before the Weyl curvature spaghettifies it.
As he says a few paragraphs above, the BH area increases when the engine enters the BH. To leading order the increase is driven by the stress-energy of the steam engine, and the stress-energy tensor encodes terms such as chemical and electrostatic potential energy, enthalpy, free energy, internal energy, virial stress, and so forth -- not just mass, although mass tends to dominate in simple systems. At early times the steam engine's entire stress-energy is outside the BH; at later times, the residual stress-energy of the steam engine (having lost some to radiation flying to infinity doing work earlier) is entirely within the horizon. I don't see why he objects to this "saving" of the second law.
Also,
> It’s all about forgetting information.
Well, it's thermodynamics that reveals the forgetting of information because of the classical irreversibility of crossing the horizon (classical BH never shrinks).
[Quantum vacuum causes classical black holes to shrink, but the Hawking radiation does not come from inside the black hole; it comes from defining a quantum vacuum as a no-particle state at an early time whereas the (Lorentzian) no-particle condition depends on the metric, and the metric for a collapsing-then-evaporating black hole is time-dependent (M is not eternally the same). By the equivalence principle, we can relate the disagreement between observers about the particle count in a state. In flat spacetime, A's no-particle state may be full of particles for observer B who is accelerated relative to A (this is the Unruh effect); in curved spacetime B who is a region with an event horizon is accelerated relative to A and C who are in regions without an event horizon. Inertial/Eulerian observers A and C may agree that each's "no-particle" vacuum state has no particles; B will see particles in A's "no-particle" vacuum and in C's "no-particle" vacuum.
> The system is really the entire space-time
Here the author is entirely correct.
In the paragraph about no-particle state, the entire spacetime has a region with some nonzero stress-energy localized as a collapsing dust and otherwise everywhere the expectation-value of the stress-energy -- <T> rather than T -- is zero and there are no-particles outside the outermost dust particles. There is another region where there is an event horizon and no dust particles outside it. But there are particles outside and near the horizon, but everywhere far from the horizon <T>=0 and we find no particles there. There is another region where there is no horizon, and an expanding cloud thermal particles with the spectrum of a blackbody radiator that started cold and got extremely hot, but with none of the initial dust particles from the first region, and beyond that a no-particle region.
The broad research programme in numerical relativity is to trace the stress-energy tensor (and its generators) through these regions. Unruh discusses this in the context of a sonic analogue of a black hole: https://pos.sissa.it/043/039/pdf
Pseudoscience refers to activities that superficially have the aesthetics of science but do not follow its constraints. For example, homeopathy is pseudoscience. They dress in white coats and use chemistry lab equipment, but they do not test their ideas with experiments that could falsify them. This has nothing to do with reasoning with the current scientific theories. What do you think theoretical physicists do?
I say this because more and more I notice people using "pseudoscience" in a sense similar to "heresy", and treating philosophy as some sort of inferior system of knowledge that is bound to be replaced by science. These people misunderstand both science and philosophy, and would benefit from a bit more of reading and thinking outside of their comfort zone.
I think that for some people science has replaced religion as a belief and social control system
For these people science defines what is possible within reality and how the world works in a fundamental way; but they don't really understand how these scientific theories work, they just accept them.
I am not saying that science is a religion (they do work under different constraints), I am just saying that science has started to serve a similar role within our civilization, like religion did back in the "dark ages"
Your wording makes it sound like there's only one "scientific" belief system, but It's in my view really only a component of one. I think constructing your worldview around at least theoretically verifiable claims is a definite improvement over religious dogma, but most important is the scientific mindset of accepting uncertainty and "I don't know" answers (until you figure them out).
I also think it's valid to trust experts, but extraordinary claims will still require extraordinary evidence.
Now, the social structures around science aren't perfect and bad science can definitely be used to mislead people. That's just another problem to solve.
And there's nothing wrong with that.
Science is "philosophical arguments" (epistemological axioms, assumptions, theories, etc) + evidence, so attacking the first still makes sense.
Evidence alone is just objects / the world unfolding, not science.
I thought he proved the opposite: that black holes (slowly) evaporate through Hawking radiation, lose mass, hence their event horizon shrinks. Could anyone please elaborate?
Also, do scientists actually say that thermodynamics and black holes are the same thing? Never heard of that. I think they only mean that laws of thermodynamics are universal and should be applicable to black holes as well.
The identification of area with entropy doesn't mean that entropy decreases as a black hole evaporates, because the emitted particles carry the entropy.
Also, I thought never-increasing entropy was only applicable to closed systems. In that sense, when particles leave black hole through Hawking radiation, total entropy of the Universe remains the same. Am I missing something?
Yes. So the entropy of the hole by itself decreases. But the entropy of the whole system, including the hole and the Hawking radiation it emits, increases.
> when particles leave black hole through Hawking radiation, total entropy of the Universe remains the same
We have no way of evaluating the total entropy of the universe. The concept itself might not make any sense, since we have no way of making measurements or running experiments on the universe from the outside.
He proved y under some set of assumptions, and then later !y under another set of assumptions. So he proved a -> y && b -> !y.
But given x -> y, if you prove !x, you can't conclude anything. And kibibu said !x, not !y.
> quantum fields near the horizon can violate some of the assumptions that are required for the area theorem
as !x
That's correct; that's all they mean.
https://en.wikipedia.org/wiki/Arieh_Ben-Naim
https://www.amazon.com/Arieh-Ben-Naim/e/B001HD1O86/ref=dp_by...
I enjoyed Ben-Naim's "Briefest History of Time, The: The History of Histories of Time and the Misconstrued Association Between Entropy and Time "
https://www.amazon.com/s?k=briefest+history+of+time
Ben-Naim takes credit for Hawking later cutting most of his "Brief History of Tme" in later editions.
... oops.
I spread out two decks of cards on the table in front of you. One is thoroughly mixed up, the other one is neatly ordered by suit from spades to diamonds and each suit is ordered from ace to king, except for 7 and 8 of clubs, they are in the wrong order. »Which deck has the higher entropy?« I ask. »The mixed up one of course.« you answer. Out of my pocket I pull a flashlight and turn it on, under the shine of its blacklight you can read the numbers 1 to 52 written onto the mixed up deck in perfect order, the numbers written onto the neatly ordered deck are in no discernible order.
This brought up another set of questions: is it possible to define entropy without a definition of simultaneity, or some other global ordering of events? Is it possible for two different observers to have differing views of entropy? What does rotation of the time-like axis do? Googling didn't find much, but it did find a discussion on physicsforums [0]. The answer seemed to be... it's complicated. Good discussion, though.
[0] https://www.physicsforums.com/threads/time-dilation-and-entr...
Once you have done that, you can start counting the microstates that satisfy those macroscopic variables.
ΔS = Q/T
The specific value for the entropy of a system depends upon the definition of the microvariables. However, regardless of the definition used, the measured value will increase over time. In the deck of cards example, above, one definition of entropy is the suites and rank, and another definition is the numbers 1 .. 52 written on the backs. Shuffling the deck will reduce the ordering, no matter which definition you look at.
This implies
a) No definition of microvariables and their ordering can result in an entropy level which is the inverse of another definition.
b) Its possible to define a set of all sets, which is the union of all possible microvariables and orderings. This uber-entropy will also increase over time.
I suppose the philosophical argument is that "objective entropy" is either referring to the uber-entropy, or the property that all entropies always increase, and "subjective entropy" is looking at specific values for one particular measurement system.
Also, c) increasing entropy over time only holds true for interaction functions which are fully transposable. For example, if microstate X is the result of an interaction "f" betweeen microstates A and B
state(x) = f(state(a), state(b))
state(a) = f'(state(x), state(b))
state(b) = f''(state(x), state(a))
f, f', and f'' are fully deterministic and yield unique results for every combination of inputs.
If your observational weakness prevents you from making any observations. Then you know nothing at all about it.
Spaghettification
What if Human are the ant?.
Proving it being right / wrong is not the point.