[gus_massa]

The Area is not equal to the Entropy, it's only proportional. I think we agree about that.

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".

[reallydude]

You continually make the grave error of framing the criticism as if I had proposed it. I am clarifying what is plainly stated in the article, by Callender, but somehow has escaped the narrow focus of individuals like yourself.

> 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 ([e-mail]) if you are still confused.

[gus_massa]

The original author always used "=" as "proportional".

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.

[raattgift]

Unfortunately, Rovelli & Christodolou show [ https://arxiv.org/abs/1411.2854 ] that: independent of coordinate choice, the spatial 3-volume of a static Schwarzschild BH departs spectacularly from Euclidean geometry (not surprising); and in a collapsar asymptoting to Schwarzschild, volume increases over time up to some maximum (depending on Hawking radiation).

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.

[reallydude]

I don't draw any "consequences", whatever that's supposed to mean. I have been following the article's claim and explaining it in simpler terms, because there was a misunderstanding of the claim. Who do you think Callender is? Not me. Talk to the guy making the claim about your explicit objections. His email is both in this thread and trivially available. I don't care that you object to the claims he makes. That isn't a debate I'm participating in (if you even read the article, or the thread). Your opinion makes no difference to me (a random guy on the internet who can read english). Good luck with whatever.