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.
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.