A revealing correspondence on laws of thermodynamics with Prof. William G. Unruh

Dear Sirs,
I note your recent paper [1] in which you derive the Bekenstein-Hawking black hole etropy equation using Boltzmann’s entropy. I point out that irrespective of how the Bekenstein-Hawking black hole entropy equation is derived it remains invalid because the black hole entropy equation violates the laws of thermodynamics. Entropy is an extensive thermodynamic coordinate whereas the area of the alleged black hole event horizon is not extensive. In any proposed thermodynamic equation not only must the units balance on each side, so too must the thermodynamic character. This is easily seen: the black hole entropy relation is:

S = πc^2kA/2hG

The entropy S must be extensive: as expressed in the second law of thermodynamics. On the right side only the area A determines the thermodynamic character in this expression because all the other terms on the right side are pure numbers or physical constants, which have no thermodynamic character. Area is not extensive so the black hole entropy equation is invalid. To amplify: since A = 4πR^2 where R = 2GM/c^2 (the so-called Schwarzschild radius),

S = 8π^2kGM^2/hc

Mass M is extensive but mass squared is not extensive. The equation is thermodynamically unbalanced and therefore invalid for violation of the laws of thermodynamics by making entropy non-extensive.

Similarly the Hawking black hole temperature equation is invalid for violation of the laws of thermodynamics by making temperature non-intensive in violation of the zeroth and second laws of thermodynamics which require that temperature must always be intensive. Recall Hawking’s black hole temperature equation:

T = hc^3/16π^2kGM

Mass M is extensive but mass squared is not extensive. The equation is   thermodynamically unbalanced and therefore invalid for violation of the laws of   thermodynamics by making entropy non-extensive. Similarly the Hawking black hole temperature equation is invalid for violation of the laws of thermodynamics by making temperature non-intensive in violation of the zeroth and second laws of thermodynamics which require that temperature must always be intensive. Recall Hawking’s black hole temperature equation:

T = hc^3/16π^2kGM

Temperature is intensive. The thermodynamic character of the right side is determined only by the mass M since all the other terms there are pure numbers or physical constants. Mass is extensive and 1/M is neither intensive nor extensive. The right side makes temperature non-intensive contrary to the zeroth and second laws of thermodynamics which require that temperature must always be intensive, so Hawking’s temperature equation is invalid. It is for the very same reason that the Unruh temperature is false.

The two attached papers explain in detail.

Yours faithfully
Stephen J. Crothers

[1] Microscopic Origin of the Entropy of Black Holes in General
Relativity, Physical
Review X 14, 011024, (2024)

You have gotten yourself confused by an approximation. Yes, for normal matter entropy goes as volume, but even in the normal case, there is a part of the entropy that goes as the area–surface effects. These are usually subdominant. They are dominant for black holes. Ie, you have been led astray by the simplied picture usually developed in textbooks.  There is absolutely no reason why entopy should go as volume. I one dimensional chain has not volume yet it has an entropy, etc.

William G. Unruh, Canadian Institute for Physics & Astronomy, Advanced Research
Tel: +1(604)822-3273, Fax: +1(604)822-5324
UBC, Vancouver, BC Program in Cosmology and Gravity, Canada V6T 1Z1
unruh@physics.ubc.ca
theory.physics.ubc.ca

Dear Mr. Unruh,

Contrary to your baseless charge, I’m not confused at all. It’s obvious that you did not carefully read the papers I provided – assuming you even bothered to read them at all since entropy and surfaces are discussed therein in detail. You don’t like it but the fact is that mass squared is not extensive whereas entropy is extensive – consequently the Bekenstein-Hawking black hole entropy is false for violation of the laws of thermodynamics. Also, you don’t like it but temperature must always be intensive. Your Unruh temperature is not intensive so it is invalid for violation of the zeroth and second laws of thermodynamics, just like Hawking’s invalid black hole temperature, etched into his gravestone in Westminster Abbey – a monumental embarrassment to cosmologists.

Cosmologists do not get to ignore or otherwise evade obeying the laws of thermodynamics and the intensive and extensive character of thermodynamic coordinates in order to advance fallacies that violate those laws.

Yours faithfully,
Stephen J. Crothers

That entropy is extensive is not a law of thermodynamics. It just happens to true often, but not always.
As you know none of the three laws says that entropy is extensive.

William G. Unruh,
Physics & Astronomy, UBC, Vancouver, BC, Canada V6T 1Z1
Canadian Institute for Advanced Research program in Cosmology and Gravity
Tel: +1(604)822-3273, Fax: +1(604)822-5324
unruh@physics.ubc.ca

Dear Mr. Unruh,

“That entropy is extensive is not a law of thermodynamics. It just happens to true often, but not always.  As you know none of the three laws says that entropy is extensive.” Unruh

Your response is rather feeble. I made it clear that extensive and intensive are characteristics of thermodynamic coordinates, and that the extensive character of entropy is expressed in the statement of the 2nd law.  I also made it clear that in any proposed thermodynamic equation the units must balance and so too the thermodynamic character of each side of the equation. Failure to maintain thermodynamic character invalidates the proposed relation and this requirement is a feature of thermodynamics. Furthermore, Landsberg has proposed that thermodynamic character balance is so important that it should be regarded as the fourth law of thermodynamics, which is also discussed in the papers I provided. It’s clear to me that you have not even bothered to read them.

I reiterate that your argument as to surfaces on black hole entropy is false because black hole entropy is directly proportional to mass squared via surface area in the so-called Schwarzschild radius. Although mass is extensive, mass squared is not extensive whereas entropy is extensive so the Bekenstein-Hawking black hole entropy is certainly invalid.

Concerning temperature, I reiterate that temperature must be intensive and this is contained within the 0th and 2nd laws of thermodynamics. The Hawking black hole temperature and your Unruh temperature are not intensive so they are certainly false.

The cosmologists’ conception of entropy is also discussed in the papers I provided. Their arguments are attempts to evade the laws of thermodynamics and the required balance of thermodynamic coordinates in thermodynamic equations. This too is discussed in the papers I provided – revealing therein that their conception of entropy is a contradiction and therefore false. As I said, cosmologists do not get to disobey or otherwise evade the laws of thermodynamics and the requirement of thermodynamic character balance in order to advance fallacies that violate those laws. You say:  “That entropy is extensive is not a law of thermodynamics. It just happens to true often, but not always.”  This is a typical example of evasion. Extensive and intensive properties exist and they cannot simply be ignored by cosmologists so that they can conjure up fallacies that do not comply with physics. 

Yours faithfully,
Stephen J. Crothers

Dear Sirs,

You have all now been made aware that your theories violate the laws of thermdynamics by making temperature non-intensive. With non-intensive temperatures you are not doing science, you are propounding only mysticism.  Bear in mind that the Bekenstein-Hawking black hole entropy, besides being able to be expressed in terms of mass squared, can also be rendered in terms of Hawking’s non-intensive black hole temperature. Thus both his expressions are invalid as is Mr. Unruh’s temperature since the latter is also non-intensive.

If you really wish to contribute something to science you are all now obliged to admit that your theories are false and to make the fact known to the scientific community and the public at large. Silence is not a scientific method. Honesty is called for. 

Yours faithfully,
Stephen J. Crothers