Geometric thermodynamics

and black holes

Bryan W Roberts, LSE

These slides: personal.lse.ac.uk/robert49/talks/munich
Join: philosophyofphysics.org

$Q_1$. When is a system really thermal?

"Beckenstein suggested on thermodynamic grounds that some multiple of $\kappa$ should be regarded as the temperature of a black hole. He did not, however, suggest that a black hole could emit particles as well as absorb them. For this reason Bardeen, Carter and I considered that the thermodynamical similarity between $\kappa$ and temperature was only an analogy. The present result seems to indicate, however, that there may be more to it than this." Hawking (1974)

"If the validity of the generalized second law is accepted, then by far the most natural interpretation of the laws of black hole thermodynamics is that they simply are the ordinary laws of thermodynamics applied to a black hole. In that case, $A/4$ truly would represent the physical entropy of a black hole" Wald (1994, p.174)

"I began this project sharing at least some of the outsiders' skepticism, and became persuaded that the evidence is enormously strong both that black holes are thermodynamical systems in the fullest sense of the word" Wallace (2018)

There are reasons to be cautious.

"What we are arguing is that the analogy may not be more than formal. ... BHT may go the way of thermoeconomics. That is why we recommend hedging one's bets." Dougherty and Callender (2016)

Whether or not black holes are truly thermodynamical is a matter of unsettled physics. — Paraphrasing Jos Uffink (MCMP, 26 June 2022)

$Q_2$. What is a model in thermodynamics?

Philosophers on models

Thermodynamics

Geometric thermodynamics

Black holes

Philosophers on models

General Relativity

Structure: Lorentzian manifold $(M,g_{ab})$ $\in$ $\mathcal{P}$ $\subseteq \mathcal{U}$
Interpretation: Structure of physical spacetime

Philosophers on models

Quantum theory

Structure: $C^*$ algebra $\mathcal{A}$, system of imprimitivity,
quantised field, etc....
Interpretation: Structure of physical states,
observables, and their dynamics

Philosophers on models

Statistical mechanics

Structure: Boltzmann system $(M,\omega,\rho,\mathcal{O})$,
von Neumann factor, etc....
Interpretation: Statistical behaviour given
many degrees of freedom

Philosophers on models

Thermodynamics

"Let $P$ be a semi-permeable membrane in contact with a heat bath."

????

Philosophers on models

Thermodynamics

"Every mathematician knows it is impossible to understand an elementary course in thermodynamics. The reason is that the thermodynamics is based — as Gibbs has explicitly proclaimed — on a rather complicated mathematical theory, on the contact geometry." Arnol'd (1990, p.163)

It is possible to understand an advanced course in thermodynamics!

Thermodynamics: An alternative

Structure: A contact manifold $(M,\theta,N)$
Interpretation: Structure of equilibrium states
when some degrees of freedom are projected out.

What does this get you?

Reduction of thermo to statmech

Statmech: Boltzmann system $(M,\omega,\rho,\mathcal{O})$
Thermodynamics: Contact manifold $(M',\theta,N)$
Reduction: An embedding of the latter into the former.

No thermodynamic arrow

...in the structure of equilibrium states.

Just like Boltzmann entropy in statistical mechanics
'Minus-first law' needed to break symmetry (Brown and Uffink 2001)
BWR (2022, Chapter 6), "There is no thermodynamic arrow"
(Open Access book, Reversing the arrow of time, out 20 September 2022)

Are black holes 'really' thermal?

Thesis. Physical Systems with radically different structure, including black holes, satisfy a basic sense of 'being thermal' when they can be represented by a model of thermodynamics.

Different properties can be added to a basic mathematical model of thermodynamics, and black holes are very different.
Stricter notions of 'thermal' may sometimes be useful.
- p-Thermal = particle creation
- c-Thermal = Carnot cycle
- e-Thermal = evaporation
For us, 'thermal' = a precise model of equilibrium thermodynamics

Black holes are very different, but still thermal

Philosophers on models

Thermodynamics

Geometric thermodynamics

Black holes

Thermodynamics

In the beginning there was Newton

$dU = PdV$

$dU = PdV \, + \,?$

$dU = PdV + \xi$

$dU = PdV + NdM + \xi$

$\xi = $ the heat one form ($= \text{đ}Q$)

$dU = \underbrace{\sum_i P_i dX_i}_{\text{'work'}} + \underbrace{\xi}_{\text{'heat'}}$

19th c. calorists: $\xi$ is energy of the material calorique.

"When one raises the temperature of air while maintaining constant pressure, only one part of the caloric it receives actually produces this effect; the other part, which becomes latent, produces an increase in volume. It is this part that disengages when one compresses the expanded air and reduces it to its original volume. The heat released by the approach of two neighboring molecules of a vibrating arial fiber elevates their temperature, and gradually diffuses into the air and the surrounding bodies." — Laplace (1816)

$dU = \underbrace{\sum_i P_i dX_i}_{\text{'work'}} + \underbrace{\xi}_{\text{'heat'}}$

Gibbs, and us: $\xi$ is energy of inaccessible degrees of freedom

What is 'inaccessible'

Small scale interpretation: Heat arises from micro degrees of freedom.
Epistemic interpretation: Heat arises from what we do not know.
Control interpretation: Heat arises from what we cannot experimentally intervene on. (Wallace 2014, Myrvold 2020)
Projective interpretation: Heat arises from projecting out degrees of freedom.
Let 🌻🌻🌻 $\cdots$

Entropy

$dU = \underbrace{\sum_i P_i dX_i}_{\text{'work'}} + \underbrace{\xi}_{\text{'heat'}}$

Q: Is there a local entropy function $S$ onto which the inaccessible degrees of freedom project? i.e. $S,T$ such that $\xi = TdS$ with $T := \partial U/\partial S$?

A: Yes, plausibly.

Carathéodory's Principle → yes (Carathéodory 1909, Bernstein 1960)
If work is conserved on closed adiabats → yes (Jauch 1972)

Projected degrees of freedom

Molecular Degrees of Freedom ($\mathbb{R}^{23}$) Thermodynamic Entropy $S$

Quantum gravity degrees of freedom Black hole area $A$

Summary

At its core, equililbrium thermodynamics requires:

Contributions to energy are 'accessible' (work) or 'inaccessible' (heat)
Missing degrees of freedom characterise the heat contribution.
An entropy function onto which those degrees of freedom project.

Thermodynamics does not necessarily require:

Macroscopic behaviour as opposed to microscopic,
Equilibriation processes described as dynamical trajectories,
The global increase of entropy to the future and not the past.

Philosophers on models

Thermodynamics

Geometric thermodynamics

Black holes

Geometric thermodynamics

Thermodynamics as differential geometry, developed by Hermann (1973), Mrugała (1978), Arnol'd (1990) and others
Not the same as Lieb and Yngvason (1999) but related
Originates in Gibbs (1873) "Graphical methods in thermodynamics"

Geometric thermodynamics

"After much reading on the subject, I would say that Gibbs understood, in about 1870, the mathematics of thermodynamics – even in its most 'modern' form – better than almost all of the authors who followed him. Unfortunately, physicists and chemists think that a reasonable statement of the mathematics which they are trying to apply is an 'axiomatization,' and therefore should be avoided as a bad thing which inhibits the creative mind." (Hermann 1973, p.261-262)

Contact manifolds

A contact manifold is a pair $(M,H)$:

$M$ a manifold, $\mathrm{dim}\,M=2n+1$
$H$ a field of hypersurfaces, given by a one-form $\eta$ with $(d\eta)^n\wedge\eta \neq 0$

'Odd dimensional analogue' of symplectic geometry

"The relations between symplectic and contact geometries are similar to those between linear algebra and projective geometry. First, the two related theories are formally more or less equivalent.... Next, all the calculations look algebraically simpler in the symplectic case, but geometrically things are usually better understood when translated into the language of contact geometry. Hence one is advised to calculate symplectically but to think rather in contact geometry terms." Arnol'd (1989, p.5)

Contact manifolds

Example: $M=\mathbb{R}^{2n}$ with $2n+1$ coordinate functions $(P_i, X_i)$ and $U$ with $P_i = \partial U/\partial X_i$
Contact structure: 'Gibbs one-form' $dU - \sum_iP_idX_i - TdS$

Legendre submanifold: surface $N$ (of dim $n$) where $\theta(v)=0$, i.e. $dU = \sum_iP_idX_i + TdS$

Thermodynamic systems

Definition. An equilibrium thermodynamic system is a triple $(M,\theta,N)$, where $M$ is a contact manifold, $\theta$ is a contact structure, and $N$ is an $n$-dimensional Legendre submanifold.

'Fundamental relation': $f:N\rightarrow\mathbb{R}$ such that $U = f(X_1,\dots,X_n)$
'Intensive' variables: $P_i = \partial U/\partial X_i$
Recover first law: $dU = P_1dX_1 + \cdots + P_ndX_n$

Thermodynamic prediction

The single, all-encompassing problem of thermodynamics is the determination of the equilibrium state that eventually results after the removal of internal constraints in a closed, composite system. Callen (1985, p.26)

Thermodynamic prediction

Components systems $(M_1,\theta_1,N_1)$ and $(M_2,\theta_2,N_2)$
Composite system $(M,\theta,N)$
An embedding of the former into the latter.
Thermodynamic prediction: the science of how the
components constrain the composite (cf. Hermann 1973, §6.7)

Philosophers on models

Thermodynamics

Geometric thermodynamics

Black holes

$\delta M= \tfrac{1}{8\pi}\kappa \delta A+\Omega \delta J$

A contact manifold is coordinatised by the parameters $M,\tfrac{1}{2\pi}\kappa,\tfrac{1}{4}A,\Omega, J$
Contact form: $\theta = dM - \tfrac{1}{8\pi}\kappa dA - \omega dJ$
First law holds on (Legendre) submanifolds $N$ where $\theta=0$.
Projective interpretation: Classical area $A$ is the projection of a large number of inaccessible degrees of freedom from a more complete theory.
This is a basic model of an equilibrium thermodynamic system.

Black holes

$\delta M= \tfrac{1}{8\pi}\kappa \delta A+\Omega \delta J$

Projective interpretation: Classical area $A$ is the projection of a large number of inaccessible degrees of freedom from a more complete theory.

"We take the area of a black hole as a measure of its entropy — entropy in the sense of inaccessibility of information about its internal configuration" Bekenstein (1972, 1973)

Classical black holes do not emit particles (they are at 'absolute zero'), but the projective interpretation is already in some sense semi-classical.

Second law

Standard Clausius form

Fact. Given a (not necessarily exact) 'heat' one-form $\text{đ}Q$, a smooth 'temperature' function $T$, and a smooth adiabatic ($\text{đ}Q(\bar{\gamma})=0$) curve $\gamma$ from $p_i$ to $p_f$ on a manifold $N$, suppose that:

$\int_c\text{đ}Q/T\leq0$ holds on all closed curves through $N$
there exists a curve $\gamma'$ from $p_f$ to $p_i$ along which $d(\text{đ}Q)=0$.

Then there is a function $S$ such that $\text{đ}Q = TdS$, and any such function must satisfy $S(p_i)\leq S(p_f)$.

Discussion of problems: Uffink (2001) "Bluff your way"

Second law

Standard Clausius form

Only holds on abiabats, unlike the area theorems.
Views dynamics as curves through thermodynamic state space, unlike in black hole thermodynamics.

Second law

Gibbs-Wightman form

Fact. Let $U = f(S, X_1, \dots, X_{n−1})$ be a smooth function such that,

(positive temperature) $T = \partial U/\partial S > 0$;
(homogeneity) $U(X) = f(S, X_1, ... , X_{n−1})$ is 1st-degree homogeneous;
(strong stability) $U = f(S, X_1, ..., X_{n−1})$ is convex.

Then $S(X) = S(U,X_1,...,X_{n-1})$ is concave, and for any fixed value of $X$,

$S(X) = \text{sup}\,\{S(\bar{X}) + S(\tilde{X})\}$

where the supremum ranges over all points $\bar{X},\tilde{X}$ such that $X = \bar{X} + \tilde{X}$.

(for more see BWR 2022 book, Chapter 6)

Second law

Gibbs-Wightman form

Black holes are not 1st degree homogeneous, but quasi-homogeneous. (Belgiorno 2003)
Question: Can a similar second law be established in this context?

Further curiosities

Transitivity of equilibrium does not take its ordinary form. (Uffink)
The second law afforded by the area theorem holds in a stronger sense than is familiar. (Uffink)
Thermal 'contact' is hard to make sense of for black holes. (Uffink)
Negative heat capacities arise in black hole thermodynamics. (Uffink)
Not about micro vs macro distinctions in any simple sense. (Uffink)
None of these challenge the basic thermodynamic model of black holes, but are indicators of the special nature of black thermodynamics.

Thank you

🌻🌻🌻

Appendix

Affanasjewa-Jauch Theorem

Theorem. Let $(U,X_1,\dots,X_{n-1})$ be a complete set of smooth coordinate functions of a manifold $N$ of dimension $n$, and let $\xi$ be a one-form on $N$, which implies that $dU=\xi+\sum_{i=1}^{n-1}P_idX_i$ for some smooth functions $(P_1,\dots,P_n)$ of $N$. Suppose that $W:=\sum_{i=1}^{n-1}P_idX_i$ is 'conserved on adiabats', in that for every closed, piecewise-smooth curve $\gamma$ with tangent vector field $\bar{\gamma}$ satisfying $\xi(\bar{\gamma})=0$, we have,

$\int_\gamma W = 0$

Then there exist smooth functions $S:N\rightarrow\mathbb{R}$ and $T:N\rightarrow\mathbb{R}$ such that $\xi = TdS$ and $T=\partial U/\partial S$.

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