4 laws of thermodynamics

0. Temperature \(T\) is constant in thermal equilibrium
1. \( T\delta S = \delta E + work \); \(S\) is entropy
2. entropy \(S\) always increases
3. \(T \to 0\) not possible in “finite” number of “physical steps”

4 laws of black hole mechanics

0. surface gravity \(\kappa\) is constant on the horizon of a stationary black hole
1. \( T_H \delta S = \delta E_{can} - \Omega_H \delta J_{can} \) where \(T_H = \frac{\kappa}{2\pi}\)
   (holds in any diffeomorphism covariant theory of gravity!)
2. entropy \(S\) always increases (holds in General Relativity)
3. \(\kappa \to 0\) not possible in “finite” number of “physical steps”

Diffeomorphism

diffeomorphism ≡ moving points or moving coordinates

Diffeomorphism Covariance

Physics does not depend on how we want to move points around/choose coordinates on spacetime

Coordinates are like deciding to use red chalk or blue chalk. Who cares! — Bob Geroch

Diffeomorphism Covariance: tensor fields

• tensor fields \(\varphi_{\mu\ldots}{}^{\nu\ldots}\) defined on spacetime know how to change under diffeomorphisms
• small diffeo given by a vector field \(X^\mu\) then \[ \delta_X \varphi_{\mu\ldots}{}^{\nu\ldots} = £_X \varphi_{\mu\ldots}{}^{\nu\ldots} \] where \(£_X\) is the Lie derivative

First law (Iyer-Wald, 1994)

Any diffeo-covariant theory of gravity described by smooth tensor fields \(\psi = (g_{\mu\nu}, \varphi_{\mu\ldots}{}^{\nu\ldots})\) on spacetime

For stationary-axisymmetric black hole solution \[ T_H \delta S = \delta E_{can} - \Omega_H \delta J_{can} \]

• \(T_H = \kappa/2\pi\) where \(\kappa\) is the surface gravity
• \(E_{can}\) canonical energy; \(J_{can}\) angular momenta at spatial infinity
• \(S\) depends only on \(\delta L/\delta R_{\mu\nu\lambda}{}^\rho\) (Wald entropy formula)

First law (Iyer-Wald): Lagrangian

1. Dynamical fields are smooth tensors \(\psi = (g_{\mu\nu}, \varphi_{\mu\ldots}{}^{\nu\ldots})\)
2. covariant Lagrangian \(L(g, R, \nabla R, \ldots, \varphi, \nabla \varphi, \ldots)\) is a \(4\)-form on spacetime
3. vary the Lagrangian, pretend to integrate-by-parts \[ \delta L = (EOM) \delta \psi + d\theta[\delta \psi] \] don’t throw away the boundary term \(\theta\)
4. Compute \( \omega (\delta_1\psi, \delta_2\psi) = \delta_1 \theta(\delta_2\psi) - \delta_2 \theta(\delta_1\psi) \)

pick a stationary-axisymmetric black hole solution

First law (Iyer-Wald): black hole

• Killing fields \(t^\mu\), \(\phi^\mu\)
• \(£_t \psi = 0 = £_\phi \psi\)
• \(K^\mu = t^\mu + \Omega_H \phi^\mu \)
  horizon Killing field
• \(K^\mu\vert_B = 0\)
• \(\Sigma\) is a Cauchy surface ≡ “now”

First law (Iyer-Wald): Lagrangian

5. \(\int_\Sigma \omega (\delta\psi, £_K\psi) = 0 \)
6. \(\int_B (\delta Q_K ) = \int_\infty (\delta Q_K - K \cdot \theta )\)
   \(Q_K\) is the Noether charge of \(K^\mu\) (Noether's theorem)

\[ T_H \delta S = \delta E_{can} - \Omega_H \delta J_{can} \] \[ T_H = \frac{\kappa}{2\pi};\quad S \sim \int _B \frac{\delta L}{\delta R_{\mu\nu\lambda}{}^\rho} \]

First law (Iyer-Wald): General Relativity

• Dynamical field is metric \(\psi = (g_{\mu\nu})\)
• Lagrangian \(L_{EH} = R \varepsilon \)
• \(T_H = \frac{\kappa}{2\pi}\); \(S = \frac{1}{4}Area(B)\)
• \(E_{can} = M_{ADM}\); \(J_{can} = J_{ADM}\) (Arnowitt, Deser, Misner in 1959–61)

Goal

• Tetrad GR, Einstein-Dirac, all of Standard Model.
• also supersymmetry, Lovelock, \(B\)-\(F\) gravity, higher derivative gravity, arbitrary charged tensor-spinor fields, …

Internal gauge freedom (“toy” electron)

A “toy” electron with electric charge described by some field \(\Psi(x)\)

• \(\Psi(x) \mapsto \Psi(x) e^{i \alpha(x)} \) Show
• can pick \(\alpha(x)\) at every point Show
• redundant description “gauge freedom”

Internal gauge freedom (Standard Model)

For Standard Model: \[ G = U(1) \times SU(2) \times SU(3) \] corresponding to electromagnetism, weak force and strong force.

• \(\Psi \equiv \) electron, neutrino, quarks, Higgs …
• \(A_\mu^I \equiv\) photon, \(W^{\pm}\), \(Z\), gluons

Internal gauge freedom (general)

In general: \(g(x) \in G\) \[ \Psi(x) \mapsto g^{-1}(x) \Psi(x) \] \[ A_\mu^I(x) \mapsto g^{-1}(x) A_\mu^I(x) g(x) + g^{-1}(x) \nabla_\mu g(x) \]

Gauge Covariance

Physics does not depend on how we want to choose a gauge

[…] Who cares! — Bob Geroch

Iyer-Wald assume all dynamical fields \(\psi\)

1. are smooth tensor fields on spacetime
2. have a well-defined group action of diffeomorphisms e.g. to decide stationarity \(£_t \psi = 0\)

Problem 1: smooth tensor fields — Dirac monopole

\[ A_\mu^{(N)} = - (\cos \theta - 1) \nabla_\mu\phi \] \[ A_\mu^{(S)} = - (\cos \theta + 1) \nabla_\mu\phi \] \[ A_\mu^{(N)} - A_\mu^{(S)} = - i e^{-i 2\phi} \nabla_\mu e^{i 2\phi} \]
• not possible to write one smooth gauge field \(A_\mu\)
• Paul Dirac in 1931

Problem 1: smooth tensor fields

In general, cannot make a gauge choice so that gauge fields \(A_\mu^I\) are smooth everywhere

physics i.e. First law should not care about this!

Problem 2: diffeomorphisms (“toy” electron)

• \(\Psi(x)\) does not know how to tranform under diffeos
• \(£_X \Psi\) is ambiguous due to gauge freedom

Problem 2: diffeomorphisms

• just a gauge transformation at fixed \(x\) is well-defined
• but diffeomorphism is only defined up to an arbitrary gauge!
• e.g. stationarity \(£_t \psi = gauge \)

Solution: work on Principal Bundle

Solution: Principal Bundle — automorphisms

First law — Principal Bundle

1. Define everything — fields \(\Psi, A^I_m\), Lagrangian \(L\) on a principal bundle. (no need to make any gauge choice)
2. Use Iyer-Wald procedure on the bundle \(P\) instead of spacetime \(M\)

Iyer-Wald on Principal Bundle

1. Dynamical fields are smooth tensors \(\psi = (g_{mn}, \Psi, A^I_m )\) on \(P\)
2. gauge-invariant Lagrangian on \(P\) \(L(g_{mn}, R, \nabla R, \ldots, \Psi, \nabla \Psi, \ldots, A, F, \nabla F \ldots)\)
3. \(\delta L = (EOM)\delta\psi + d\theta(\delta \psi)\) (\(\theta\) is gauge-invariant)
4. \( \omega (\delta_1\psi, \delta_2\psi) = \delta_1 \theta(\delta_2\psi) - \delta_2 \theta(\delta_1\psi) \)
5. \(\int_\Sigma \omega (\delta\psi, £_K\psi) = 0 \)
   (\(K^m\) on the bundle which projects to \(K^\mu\) on spacetime)
6. \(\int_B (\delta Q_K ) = \int_\infty (\delta Q_K - K \cdot \theta )\) (\(Q_K\) is gauge-invariant)

First law — Einstein-Yang-Mills

Dynamical fields: \(g_{\mu\nu}\) for gravity and connection \(A_m^I\) on bundle for Yang-Mills; curvature \(F^I = DA^I\)

\(L = R\varepsilon + \tfrac{1}{4g^2}(\star F^I) \wedge F_I + \vartheta ( F^I\wedge F_I ) \)

plug into Iyer-Wald procedure to get first law

• \(\delta E_{can} = \delta M_{ADM} + (\mathscr V^\Lambda \delta\mathscr Q_\Lambda)\vert_\infty\)
• \(\delta J_{can} = \delta J_{ADM} - \tfrac{1}{2g^2} \int_\infty (\phi^m A_m^I) (\star F_I) \)

Yang-Mills potentials & charges

• \(\mathscr V^\Lambda\) potentials; depend explicity only on connection \(A_m^I\) and are constant on the horizon (zeroth law)
• e.g. \(n\) independent potentials for \(U(1)^n\) or \(SU(n+1)\) (dimension of Cartan subalgebra)
• charges \(\mathscr Q_\Lambda \sim \int_B \delta L / \delta {F_{mn}^I}\) (just like Wald entropy formula)
• \(\mathscr Q_\Lambda = \tfrac{1}{2g^2}\int \star F_I h^I_\Lambda\) and \(\tilde{\mathscr Q}_\Lambda = \vartheta \int F_I h^I_\Lambda\) electric and magnetic charges
• Sudarsky-Wald (1992) get zero potential at horizon — assuming \(£_t A = 0\), in general horizon potential is not zero.
• Magnetic charge is topological and does not contribute to first law

Temperature & Entropy

• Only one non-zero potential ≡ boosts along the horizon
• \(\mathscr V_{grav} = \kappa \implies T_H = \tfrac{1}{2\pi}\mathscr V_{grav}\)
• and perturbed entropy \(\delta S = 2\pi\delta \mathscr Q_{grav}\)
• for General Relativity: \(\mathscr Q_{grav} = \tfrac{1}{8\pi}{\rm Area}(B)\)

Einstein-Dirac

• no contribution at the horizon (only connections \(A_m^I\) contribute)
• no contribution at infinity due to fall-off conditions
• usual form of first law!

Not Covered

• \(p\)-form gauge fields \(A_{\mu\nu\ldots}\) with magnetic charge
• Chern-Simons Lagrangians — since Lagrangian is not gauge-invariant (coming soon)