Simpler version

# The first law ofblack hole mechanicsfor fields with internal gauge freedom

Kartik Prabhu (arXiv:1511.00388)

# 4 laws of thermodynamics

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

# 4 laws of black hole mechanics

1. surface gravity $$\kappa$$ is constant on the horizon of a stationary black hole
2. $$T_H \delta S = \delta E_{can} - \Omega_H \delta J_{can}$$ where $$T_H = \frac{\kappa}{2\pi}$$
(holds in any diffeomorphism covariant theory of gravity!)
3. entropy $$S$$ always increases (holds in General Relativity)
4. $$\kappa \to 0$$ 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

1. $$\int_\Sigma \omega (\delta\psi, £_K\psi) = 0$$
2. $$\int_B (\delta Q_K ) = \int_\infty (\delta Q_K - K \cdot \theta )$$
$$Q_K$$ 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)
$T_H \delta S = \delta E_{can} - \Omega_H \delta J_{can}$

# Goal

Derive the First Law of Black Hole Mechanics for Einstein-Yang-Mills
• 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 $$\Psi(x)$$

• $$\Psi(x) \mapsto \Psi(x) e^{i \alpha(x)}$$
• can pick $$\alpha(x)$$ at every point
• 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

# 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

$\Psi(x) \mapsto g^{-1}(x) \Psi(x)$ $A_\mu^I(x) \mapsto g^{-1}(x) A_\mu^I(x) g(x) + g^{-1}(x) \nabla_\mu g(x)$
• 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

$$\pi: P \to M$$; $$\pi^{-1}(x) \cong G$$

all fields smooth on $$P$$
• $$\Psi$$ are smooth equivariant tensor fields
• gauge field $$A_\mu^I$$ is a smooth connection $$A_m^I$$

# Example: Dirac monopole — circles on a circle

• Standard torus: unlinked circle, no magentic monople
• “twisted” torus: circles linked once, unit monopole

# Example: Dirac monopole — Hopf fibration

• $$G = S^1 \cong U(1)$$
• $$M = S^2$$
• $$P = S^3$$

Heinz Hopf in 1931

# Solution: Principal Bundle — automorphisms

automorphism $$f: P \to P$$
• vertical $$f$$ ≡ gauge transformation
• general $$f$$ ≡ diffeo + gauge
• All fields $$\Psi$$, $$A$$ know how to transform under $$f$$
• stationary $$£_X \psi = 0$$ where $$X \in TP$$ and $$\pi_*X = t$$

# 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)
$T_H \delta S + \left.\mathscr{V}^\Lambda \delta\mathscr{Q}_\Lambda\right\vert_B = \delta E_{can} - \Omega_H \delta J_{can}$

# 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

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

# 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

Gravity: tetrads $$e_\mu^a$$ and spin connection $${\omega_\mu}^a{}_b$$ on the bundle with $$G = SO(1,3)$$
Compute $$\mathscr V^\Lambda$$ for spin connection for any covariant Lagrangian. (just like Yang-Mills)
• 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

For spinor fields $$\Psi$$ with Dirac Lagrangian on bundle
• 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)