The first law of
black hole mechanics
for fields with
internal gauge freedom
(arXiv:1511.00388)
4 laws of thermodynamics
 Temperature \(T\) is constant in thermal equilibrium
 \( T\delta S = \delta E + work \); \(S\) is entropy
 entropy \(S\) always increases
 \(T \to 0\) not possible in “finite” number of “physical steps”
4 laws of black hole mechanics
 surface gravity \(\kappa\) is constant on the horizon of a stationary black hole
 \( T_H \delta S = \delta E_{can}  \Omega_H \delta J_{can} \) where \(T_H = \frac{\kappa}{2\pi}\)
(holds in any diffeomorphism covariant theory of gravity!)  entropy \(S\) always increases (holds in General Relativity)
 \(\kappa \to 0\) 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 (IyerWald, 1994)
Any diffeocovariant theory of gravity described by smooth tensor fields \(\psi = (g_{\mu\nu}, \varphi_{\mu\ldots}{}^{\nu\ldots})\) on spacetime
For stationaryaxisymmetric 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 (IyerWald): Lagrangian
 Dynamical fields are smooth tensors \(\psi = (g_{\mu\nu}, \varphi_{\mu\ldots}{}^{\nu\ldots})\)
 covariant Lagrangian \(L(g, R, \nabla R, \ldots, \varphi, \nabla \varphi, \ldots)\) is a \(4\)form on spacetime
 vary the Lagrangian, pretend to integratebyparts \[ \delta L = (EOM) \delta \psi + d\theta[\delta \psi] \] don’t throw away the boundary term \(\theta\)
 Compute \( \omega (\delta_1\psi, \delta_2\psi) = \delta_1 \theta(\delta_2\psi)  \delta_2 \theta(\delta_1\psi) \)
pick a stationaryaxisymmetric black hole solution
First law (IyerWald): 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 (IyerWald): Lagrangian
 \(\int_\Sigma \omega (\delta\psi, £_K\psi) = 0 \)
 \(\int_B (\delta Q_K ) = \int_\infty (\delta Q_K  K \cdot \theta )\)
\(Q_K\) 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 (IyerWald): 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
Derive the First Law of Black Hole Mechanics for EinsteinYangMills Tetrad GR, EinsteinDirac, all of Standard Model.
 also supersymmetry, Lovelock, \(B\)\(F\) gravity, higher derivative gravity, arbitrary charged tensorspinor 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
IyerWald assume all dynamical fields \(\psi\)
 are smooth tensor fields on spacetime
 have a welldefined group action of diffeomorphisms e.g. to decide stationarity \(£_t \psi = 0\)
Problem 1: smooth tensor fields — Dirac monopole
 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 welldefined
 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
 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
 Define everything — fields \(\Psi, A^I_m\), Lagrangian \(L\) on a principal bundle. (no need to make any gauge choice)
 Use IyerWald procedure on the bundle \(P\) instead of spacetime \(M\)
IyerWald on Principal Bundle
 Dynamical fields are smooth tensors \(\psi = (g_{mn}, \Psi, A^I_m )\) on \(P\)
 gaugeinvariant Lagrangian on \(P\) \(L(g_{mn}, R, \nabla R, \ldots, \Psi, \nabla \Psi, \ldots, A, F, \nabla F \ldots)\)
 \(\delta L = (EOM)\delta\psi + d\theta(\delta \psi)\) (\(\theta\) is gaugeinvariant)
 \( \omega (\delta_1\psi, \delta_2\psi) = \delta_1 \theta(\delta_2\psi)  \delta_2 \theta(\delta_1\psi) \)
 \(\int_\Sigma \omega (\delta\psi, £_K\psi) = 0 \)
(\(K^m\) on the bundle which projects to \(K^\mu\) on spacetime)  \(\int_B (\delta Q_K ) = \int_\infty (\delta Q_K  K \cdot \theta )\) (\(Q_K\) is gaugeinvariant)
First law — EinsteinYangMills
Dynamical fields: \(g_{\mu\nu}\) for gravity and connection \(A_m^I\) on bundle for YangMills; 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 IyerWald procedure to get first law
\[ T_H\delta S + (\mathscr V^\Lambda \delta\mathscr Q_\Lambda)\vert_B = \delta E_{can}  \Omega_H \delta J_{can} \] \(\delta E_{can} = \delta M_{ADM} + (\mathscr V^\Lambda \delta\mathscr Q_\Lambda)\vert_\infty\)
 \(\delta J_{can} = \delta J_{ADM}  \tfrac{1}{2g^2} \int_\infty (\phi^m A_m^I) (\star F_I) \)
YangMills 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
 SudarskyWald (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 YangMills)
 Only one nonzero 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: \(\mathscr Q_{grav} = \tfrac{1}{8\pi}{\rm Area}(B)\)
EinsteinDirac
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 falloff conditions
 usual form of first law!
Not Covered
 \(p\)form gauge fields \(A_{\mu\nu\ldots}\) with magnetic charge
 ChernSimons Lagrangians — since Lagrangian is not gaugeinvariant (coming soon)