Simpler version

The first law of
black hole mechanics
for 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 ≡ 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)
\[ T_H \delta S = \delta E_{can} - \Omega_H \delta J_{can} \]


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

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

\[ \Psi(x) \mapsto g^{-1}(x) \Psi(x) \] \[ A_\mu^I(x) \mapsto g^{-1}(x) A_\mu^I(x) g(x) + g^{-1}(x) \nabla_\mu g(x) \]
  • 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: \(\mathscr Q_{grav} = \tfrac{1}{8\pi}{\rm Area}(B)\)


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)