First law (Iyer-Wald)

Lagrangian \(L(g_{\mu\nu}, R, \nabla R, \ldots, \varphi, \nabla \varphi, \ldots)\) on spacetime.
For stationary axisymmetric black hole solution \[ T_H \delta S = \delta E_{can} - \Omega_H \delta J_{can} \]

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

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

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

Problem 1: smooth tensor fields

In general, gauge fields \(A_\mu^I\) cannot be chosen to be smooth everywhere

• E.g. magnetic monopole in Electrodynamics (Dirac string singularity)

would be nice to have a first law without gauge-fixing

Problem 2: diffeomorphisms

Charged fields have internal gauge transformations \(g \in G\)

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

Goal

Derive the First Law of Black Hole Mechanics for Einstein-Yang-Mills

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

Solution: work on Principal Bundle

• \(\pi: P \to M\); \(\pi^{-1}(x) \cong G\)
• all fields smooth on \(P\); gauge fields ≡ connection
• \(f: P \to P\) automorphism of \(P\) ≡ combined diffeo & gauge
• stationary \(£_X \psi = 0\) where \(X \in TP\) and \(\pi_*X = t\)
• apply Iyer-Wald procedure but on \(P\)

First law — Einstein-Yang Mills

Gauge field as connection \(A_\mu^I\) on bundle with \(L = L_{EH} + \star F \wedge F + \theta ( F\wedge F ) \)

\[ T_H\delta S + (\mathscr V^\Lambda \delta\mathscr Q_\Lambda)\vert_B = \delta E_{can} - \Omega_H \delta J_{can} \]

• \(\delta E_{can} = \delta M_{ADM} + (\mathscr V^\Lambda \delta\mathscr Q_\Lambda)\vert_\infty\)
• \(\mathscr V^\Lambda\) potentials; depend explicity only on connection \(A_\mu^I\)
• charges \(\mathscr Q_\Lambda\) depend only on \(\delta L / \delta {F_{\mu\nu}^I}\)

Yang-Mills charges

• \(\mathscr Q_\Lambda = \int * F_I h^I_\Lambda\) and \(\tilde{\mathscr Q}_\Lambda = \theta \int F_I h^I_\Lambda\) electric and magnetic charges
• e.g. \(n\) independent charges for \(U(1)^n\) or \(SU(n+1)\)
• Sudarsky-Wald get zero potential at horizon due to 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. Compute \(\mathscr V^\Lambda\) for spin connection for any covariant Lagrangian.

• 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 Einstein-Hilbert Lagrangian: \(\mathscr Q_{grav} = \tfrac{1}{8\pi}{\rm Area}\)

Einstein-Dirac

For spinor fields \(\Psi\) with Dirac Lagrangian on bundle

• no contribution at the horizon
• no contribution at infinity due to fall-off conditions
• usual form of first law!

Not Covered

• \(p\)-form gauge fields with magnetic charge
• Chern-Simons Lagrangians (coming soon)