First law (Iyer-Wald)

Lagrangian \(L(g_{\mu\nu}, R, \nabla R, \ldots, H, \nabla H, \ldots)\) 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
• \(\delta S \) depends on \(\delta L/\delta {R_{\mu\nu\rho}}^\lambda\)

Goal

Derive the First Law of Black Hole Mechanics for Tetrad GR, Einstein-Yang-Mills, Einstein-Dirac

• covers Lovelock, \(B\)-\(F\) gravity, higher derivative gravity, arbitrary charged tensor-spinor fields, all of Standard Model…
• metric-affine, non-metricity, Poincaré gauge theory by simple extension

why not use Iyer-Wald?

Iyer-Wald assume dynamical fields

• are spacetime metric for gravity
  but need tetrads to define spinors!
• are smooth tensor fields on spacetime
• have a well-defined group action of diffeomorphisms to decide stationary and axisymmetric i.e. \(£_t \psi = 0\)

Problems (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)
• cannot choose a gauge smoothly everywhere!
• would be nice to have a first law without gauge-fixing

Problems (diffeomorphisms)

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

• just a gauge transformation is well-defined
• but diffeomorphism is only defined up to gauge!
• stationary and axisymmetric \(£_t \psi = gauge \)

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

covariant Lagrangian \(L(e^a, {A^a}_b, F, \nabla F, \ldots, T, \nabla T, \ldots)\) on bundle

\[ \mathscr V^\Lambda \delta\mathscr Q_\Lambda = \delta E_{can} - \Omega_H \delta J_{can} \]

• \(E_{can}\) canonical energy; \(J_{can}\) angular momenta at spatial infinity (compute for a given \(L\))
• \(\mathscr V^\Lambda\) potentials; depend explicity only on connection \({A_\mu^a}_b\)
• charges \(\mathscr Q_\Lambda\) depend only on \(\delta L / \delta {F_{\mu\nu}^a}_b\)

Temperature & Entropy

Compute \(\mathscr V^\Lambda\) for gravitational Lorentz (spin) connection \({A^a}_b\)

• 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 tetrad GR: \(\mathscr Q_{grav} = \tfrac{1}{8\pi}{\rm Area}\); \(E_{can} = M_{ADM}\) and \(J_{can} = J_{ADM}\)

Einstein-Yang-Mills

Yang-Mills connection \(A^I\) on bundle. Compute for Yang-Mills theory \(L = \star F \wedge F + () F\wedge F\)

• \(\mathscr Q_\Lambda = \int * F_I h^I_\Lambda\) and \(\tilde{\mathscr Q}_\Lambda = \int F_I h^I_\Lambda\)
• \(E_{can}\) similar term at infinity
• 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

Einstein-Dirac

For Dirac spinor fields \(\Psi\) 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)