# First Law

for fields with

Internal Gauge

Evanston, IL

MWRM2015, (coming soon to an arXiv near you!)
# 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 \(g \in G\)

\[
\Psi(x) \mapsto g^{-1}(x) \Psi(x)
\]
\[
A(x) \mapsto g^{-1}(x) A(x) g(x) + g^{-1}(x) d g(x)
\]

- 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)