Simpler version

# First Lawfor fields with Internal Gauge

Kartik PrabhuAPS2016, Salt Lake City
(arXiv:1511.00388)

# 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 $$g \in G$$

$\Psi(x) \mapsto g^{-1}(x) \Psi(x)$ $A(x) \mapsto g^{-1}(x) A(x) g(x) + g^{-1}(x) d g(x)$
• just a gauge transformation 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)