Simpler version

# First Lawfor fields with Internal Gauge

Kartik PrabhuMWRM2015, Evanston, IL
(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)