Simpler version

First Law
for fields with

Kartik PrabhuAPS2016, Salt Lake City

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 \(£_t \psi = gauge \)


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: \(\mathscr Q_{grav} = \tfrac{1}{8\pi}{\rm Area}\)


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)