Dynamical entropy of charged black objects

Abstract

We develop a general framework for electromagnetic potential-charge contributions to the first law of black hole mechanics, applicable to dynamical first-order perturbations of stationary black objects with possibly non-compact bifurcate Killing horizons. Working in the covariant phase space formalism, we derive both comparison and physical process versions of the first law. We consider generic diffeomorphism-invariant theories of gravity in $D$ spacetime dimensions, containing non-minimally coupled abelian $p$-form gauge fields. The pullback of the gauge field to the horizon is allowed to diverge while its field strength remains smooth, yielding gauge-invariant electric potential-charge pairs in the first law. We further extend the construction to include magnetic charges by developing a bundle-covariant, gauge-invariant prescription that fixes the Jacobson-Kang-Myers ambiguity in the improved Noether charge. Electric and magnetic charges are, respectively, associated with non-trivial $(D - p - 1)$- and $(p + 1)$-cycles of the horizon cross-section, whose homology classes determine the number of independent potential-charge pairs through the Betti numbers $b_{D - p - 1}$ and $b_{p + 1}$. Further, the dynamical gravitational entropy entering the first law is identified with the gauge-invariant part of the improved Noether charge, giving a gauge-invariant extension of the recent proposal by Hollands, Wald and Zhang. We illustrate our framework with dyonic AdS black holes, dipole black rings, and charged black branes.

Figures (3)

• Figure 1: Bifurcate Killing horizon $\mathcal{H}$, consisting of a future horizon $\mathcal{H}^+$ at $u=0$ and a past horizon $\mathcal{H}^-$ at $v=0$ that intersect at the bifurcation surface $\mathcal{B}$. The Gaussian null coordinate system around $\mathcal{H}^+$ is $(v,u,x^i)$, where $v$ is the affine parameter on the null geodesics of $\mathcal{H}^+$, with $\mathcal{B}$ at $v=0$, $u$ is the affine null geodesic distance away from $\mathcal{H}^+$, and $x^i$ are the $(D-2)$ spatial coordinates on the cross-section of $\mathcal{H}^+$. This cross-section $\mathcal{C}$ is the product of a compact subspace $\tilde{\mathcal{C}}$ and a non-compact manifold $\Sigma_k$.
• Figure 5: Gauge field as a connection on a non-trivial principal bundle. $\mathbf A$ can only be expressed patch by patch, and on overlapping patches the connections are related by a gauge transformation.
• Figure 6: The relative cycle $\sigma'$ of the horizon segment $\mathcal{H}_{v_1}^{v_2}$ has the topology of $\sigma \times [0,1]$, and it has two boundaries $\sigma(v_1) \subset \tilde{\mathcal{C}}(v_1)$ and $\sigma(v_2) \subset \tilde{\mathcal{C}}(v_2)$.

Theorems & Definitions (8)

• Theorem 2.1
• Theorem 2.2
• proof
• Lemma A.1
• proof
• Corollary A.2
• Lemma A.3: Double null decomposition of Hodge dual
• proof