[Paper Review] Angular momentum and conservation laws for dynamical black holes

Sean A. Hayward|ArXiv.org|Jan 15, 2006

Experimental and Theoretical Physics Studies3 citations

TL;DR

This paper formulates a consistent dynamical framework for black hole thermodynamics using trapping horizons, defining unique angular momentum via a Komar integral with a conservation law. It derives a first law for dynamical black holes involving energy, angular momentum, and charge fluxes, generalizing the classical Kerr-Newman laws to non-stationary, evolving horizons with gravitational radiation encoded in shear tensors.

ABSTRACT

Black holes can be practically located (e.g. in numerical simulations) by trapping horizons, hypersurfaces foliated by marginal surfaces, and one desires physically sound measures of their mass and angular momentum. A generically unique angular momentum can be obtained from the Komar integral by demanding that it satisfy a simple conservation law. With the irreducible (Hawking) mass as the measure of energy, the conservation laws of energy and angular momentum take a similar form, expressing the rate of change of mass and angular momentum of a black hole in terms of fluxes of energy and angular momentum, obtained from the matter energy tensor and an effective energy tensor for gravitational radiation. Adding charge conservation for generality, one can use Kerr-Newman formulas to define combined energy, surface gravity, angular speed and electric potential, and derive a dynamical version of the so-called "first law" for black holes. A generalization of the "zeroth law" to local equilibrium follows. Combined with an existing version of the "second law", all the key quantities and laws of the classical paradigm for black holes (in terms of Killing or event horizons) have now been formulated coherently in a general dynamical paradigm in terms of trapping horizons.

Motivation & Objective

• To establish a physically sound, unique measure of angular momentum for dynamical black holes in non-stationary spacetimes.
• To derive a conservation law for angular momentum analogous to energy conservation, incorporating matter and gravitational radiation fluxes.
• To generalize the first and zeroth laws of black hole mechanics to dynamical, trapping horizon-based black holes without requiring Killing vectors.
• To unify energy, angular momentum, and charge conservation in a quasi-local framework using effective energy tensors for gravitational radiation.
• To define thermodynamic quantities (energy, surface gravity, angular velocity, electric potential) via Kerr-Newman formulas adapted to marginal surfaces on trapping horizons.

Proposed method

• Uses the Komar integral with a twist form to define angular momentum, ensuring uniqueness via a transverse divergence-free axial vector field on marginal surfaces.
• Imposes a Lie derivative condition $ L_{\xi}\psi^{a} = 0 $ along a foliating vector field $ \xi^{a} $, ensuring consistency with conservation laws.
• Derives a conservation law for angular momentum: $ L_{\xi}J = -\oint_{S} {*} (T_{ab} + \Theta_{ab}) \psi^{a} \tau^{b} $, where $ \Theta_{ab} $ is an effective energy tensor for gravitational radiation.
• Defines the effective energy tensor $ \Theta_{ab} $ from the shear $ \sigma_{\pm ij} $ of the marginal surfaces, with $ \Theta_{i\pm} = -\frac{1}{16\pi} h^{jk} D_k \sigma_{\pm ij} $.
• Constructs a generalized energy $ E $ from the Kerr-Newman formula, using irreducible mass $ M $, angular momentum $ J $, and charge $ Q $, to define thermodynamic potentials.
• Establishes a dynamical first law: $ L_{\xi}E = \frac{\kappa}{8\pi} L_{\xi}A + \Omega L_{\xi}J + \Phi L_{\xi}Q $, with $ \kappa, \Omega, \Phi $ defined via thermodynamic derivatives of $ E $.

Experimental results

Research questions

• RQ1 How can angular momentum be uniquely defined for a dynamical black hole without relying on Killing symmetries?
• RQ2 What is the form of the angular momentum conservation law in the presence of gravitational radiation and matter fluxes?
• RQ3 Can the first law of black hole mechanics be generalized to dynamical, trapping horizon-based black holes?
• RQ4 How is gravitational radiation encoded in the geometry of a trapping horizon?
• RQ5 What thermodynamic quantities (energy, surface gravity, angular velocity, electric potential) can be consistently defined on a dynamical horizon?

Key findings

• A unique angular momentum $ J $ is defined via the Komar integral with a transverse, divergence-free axial vector field $ \psi^{a} $, fixed by geometric constraints on the marginal surface.
• The conservation law $ L_{\xi}J = -\oint_{S} {*} (T_{ab} + \Theta_{ab}) \psi^{a} \tau^{b} $ holds for trapping horizons and uniformly expanding flows, with $ \Theta_{ab} $ encoding gravitational radiation flux.
• The effective energy tensor $ \Theta_{ab} $ is derived from the shear $ \sigma_{\pm ij} $, and its components $ \Theta_{\pm\pm} = ||\sigma_{\pm}||^2 / 32\pi $ recover known results for Bondi energy flux and Isaacson radiation density.
• A generalized energy $ E $ is defined via the Kerr-Newman formula, with $ E \geq M $, including contributions from irreducible mass, rotational kinetic energy, and electrostatic energy.
• The dynamical first law $ L_{\xi}E = \frac{\kappa}{8\pi} L_{\xi}A + \Omega L_{\xi}J + \Phi L_{\xi}Q $ is derived, with thermodynamic potentials $ \kappa, \Omega, \Phi $ defined as derivatives of $ E $, generalizing the classical first law.
• Local equilibrium corresponds to $ (j_M, j_J, j_Q)^a \tau_a = 0 $, implying constant $ (M, J, Q) $ and constant surface gravity $ \kappa $, generalizing the zeroth law to dynamical horizons.