[Paper Review] Asymptotically flat black holes and gravitational waves in three-dimensional massive gravity

Cédric Troessaert, David Tempo+1 more|arXiv (Cornell University)|Dec 30, 2015

Black Holes and Theoretical Physics96 references3 citations

TL;DR

This paper constructs exact solutions for three-dimensional massive gravity with a purely quadratic Lagrangian, introducing two classes of asymptotically flat black holes: Kerr-Schild-type pp-waves and deformed 'black flowers' breaking spherical symmetry. The key result is that despite time-dependent gravitational radiation, the black flowers possess finite energy and obey the first law of thermodynamics via conserved BMS₃ charges without central charges.

ABSTRACT

Different classes of exact solutions for the BHT massive gravity theory are constructed and analyzed. We focus in the special case of the purely quadratic Lagrangian, whose field equations are irreducibly of fourth order and are known to admit asymptotically locally flat black holes endowed with gravitational hair. The first class corresponds to a Kerr-Schild deformation of Minkowski spacetime along a covariantly constant null vector. As in the case of General Relativity, the field equations linearize so that the solution can be easily shown to be described by four arbitrary functions of a single null coordinate. These solutions can be regarded as a new sort of pp-waves. The second class is obtained from a deformation of the static asymptotically locally flat black hole, that goes along the spacelike (angular) Killing vector. Remarkably, although the deformation is not of Kerr-Schild type, the field equations also linearize, and hence the generic solution can be readily integrated. It is neither static nor spherically symmetric, being described by two integration constants and two arbitrary functions of the angular coordinate. In the static case it describes "black flowers" whose event horizons break the spherical symmetry. The generic time-dependent solution appears to describe a graviton that moves away from a black flower. Despite the asymptotic behaviour of these solutions at null infinity is relaxed with respect to the one for General Relativity, the asymptotic symmetries coincide. However, the algebra of the conserved charges corresponds to BMS$_{3}$, but devoid of central extensions. The "dynamical black flowers" are shown to possess a finite energy. The surface integrals that define the global charges also turn out to be useful in the description of the thermodynamics of solutions with event horizons.

Motivation & Objective

• To construct exact solutions for three-dimensional massive gravity with a purely quadratic Lagrangian, focusing on asymptotically flat spacetimes.
• To analyze the field equations, which are fourth-order and linearize under specific deformations, enabling exact integration.
• To explore the physical properties of new black hole solutions, including their thermodynamics and conserved charges.
• To investigate the asymptotic structure and symmetry algebra, particularly the absence of central charges in the BMS₃ algebra.
• To establish a thermodynamic framework for black flowers using surface integrals and Wald’s entropy formula.

Proposed method

• Construct solutions via Kerr-Schild deformation of Minkowski spacetime along a covariantly constant null vector, leading to linearized field equations.
• Apply a deformation along the spacelike (angular) Killing vector of a static asymptotically locally flat black hole, which also leads to linearized field equations.
• Use the superpotential formalism to compute conserved charges from surface integrals, enabling evaluation of mass and entropy.
• Evaluate the conserved charges at both null infinity and the horizon to derive the first law of thermodynamics.
• Employ the Wald entropy formula with the binormal to the bifurcation surface to compute black flower entropy.
• Analyze the asymptotic symmetry algebra, showing it is BMS₃ without central extensions, despite relaxed asymptotic behavior.

Experimental results

Research questions

• RQ1 Can exact solutions be constructed in three-dimensional massive gravity with a purely quadratic Lagrangian that describe asymptotically flat black holes with gravitational hair?
• RQ2 Do the field equations linearize under Kerr-Schild and non-Kerr-Schild deformations, enabling exact integration?
• RQ3 What is the asymptotic symmetry algebra of these solutions, and how does it compare to General Relativity?
• RQ4 Do the time-dependent solutions, such as 'black flowers' emitting gravitons, possess finite energy and consistent thermodynamics?
• RQ5 Can the first law of thermodynamics be recovered for these black hole solutions using conserved surface charges?

Key findings

• The paper constructs two classes of exact solutions: pp-wave-like solutions from Kerr-Schild deformations and 'black flowers' from spacelike Killing vector deformations, both with linearized field equations.
• The generic solution for the Kerr-Schild-type deformation is described by four arbitrary functions of a single null coordinate, forming a new class of pp-waves.
• The black flower solutions break spherical symmetry and are described by two integration constants and two arbitrary functions of the angular coordinate.
• Despite time dependence and outgoing gravitational radiation, the total energy at null infinity remains constant, indicating no news in the Bondi sense.
• The conserved global charges are finite and match those of the static black hole, with mass $ M = rac{b^2}{32G} $ and angular momentum $ J = 0 $, even in the time-dependent case.
• The first law of thermodynamics is recovered via surface integrals: $ dM = TdS $, with $ ilde{k}_{ heta}^{[ur]} = rac{b heta b}{32 heta G} $, and entropy derived from Wald’s formula.