Unified first law of black-hole dynamics and relativistic thermodynamics

A unified first law of black-hole dynamics and relativistic thermodynamics is derived in spherically symmetric general relativity. This equation expresses the gradient of the active gravitational energy E according to the Einstein equation, divided into energy-supply and work terms. Projecting the equation along the flow of thermodynamic matter and along the trapping horizon of a blackhole yield, respectively, first laws of relativistic thermodynamics and black-hole dynamics. In the black-hole case, this first law has the same form as the first law of black-hole statics, with static perturbations replaced by the derivative along the horizon. There is the expected term involving the area and surface gravity, where the dynamic surface gravity is defined as in the static case but using the Kodama vector and trapping horizon. This surface gravity vanishes for degenerate trapping horizons and satisfies certain expected inequalities involving the area and energy. In the thermodynamic case, the quasi-local first law has the same form, apart from a relativistic factor, as the classical first law of thermodynamics, involving heat supply and hydrodynamic work, but with E replacing the internal energy. Expanding E in the Newtonian limit shows that it incorporates the Newtonian mass, kinetic energy, gravitational potential energy and thermal energy. There is also a weak type of unified zeroth law: a Gibbs-like definition of thermal equilibrium requires constancy of an effective temperature, generalising the Tolman condition and the particular case of Hawking radiation, while gravithermal equilibrium further requires constancy of surface gravity. Finally, it is suggested that the energy operator of spherically symmetric quantum gravity is determined by the Kodama vector, which encodes a dynamic time related to E.Comment: 18 pages, TeX, expanded somewhat, to appear in Class. Quantum Gra

A Cosmological Constant Limits the Size of Black Holes

In a space-time with cosmological constant $\Lambda>0$ and matter satisfying the dominant energy condition, the area of a black or white hole cannot exceed $4\pi/\Lambda$. This applies to event horizons where defined, i.e. in an asymptotically deSitter space-time, and to outer trapping horizons (cf. apparent horizons) in any space-time. The bound is attained if and only if the horizon is identical to that of the degenerate `Schwarzschild-deSitter' solution. This yields a topological restriction on the event horizon, namely that components whose total area exceeds $4\pi/\Lambda$ cannot merge. We discuss the conjectured isoperimetric inequality and implications for the cosmic censorship conjecture.Comment: 10 page

Gravitational radiation from dynamical black holes

An effective energy tensor for gravitational radiation is identified for uniformly expanding flows of the Hawking mass-energy. It appears in an energy conservation law expressing the change in mass due to the energy densities of matter and gravitational radiation, with respect to a Killing-like vector encoding a preferred flow of time outside a black hole. In a spin-coefficient formulation, the components of the effective energy tensor can be understood as the energy densities of ingoing and outgoing, transverse and longitudinal gravitational radiation. By anchoring the flow to the trapping horizon of a black hole in a given sequence of spatial hypersurfaces, there is a locally unique flow and a measure of gravitational radiation in the strong-field regime.Comment: 5 revtex4 pages. Additional comment

Gravitational waves from quasi-spherical black holes

A quasi-spherical approximation scheme, intended to apply to coalescing black holes, allows the waveforms of gravitational radiation to be computed by integrating ordinary differential equations.Comment: 4 revtex pages, 2 eps figure

Quasi-local first law of black-hole dynamics

A property well known as the first law of black hole is a relation among infinitesimal variations of parameters of stationary black holes. We consider a dynamical version of the first law, which may be called the first law of black hole dynamics. The first law of black hole dynamics is derived without assuming any symmetry or any asymptotic conditions. In the derivation, a definition of dynamical surface gravity is proposed. In spherical symmetry it reduces to that defined recently by one of the authors (SAH).Comment: Latex, 8 pages; version to appear in Class. Quantum Gra

On the Definition of Averagely Trapped Surfaces

Previously suggested definitions of averagely trapped surfaces are not well-defined properties of 2-surfaces, and can include surfaces in flat space-time. A natural definition of averagely trapped surfaces is that the product of the null expansions be positive on average. A surface is averagely trapped in the latter sense if and only if its area $A$ and Hawking mass $M$ satisfy the isoperimetric inequality $16\pi M^2 > A$, with similar inequalities existing for other definitions of quasi-local energy.Comment: 4 page

Construction and enlargement of traversable wormholes from Schwarzschild black holes

Analytic solutions are presented which describe the construction of a traversable wormhole from a Schwarzschild black hole, and the enlargement of such a wormhole, in Einstein gravity. The matter model is pure radiation which may have negative energy density (phantom or ghost radiation) and the idealization of impulsive radiation (infinitesimally thin null shells) is employed.Comment: 22 pages, 7 figure

Energy conservation for dynamical black holes

An energy conservation law is described, expressing the increase in mass-energy of a general black hole in terms of the energy densities of the infalling matter and gravitational radiation. For a growing black hole, this first law of black-hole dynamics is equivalent to an equation of Ashtekar & Krishnan, but the new integral and differential forms are regular in the limit where the black hole ceases to grow. An effective gravitational-radiation energy tensor is obtained, providing measures of both ingoing and outgoing, transverse and longitudinal gravitational radiation on and near a black hole. Corresponding energy-tensor forms of the first law involve a preferred time vector which plays the role for dynamical black holes which the stationary Killing vector plays for stationary black holes. Identifying an energy flux, vanishing if and only if the horizon is null, allows a division into energy-supply and work terms, as in the first law of thermodynamics. The energy supply can be expressed in terms of area increase and a newly defined surface gravity, yielding a Gibbs-like equation, with a similar form to the so-called first law for stationary black holes.Comment: 4 revtex4 pages. Many (mostly presentational) changes; emphasizes the definition of gravitational radiation in the strong-field regim

Black holes, cosmological singularities and change of signature

There exists a widespread belief that signature type change could be used to avoid spacetime singularities. We show that signature change cannot be utilised to this end unless the Einstein equation is abandoned at the suface of signature type change. We also discuss how to solve the initial value problem and show to which extent smooth and discontinuous signature changing solutions are equivalent.Comment: 14pages, Latex, no figur

Kerr black holes in horizon-generating form

New coordinates are given which describe non-degenerate Kerr black holes in dual-null foliations based on the outer (or inner) horizons, generalizing the Kruskal form for Schwarzschild black holes. The construction involves an area radius for the transverse surfaces and a generalization of the Regge-Wheeler radial function, both functions of the original radial coordinate only.Comment: 4 revtex4 page