2,162 research outputs found
Action and Energy of the Gravitational Field
We present a detailed examination of the variational principle for metric
general relativity as applied to a ``quasilocal'' spacetime region \M (that
is, a region that is both spatially and temporally bounded). Our analysis
relies on the Hamiltonian formulation of general relativity, and thereby
assumes a foliation of \M into spacelike hypersurfaces . We allow for
near complete generality in the choice of foliation. Using a field--theoretic
generalization of Hamilton--Jacobi theory, we define the quasilocal
stress-energy-momentum of the gravitational field by varying the action with
respect to the metric on the boundary \partial\M. The gravitational
stress-energy-momentum is defined for a two--surface spanned by a spacelike
hypersurface in spacetime. We examine the behavior of the gravitational
stress-energy-momentum under boosts of the spanning hypersurface. The boost
relations are derived from the geometrical and invariance properties of the
gravitational action and Hamiltonian. Finally, we present several new examples
of quasilocal energy--momentum, including a novel discussion of quasilocal
energy--momentum in the large-sphere limit towards spatial infinity.Comment: To be published in Annals of Physics. This final version includes two
new sections, one giving examples of quasilocal energy and the other
containing a discussion of energy at spatial infinity. References have been
added to papers by Bose and Dadhich, Anco and Tun
Canonical Quasilocal Energy and Small Spheres
Consider the definition E of quasilocal energy stemming from the
Hamilton-Jacobi method as applied to the canonical form of the gravitational
action. We examine E in the standard "small-sphere limit," first considered by
Horowitz and Schmidt in their examination of Hawking's quasilocal mass. By the
term "small sphere" we mean a cut S(r), level in an affine radius r, of the
lightcone belonging to a generic spacetime point. As a power series in r, we
compute the energy E of the gravitational and matter fields on a spacelike
hypersurface spanning S(r). Much of our analysis concerns conceptual and
technical issues associated with assigning the zero-point of the energy. For
the small-sphere limit, we argue that the correct zero-point is obtained via a
"lightcone reference," which stems from a certain isometric embedding of S(r)
into a genuine lightcone of Minkowski spacetime. Choosing this zero-point, we
find agreement with Hawking's quasilocal mass expression, up to and including
the first non-trivial order in the affine radius. The vacuum limit relates the
quasilocal energy directly to the Bel-Robinson tensor.Comment: revtex, 22 p, uses amssymb option (can be removed
Positivity of Entropy in the Semi-Classical Theory of Black Holes and Radiation
Quantum stress-energy tensors of fields renormalized on a Schwarzschild
background violate the classical energy conditions near the black hole.
Nevertheless, the associated equilibrium thermodynamical entropy by
which such fields augment the usual black hole entropy is found to be positive.
More precisely, the derivative of with respect to radius, at fixed
black hole mass, is found to vanish at the horizon for {\it all} regular
renormalized stress-energy quantum tensors. For the cases of conformal scalar
fields and U(1) gauge fields, the corresponding second derivative is positive,
indicating that has a local minimum there. Explicit calculation
shows that indeed increases monotonically for increasing radius and
is positive. (The same conclusions hold for a massless spin 1/2 field, but the
accuracy of the stress-energy tensor we employ has not been confirmed, in
contrast to the scalar and vector cases). None of these results would hold if
the back-reaction of the radiation on the spacetime geometry were ignored;
consequently, one must regard as arising from both the radiation
fields and their effects on the gravitational field. The back-reaction, no
matter how "small",Comment: 19 pages, RevTe
The Microcanonical Functional Integral. I. The Gravitational Field
The gravitational field in a spatially finite region is described as a
microcanonical system. The density of states is expressed formally as a
functional integral over Lorentzian metrics and is a functional of the
geometrical boundary data that are fixed in the corresponding action. These
boundary data are the thermodynamical extensive variables, including the energy
and angular momentum of the system. When the boundary data are chosen such that
the system is described semiclassically by {\it any} real stationary
axisymmetric black hole, then in this same approximation is shown to
equal 1/4 the area of the black hole event horizon. The canonical and grand
canonical partition functions are obtained by integral transforms of that
lead to "imaginary time" functional integrals. A general form of the first law
of thermodynamics for stationary black holes is derived. For the simpler case
of nonrelativistic mechanics, the density of states is expressed as a real-time
functional integral and then used to deduce Feynman's imaginary-time functional
integral for the canonical partition function.Comment: 29 pages, plain Te
Action Principle for the Generalized Harmonic Formulation of General Relativity
An action principle for the generalized harmonic formulation of general
relativity is presented. The action is a functional of the spacetime metric and
the gauge source vector. An action principle for the Z4 formulation of general
relativity has been proposed recently by Bona, Bona--Casas and Palenzuela
(BBP). The relationship between the generalized harmonic action and the BBP
action is discussed in detail.Comment: This version is contains more thorough presentations and discussions
of the key results. To be published in PRD. (8 pages, no figures
A Liquid Model Analogue for Black Hole Thermodynamics
We are able to characterize a 2--dimensional classical fluid sharing some of
the same thermodynamic state functions as the Schwarzschild black hole. This
phenomenological correspondence between black holes and fluids is established
by means of the model liquid's pair-correlation function and the two-body
atomic interaction potential. These latter two functions are calculated exactly
in terms of the black hole internal (quasilocal) energy and the isothermal
compressibility. We find the existence of a ``screening" like effect for the
components of the liquid.Comment: 20 pages and 6 Encapsulated PostScript figure
New variables, the gravitational action, and boosted quasilocal stress-energy-momentum
This paper presents a complete set of quasilocal densities which describe the
stress-energy-momentum content of the gravitational field and which are built
with Ashtekar variables. The densities are defined on a two-surface which
bounds a generic spacelike hypersurface of spacetime. The method used
to derive the set of quasilocal densities is a Hamilton-Jacobi analysis of a
suitable covariant action principle for the Ashtekar variables. As such, the
theory presented here is an Ashtekar-variable reformulation of the metric
theory of quasilocal stress-energy-momentum originally due to Brown and York.
This work also investigates how the quasilocal densities behave under
generalized boosts, i. e. switches of the slice spanning . It is
shown that under such boosts the densities behave in a manner which is similar
to the simple boost law for energy-momentum four-vectors in special relativity.
The developed formalism is used to obtain a collection of two-surface or boost
invariants. With these invariants, one may ``build" several different mass
definitions in general relativity, such as the Hawking expression. Also
discussed in detail in this paper is the canonical action principle as applied
to bounded spacetime regions with ``sharp corners."Comment: Revtex, 41 Pages, 4 figures added. Final version has been revised and
improved quite a bit. To appear in Classical and Quantum Gravit
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