# Microcanonical functional integral for the gravitational field

## Abstract

The gravitational field in a spatially finite region is described as a microcanonical system. The density of states [nu] 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 [ital any] real stationary axisymmetric black hole, then in this same approximation ln[nu] 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 [nu] 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.

- Authors:

- (Institute of Field Physics and Theoretical Astrophysics and Relativity Group, Department of Physics and Astronomy, The University of North Carolina, Chapel Hill, North Carolina 27599-3255 (United States))

- Publication Date:

- OSTI Identifier:
- 6993692

- Alternate Identifier(s):
- OSTI ID: 6993692

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review, D (Particles Fields); (United States)

- Additional Journal Information:
- Journal Volume: 47:4; Journal ID: ISSN 0556-2821

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; GRAVITATIONAL FIELDS; FUNCTIONAL ANALYSIS; ANGULAR MOMENTUM; BLACK HOLES; BOUNDARY CONDITIONS; ENERGY LEVELS; ENTROPY; FEYNMAN PATH INTEGRAL; METRICS; PARTITION FUNCTIONS; SEMICLASSICAL APPROXIMATION; THERMODYNAMIC PROPERTIES; FUNCTIONS; INTEGRALS; MATHEMATICS; PHYSICAL PROPERTIES 662110* -- General Theory of Particles & Fields-- Theory of Fields & Strings-- (1992-); 661100 -- Classical & Quantum Mechanics-- (1992-)

### Citation Formats

```
Brown, J.D., and York, J.W. Jr.
```*Microcanonical functional integral for the gravitational field*. United States: N. p., 1993.
Web. doi:10.1103/PhysRevD.47.1420.

```
Brown, J.D., & York, J.W. Jr.
```*Microcanonical functional integral for the gravitational field*. United States. doi:10.1103/PhysRevD.47.1420.

```
Brown, J.D., and York, J.W. Jr. Mon .
"Microcanonical functional integral for the gravitational field". United States. doi:10.1103/PhysRevD.47.1420.
```

```
@article{osti_6993692,
```

title = {Microcanonical functional integral for the gravitational field},

author = {Brown, J.D. and York, J.W. Jr.},

abstractNote = {The gravitational field in a spatially finite region is described as a microcanonical system. The density of states [nu] 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 [ital any] real stationary axisymmetric black hole, then in this same approximation ln[nu] 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 [nu] 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.},

doi = {10.1103/PhysRevD.47.1420},

journal = {Physical Review, D (Particles Fields); (United States)},

issn = {0556-2821},

number = ,

volume = 47:4,

place = {United States},

year = {1993},

month = {2}

}