# Transgressing the horizons: Time operator in two-dimensional dilaton gravity

## Abstract

We present a Dirac quantization of generic single-horizon black holes in two-dimensional dilaton gravity. The classical theory is first partially reduced by a spatial gauge choice under which the spatial surfaces extend from a black or white hole singularity to a spacelike infinity. The theory is then quantized in a metric representation, solving the quantum Hamiltonian constraint in terms of (generalized) eigenstates of the ADM mass operator and specifying the physical inner product by self-adjointness of a time operator that is affinely conjugate to the ADM mass. Regularity of the time operator across the horizon requires the operator to contain a quantum correction that distinguishes the future and past horizons and gives rise to a quantum correction in the hole's surface gravity. We expect a similar quantum correction to be present in systems whose dynamics admits black hole formation by gravitational collapse.

- Authors:

- Department of Physics and Winnipeg Institute of Theoretical Physics, University of Winnipeg, Winnipeg, Manitoba, R3B 2E9 (Canada)
- School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 21010906

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 2; Other Information: DOI: 10.1103/PhysRevD.75.024036; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; BLACK HOLES; CORRECTIONS; COSMOLOGY; EIGENSTATES; GRAVITATION; GRAVITATIONAL COLLAPSE; HAMILTONIANS; MASS; QUANTIZATION; QUANTUM FIELD THEORY; QUANTUM OPERATORS; SINGULARITY; TWO-DIMENSIONAL CALCULATIONS; WHITE HOLES

### Citation Formats

```
Kunstatter, Gabor, and Louko, Jorma.
```*Transgressing the horizons: Time operator in two-dimensional dilaton gravity*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVD.75.024036.

```
Kunstatter, Gabor, & Louko, Jorma.
```*Transgressing the horizons: Time operator in two-dimensional dilaton gravity*. United States. doi:10.1103/PHYSREVD.75.024036.

```
Kunstatter, Gabor, and Louko, Jorma. Mon .
"Transgressing the horizons: Time operator in two-dimensional dilaton gravity". United States.
doi:10.1103/PHYSREVD.75.024036.
```

```
@article{osti_21010906,
```

title = {Transgressing the horizons: Time operator in two-dimensional dilaton gravity},

author = {Kunstatter, Gabor and Louko, Jorma},

abstractNote = {We present a Dirac quantization of generic single-horizon black holes in two-dimensional dilaton gravity. The classical theory is first partially reduced by a spatial gauge choice under which the spatial surfaces extend from a black or white hole singularity to a spacelike infinity. The theory is then quantized in a metric representation, solving the quantum Hamiltonian constraint in terms of (generalized) eigenstates of the ADM mass operator and specifying the physical inner product by self-adjointness of a time operator that is affinely conjugate to the ADM mass. Regularity of the time operator across the horizon requires the operator to contain a quantum correction that distinguishes the future and past horizons and gives rise to a quantum correction in the hole's surface gravity. We expect a similar quantum correction to be present in systems whose dynamics admits black hole formation by gravitational collapse.},

doi = {10.1103/PHYSREVD.75.024036},

journal = {Physical Review. D, Particles Fields},

number = 2,

volume = 75,

place = {United States},

year = {Mon Jan 15 00:00:00 EST 2007},

month = {Mon Jan 15 00:00:00 EST 2007}

}