2+1 gravity on the conformal sphere

We show that there are two equivalent first order descriptions of 2+1 gravity with a nonzero cosmological constant. One is the well-known spacetime description, and the other is in terms of evolving conformal geometry. The key tool that links these pictures is Cartan geometry, a generalization of Riemannian geometry that allows for geometries locally modeled off arbitrary homogeneous spaces. The two different interpretations suggest two distinct phase space reductions. The spacetime picture leads to the 2+1 formulation of general relativity due to Arnowitt, Deser, and Misner, while the conformal picture leads to shape dynamics. Cartan geometry thus provides an alternative to symmetry trading for explaining the equivalence of general relativity and shape dynamics.

Note:

29 preprint pages, 3 figures

04.60.-m
04.20.Fy
11.15.Yc
11.25.Hf
space: sphere
dimension: 3
geometry: Cartan
geometry: conformal
Chern-Simons term

References(32)

Figures(3)

[1]

La mesure du temps

H. Poincaré

[2]

La théorie de Lorentz et le principe de réaction

H. Poincaré

[3]

The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity, ch. Space and Time, pp. 75-91. New York: 1952

H. Minkowski

[4]

The Physical gravitational degrees of freedom

Edward Anderson
(
- Queen Mary, U. of London, Math. Sci.
)
,
Julian Barbour
(
- Unlisted, UK
)
,
Brendan Z. Foster
(
- Paris, Inst. Astrophys. and
- Maryland U.
)
,
Bryan Kelleher
(
- University Coll., Cork
)
,
Niall O. Murchadha
(
- University Coll., Cork
)

- Class.Quant.Grav. 22 (2005) 1795-1802
•
e-Print:
- gr-qc/0407104
•
DOI:
- 10.1088/0264-9381/22/9/020

[5]

Einstein gravity as a 3D conformally invariant theory

Henrique Gomes
(
- Nottingham U.
)
,
Sean Gryb
(
- Perimeter Inst. Theor. Phys. and
- Waterloo U.
)
,
Tim Koslowski
(
- Perimeter Inst. Theor. Phys.
)

- Class.Quant.Grav. 28 (2011) 045005
•
e-Print:
- 1010.2481
•
DOI:
- 10.1088/0264-9381/28/4/045005

[6]

The Link between General Relativity and Shape Dynamics

Henrique Gomes
(
- Nottingham U. and
- Imperial Coll., London
)
,
Tim Koslowski
(
- Perimeter Inst. Theor. Phys.
)

- Class.Quant.Grav. 29 (2012) 075009
•
e-Print:
- 1101.5974
•
DOI:
- 10.1088/0264-9381/29/7/075009

[7]

Dynamics of pure shape, relativity and the problem of time

J. Barbour

[8]

Shape Dynamics: An Introduction

Julian Barbour
(
- Unlisted
)

[9]

Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program New York, 1997

R.W. Sharpe

[10]

Fixation of coordinates in the Hamiltonian theory of gravitation

P.A.M. Dirac
(
- Princeton, Inst. Advanced Study
)

- Phys.Rev. 114 (1959) 924-930,
•
DOI:
- 10.1103/PhysRev.114.924

[11]

Gravitational degrees of freedom and the initial-value problem

James W. York, Jr.
(
- Princeton U.
)

- Phys.Rev.Lett. 26 (1971) 1656-1658
•
DOI:
- 10.1103/PhysRevLett.26.1656

[12]

Role of conformal three geometry in the dynamics of gravitation

James W. York, Jr.
(
- Princeton U.
)

- Phys.Rev.Lett. 28 (1972) 1082-1085
•
DOI:
- 10.1103/PhysRevLett.28.1082

[13]

Reduction of the Einstein equations in (2+1)-dimensions to a Hamiltonian system over Teichmuller space

V. Moncrief
(
- Yale U.
)

- J.Math.Phys. 30 (1989) 2907-2914
•
DOI:
- 10.1063/1.528475

[14]

Quantum gravity in 2+1 dimensions. Cambridge, UK: r., 1998

S. Carlip

[15]

Conformatlly invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial value problem of general relativity

James W. York, Jr.
(
- Princeton U.
)

- J.Math.Phys. 14 (1973) 456-464
•
DOI:
- 10.1063/1.1666338

[16]

Existence and uniqueness of solutions of the Hamiltonian constraint of general relativity on compact manifolds

N. O'Murchadha
,
J.J.W. York

- J.Math.Phys. 4 (1973) 1551

[17]

Loop quantum gravity without the Hamiltonian constraint

- Class.Quant.Grav. 30 (2013) 082001,
- Class.Quant.Grav. 30 (2013) 119501 (erratum)
•
e-Print:
- 1203.6525
•
DOI:
- 10.1088/0264-9381/30/8/082001

[18]

On a partially reduced phase space quantisation of general relativity conformally coupled to a scalar field

- Class.Quant.Grav. 30 (2013) 115017
•
e-Print:
- 1203.6526
•
DOI:
- 10.1088/0264-9381/30/11/115017

[19]

Quantum Gravity at a Lifshitz Point

Petr Horava
(
- UC, Berkeley and
- LBL, Berkeley
)

- Phys.Rev.D 79 (2009) 084008
•
e-Print:
- 0901.3775
•
DOI:
- 10.1103/PhysRevD.79.084008

[20]

The Large $N$ limit of superconformal field theories and supergravity

Juan Martin Maldacena
(
- Harvard U.
)

- Int.J.Theor.Phys. 38 (1999) 1113-1133 (reprint),
- Adv.Theor.Math.Phys. 2 (1998) 231-252
•
e-Print:
- hep-th/9711200
•
DOI:
- 10.4310/ATMP.1998.v2.n2.a1