Gravity in (2+1)-dimensions Darío Alberto Castro Castro 1 1
Universidad Nacional de Colombia Departamento de Física Qualify Exam Doctoral Program in Physics November, 2005
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
1 / 71
Outline
Outline 1
Abstract
2
Introduction Sumary History The pedagogical value of 2+1 dimensions
3
Why (2+1)-dimensional gravity? Sumary General relativity in 2+1 dimensions
4
Classical Gravity in (2+1)-dimensions Introduction The ADM decomposition The first-order formalism: The Chern-Simons formulation
5
Quantum Gravity in (2+1)-dimensions
6
The (2+1)-Dimensional Black Hole: BTZ BTZ Geometry
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
2 / 71
Outline
Outline 1
Abstract
2
Introduction Sumary History The pedagogical value of 2+1 dimensions
3
Why (2+1)-dimensional gravity? Sumary General relativity in 2+1 dimensions
4
Classical Gravity in (2+1)-dimensions Introduction The ADM decomposition The first-order formalism: The Chern-Simons formulation
5
Quantum Gravity in (2+1)-dimensions
6
The (2+1)-Dimensional Black Hole: BTZ BTZ Geometry
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
2 / 71
Outline
Outline 1
Abstract
2
Introduction Sumary History The pedagogical value of 2+1 dimensions
3
Why (2+1)-dimensional gravity? Sumary General relativity in 2+1 dimensions
4
Classical Gravity in (2+1)-dimensions Introduction The ADM decomposition The first-order formalism: The Chern-Simons formulation
5
Quantum Gravity in (2+1)-dimensions
6
The (2+1)-Dimensional Black Hole: BTZ BTZ Geometry
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
2 / 71
Outline
Outline 1
Abstract
2
Introduction Sumary History The pedagogical value of 2+1 dimensions
3
Why (2+1)-dimensional gravity? Sumary General relativity in 2+1 dimensions
4
Classical Gravity in (2+1)-dimensions Introduction The ADM decomposition The first-order formalism: The Chern-Simons formulation
5
Quantum Gravity in (2+1)-dimensions
6
The (2+1)-Dimensional Black Hole: BTZ BTZ Geometry
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
2 / 71
Outline
Outline 1
Abstract
2
Introduction Sumary History The pedagogical value of 2+1 dimensions
3
Why (2+1)-dimensional gravity? Sumary General relativity in 2+1 dimensions
4
Classical Gravity in (2+1)-dimensions Introduction The ADM decomposition The first-order formalism: The Chern-Simons formulation
5
Quantum Gravity in (2+1)-dimensions
6
The (2+1)-Dimensional Black Hole: BTZ BTZ Geometry
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
2 / 71
Outline
Outline 1
Abstract
2
Introduction Sumary History The pedagogical value of 2+1 dimensions
3
Why (2+1)-dimensional gravity? Sumary General relativity in 2+1 dimensions
4
Classical Gravity in (2+1)-dimensions Introduction The ADM decomposition The first-order formalism: The Chern-Simons formulation
5
Quantum Gravity in (2+1)-dimensions
6
The (2+1)-Dimensional Black Hole: BTZ BTZ Geometry
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
2 / 71
Abstract
Abstract
Abstract General relativity is a highly complicated field theory both classically and quantum mechanically. Even though a large number of classical solutions exists, a general classification of the space of solutions has never been achieved. These lectures briefly review our current understanding of classical and quantum gravity in three spacetime dimension, general relativity in two spatial dimensions plus time. In three spacetime dimensions, general relativity drastically simplifies. Nevertheless, many of the difficult conceptual problems of quantizing gravity are still present. In this review, I summarize the rather large body of work that has gone towards quantizing (2+1)-dimensional vacuum gravity.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
3 / 71
Abstract
Abstract: Graphics Review Gravity in (2+1)-dimensions
Quantum Models
Classical Models
Chern-Simons
c 2005
ADM
Other
Covariant
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
Canonical
Sum
November, 2005
4 / 71
Introduction
Sumary
Outline 1
Abstract
2
Introduction Sumary History The pedagogical value of 2+1 dimensions
3
Why (2+1)-dimensional gravity? Sumary General relativity in 2+1 dimensions
4
Classical Gravity in (2+1)-dimensions Introduction The ADM decomposition The first-order formalism: The Chern-Simons formulation
5
Quantum Gravity in (2+1)-dimensions
6
The (2+1)-Dimensional Black Hole: BTZ BTZ Geometry
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
5 / 71
Introduction
Sumary
Introduction Sumary Studying physics in spacetimes with dimensions less than four has often proved useful. Intrinsic interest. Continuing search for a realistic quantum theory of gravity. Useful to explore simpler models that share the basic conceptual features of general relativity. Gravity in (2+1)-dimensions —general relativity in two spatial dimensions plus time— A fully theory of spacetime geometry that nevertheless avoids many of the technical difficulties of realistic (3+1)-dimensional gravity.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
6 / 71
Introduction
Sumary
Introduction Sumary Studying physics in spacetimes with dimensions less than four has often proved useful. Intrinsic interest. Continuing search for a realistic quantum theory of gravity. Useful to explore simpler models that share the basic conceptual features of general relativity. Gravity in (2+1)-dimensions —general relativity in two spatial dimensions plus time— A fully theory of spacetime geometry that nevertheless avoids many of the technical difficulties of realistic (3+1)-dimensional gravity.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
6 / 71
Introduction
Sumary
Introduction Sumary Studying physics in spacetimes with dimensions less than four has often proved useful. Intrinsic interest. Continuing search for a realistic quantum theory of gravity. Useful to explore simpler models that share the basic conceptual features of general relativity. Gravity in (2+1)-dimensions —general relativity in two spatial dimensions plus time— A fully theory of spacetime geometry that nevertheless avoids many of the technical difficulties of realistic (3+1)-dimensional gravity.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
6 / 71
Introduction
Sumary
Introduction Sumary Studying physics in spacetimes with dimensions less than four has often proved useful. Intrinsic interest. Continuing search for a realistic quantum theory of gravity. Useful to explore simpler models that share the basic conceptual features of general relativity. Gravity in (2+1)-dimensions —general relativity in two spatial dimensions plus time— A fully theory of spacetime geometry that nevertheless avoids many of the technical difficulties of realistic (3+1)-dimensional gravity.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
6 / 71
Introduction
Sumary
Introduction Sumary Studying physics in spacetimes with dimensions less than four has often proved useful. Intrinsic interest. Continuing search for a realistic quantum theory of gravity. Useful to explore simpler models that share the basic conceptual features of general relativity. Gravity in (2+1)-dimensions —general relativity in two spatial dimensions plus time— A fully theory of spacetime geometry that nevertheless avoids many of the technical difficulties of realistic (3+1)-dimensional gravity.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
6 / 71
Introduction
Sumary
Introduction Sumary Studying physics in spacetimes with dimensions less than four has often proved useful. Intrinsic interest. Continuing search for a realistic quantum theory of gravity. Useful to explore simpler models that share the basic conceptual features of general relativity. Gravity in (2+1)-dimensions —general relativity in two spatial dimensions plus time— A fully theory of spacetime geometry that nevertheless avoids many of the technical difficulties of realistic (3+1)-dimensional gravity.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
6 / 71
Introduction
History
Outline 1
Abstract
2
Introduction Sumary History The pedagogical value of 2+1 dimensions
3
Why (2+1)-dimensional gravity? Sumary General relativity in 2+1 dimensions
4
Classical Gravity in (2+1)-dimensions Introduction The ADM decomposition The first-order formalism: The Chern-Simons formulation
5
Quantum Gravity in (2+1)-dimensions
6
The (2+1)-Dimensional Black Hole: BTZ BTZ Geometry
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
7 / 71
Introduction
History
Introduction History 1963 — Staruszkiewicz: showed that point particles in a (2+1)-dimensional spacetime could be given a simple and elegant geometrical description. 1966 — Leutwyler: papers on quantum mechanical aspects 1976 — Clement: papers on classical mechanical aspects 1977 — Collas: papers on classical mechanical aspects 1984 — Deser, Jackiw, and ’t Hooft: began a systematic investigation 1988 — Witten: showed that (2+1)-dimensional general relativity could be rewritten as a Chern-Simons theory 1989 -... — over 500 papers have been published on various features of quantum gravity in 2+1 dimensions
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
8 / 71
Introduction
History
Introduction History 1963 — Staruszkiewicz: showed that point particles in a (2+1)-dimensional spacetime could be given a simple and elegant geometrical description. 1966 — Leutwyler: papers on quantum mechanical aspects 1976 — Clement: papers on classical mechanical aspects 1977 — Collas: papers on classical mechanical aspects 1984 — Deser, Jackiw, and ’t Hooft: began a systematic investigation 1988 — Witten: showed that (2+1)-dimensional general relativity could be rewritten as a Chern-Simons theory 1989 -... — over 500 papers have been published on various features of quantum gravity in 2+1 dimensions
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
8 / 71
Introduction
History
Introduction History 1963 — Staruszkiewicz: showed that point particles in a (2+1)-dimensional spacetime could be given a simple and elegant geometrical description. 1966 — Leutwyler: papers on quantum mechanical aspects 1976 — Clement: papers on classical mechanical aspects 1977 — Collas: papers on classical mechanical aspects 1984 — Deser, Jackiw, and ’t Hooft: began a systematic investigation 1988 — Witten: showed that (2+1)-dimensional general relativity could be rewritten as a Chern-Simons theory 1989 -... — over 500 papers have been published on various features of quantum gravity in 2+1 dimensions
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
8 / 71
Introduction
History
Introduction History 1963 — Staruszkiewicz: showed that point particles in a (2+1)-dimensional spacetime could be given a simple and elegant geometrical description. 1966 — Leutwyler: papers on quantum mechanical aspects 1976 — Clement: papers on classical mechanical aspects 1977 — Collas: papers on classical mechanical aspects 1984 — Deser, Jackiw, and ’t Hooft: began a systematic investigation 1988 — Witten: showed that (2+1)-dimensional general relativity could be rewritten as a Chern-Simons theory 1989 -... — over 500 papers have been published on various features of quantum gravity in 2+1 dimensions
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
8 / 71
Introduction
History
Introduction History 1963 — Staruszkiewicz: showed that point particles in a (2+1)-dimensional spacetime could be given a simple and elegant geometrical description. 1966 — Leutwyler: papers on quantum mechanical aspects 1976 — Clement: papers on classical mechanical aspects 1977 — Collas: papers on classical mechanical aspects 1984 — Deser, Jackiw, and ’t Hooft: began a systematic investigation 1988 — Witten: showed that (2+1)-dimensional general relativity could be rewritten as a Chern-Simons theory 1989 -... — over 500 papers have been published on various features of quantum gravity in 2+1 dimensions
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
8 / 71
Introduction
History
Introduction History 1963 — Staruszkiewicz: showed that point particles in a (2+1)-dimensional spacetime could be given a simple and elegant geometrical description. 1966 — Leutwyler: papers on quantum mechanical aspects 1976 — Clement: papers on classical mechanical aspects 1977 — Collas: papers on classical mechanical aspects 1984 — Deser, Jackiw, and ’t Hooft: began a systematic investigation 1988 — Witten: showed that (2+1)-dimensional general relativity could be rewritten as a Chern-Simons theory 1989 -... — over 500 papers have been published on various features of quantum gravity in 2+1 dimensions
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
8 / 71
Introduction
History
Introduction History 1963 — Staruszkiewicz: showed that point particles in a (2+1)-dimensional spacetime could be given a simple and elegant geometrical description. 1966 — Leutwyler: papers on quantum mechanical aspects 1976 — Clement: papers on classical mechanical aspects 1977 — Collas: papers on classical mechanical aspects 1984 — Deser, Jackiw, and ’t Hooft: began a systematic investigation 1988 — Witten: showed that (2+1)-dimensional general relativity could be rewritten as a Chern-Simons theory 1989 -... — over 500 papers have been published on various features of quantum gravity in 2+1 dimensions
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
8 / 71
Introduction
The pedagogical value of 2+1 dimensions
Outline 1
Abstract
2
Introduction Sumary History The pedagogical value of 2+1 dimensions
3
Why (2+1)-dimensional gravity? Sumary General relativity in 2+1 dimensions
4
Classical Gravity in (2+1)-dimensions Introduction The ADM decomposition The first-order formalism: The Chern-Simons formulation
5
Quantum Gravity in (2+1)-dimensions
6
The (2+1)-Dimensional Black Hole: BTZ BTZ Geometry
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
9 / 71
Introduction
The pedagogical value of 2+1 dimensions
Introduction
The pedagogical value of 2+1 dimensions One hopes that lower dimensional gravitational physics can provide insight into problems in (3+1)-dimensions by yielding a greater measure of computational simplicity without sacrificing too much of the conceptual complexity of the original problem. A number of physical problems have been approached in this manner, clarification of the conceptual issues associated with black hole physics, black hole information loss, and the high–temperature behavior of (3+1) dimensions theories also motivates the study of 2+1 dimensions theories.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
10 / 71
Why (2+1)-dimensional gravity?
Sumary
Outline 1
Abstract
2
Introduction Sumary History The pedagogical value of 2+1 dimensions
3
Why (2+1)-dimensional gravity? Sumary General relativity in 2+1 dimensions
4
Classical Gravity in (2+1)-dimensions Introduction The ADM decomposition The first-order formalism: The Chern-Simons formulation
5
Quantum Gravity in (2+1)-dimensions
6
The (2+1)-Dimensional Black Hole: BTZ BTZ Geometry
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
11 / 71
Why (2+1)-dimensional gravity?
Sumary
Why (2+1)-dimensional gravity? Sumary
The past 25 years have witnessed remarkable growth in our understanding of fundamental physics. The task of quantizing general relativity remains one of the outstanding problems of theoretical physics. The obstacles to quantizing gravity are in part technical. Ordinary quantum field theory is local. Ordinary quantum field theory takes causality as a fundamental postulate. I Time evolution in quantum field theory is determined by a Hamiltonian operator.
I
I
Simpler models. (2+1)-dimensional gravity has the same conceptual foundation.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
12 / 71
Why (2+1)-dimensional gravity?
Sumary
Why (2+1)-dimensional gravity? Sumary
The past 25 years have witnessed remarkable growth in our understanding of fundamental physics. The task of quantizing general relativity remains one of the outstanding problems of theoretical physics. The obstacles to quantizing gravity are in part technical. Ordinary quantum field theory is local. Ordinary quantum field theory takes causality as a fundamental postulate. I Time evolution in quantum field theory is determined by a Hamiltonian operator. I I
Simpler models. (2+1)-dimensional gravity has the same conceptual foundation.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
12 / 71
Why (2+1)-dimensional gravity?
Sumary
Why (2+1)-dimensional gravity? Sumary
The past 25 years have witnessed remarkable growth in our understanding of fundamental physics. The task of quantizing general relativity remains one of the outstanding problems of theoretical physics. The obstacles to quantizing gravity are in part technical. Ordinary quantum field theory is local. Ordinary quantum field theory takes causality as a fundamental postulate. I Time evolution in quantum field theory is determined by a Hamiltonian operator. I I
Simpler models. (2+1)-dimensional gravity has the same conceptual foundation.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
12 / 71
Why (2+1)-dimensional gravity?
Sumary
Why (2+1)-dimensional gravity? Sumary
The past 25 years have witnessed remarkable growth in our understanding of fundamental physics. The task of quantizing general relativity remains one of the outstanding problems of theoretical physics. The obstacles to quantizing gravity are in part technical. Ordinary quantum field theory is local. Ordinary quantum field theory takes causality as a fundamental postulate. I Time evolution in quantum field theory is determined by a Hamiltonian operator. I I
Simpler models. (2+1)-dimensional gravity has the same conceptual foundation.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
12 / 71
Why (2+1)-dimensional gravity?
Sumary
Why (2+1)-dimensional gravity? Sumary
The past 25 years have witnessed remarkable growth in our understanding of fundamental physics. The task of quantizing general relativity remains one of the outstanding problems of theoretical physics. The obstacles to quantizing gravity are in part technical. Ordinary quantum field theory is local. Ordinary quantum field theory takes causality as a fundamental postulate. I Time evolution in quantum field theory is determined by a Hamiltonian operator. I I
Simpler models. (2+1)-dimensional gravity has the same conceptual foundation.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
12 / 71
Why (2+1)-dimensional gravity?
Sumary
Why (2+1)-dimensional gravity? Sumary
The past 25 years have witnessed remarkable growth in our understanding of fundamental physics. The task of quantizing general relativity remains one of the outstanding problems of theoretical physics. The obstacles to quantizing gravity are in part technical. Ordinary quantum field theory is local. Ordinary quantum field theory takes causality as a fundamental postulate. I Time evolution in quantum field theory is determined by a Hamiltonian operator. I I
Simpler models. (2+1)-dimensional gravity has the same conceptual foundation.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
12 / 71
Why (2+1)-dimensional gravity?
Sumary
Why (2+1)-dimensional gravity? Sumary
The past 25 years have witnessed remarkable growth in our understanding of fundamental physics. The task of quantizing general relativity remains one of the outstanding problems of theoretical physics. The obstacles to quantizing gravity are in part technical. Ordinary quantum field theory is local. Ordinary quantum field theory takes causality as a fundamental postulate. I Time evolution in quantum field theory is determined by a Hamiltonian operator. I I
Simpler models. (2+1)-dimensional gravity has the same conceptual foundation.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
12 / 71
Why (2+1)-dimensional gravity?
Sumary
Why (2+1)-dimensional gravity? Sumary
The past 25 years have witnessed remarkable growth in our understanding of fundamental physics. The task of quantizing general relativity remains one of the outstanding problems of theoretical physics. The obstacles to quantizing gravity are in part technical. Ordinary quantum field theory is local. Ordinary quantum field theory takes causality as a fundamental postulate. I Time evolution in quantum field theory is determined by a Hamiltonian operator. I I
Simpler models. (2+1)-dimensional gravity has the same conceptual foundation.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
12 / 71
Why (2+1)-dimensional gravity?
Sumary
Basic Concepts
Definition The spacetime constitutes it all the physical events, which will be described by the couple (M , g), where M is a smooth manifold (C∞ ) 3-dimensional connected of Hausdorf and g it is a metric Lorentziana M .
Definition A slice Σ of an n-dimensional spacetime M is an embedded spacelike (n − 1) -dimensional submanifold that is closed as a subset of M .
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
13 / 71
Why (2+1)-dimensional gravity?
Sumary
Basic Concepts
Definition The spacetime constitutes it all the physical events, which will be described by the couple (M , g), where M is a smooth manifold (C∞ ) 3-dimensional connected of Hausdorf and g it is a metric Lorentziana M .
Definition A slice Σ of an n-dimensional spacetime M is an embedded spacelike (n − 1) -dimensional submanifold that is closed as a subset of M .
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
13 / 71
Why (2+1)-dimensional gravity?
Sumary
...Basic Concepts
Definition Let Σ ⊂ M be any closed set which no pair of points p, q ∈ Σ can be joined by timelike curve. We define the domain of dependence of Σ by D (Σ) = {p ∈ M | every causal curve through p intersects Σ} If D (Σ) = M , then Σ is said to be a Cauchy surface for the spacetime (M , g). If a spacetime admits a Cauchy surface, then spacetime is said to be globally hyperbolic.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
14 / 71
Why (2+1)-dimensional gravity?
Sumary
...Basic Concepts
Theorem (Geroch 1970; Dieckman 1988): If (M , g) is globally hyperbolic with Cauchy surface Σ, then M has topology R×Σ. Furthemore, M can be foliated by a one-parameter family of smooth Cauchy surfaces Σt , i.e., a smooth “time coordinate” t can be chosen on M such that each surface of constant t is a Cauchy surface.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
15 / 71
Why (2+1)-dimensional gravity?
General relativity in 2+1 dimensions
Outline 1
Abstract
2
Introduction Sumary History The pedagogical value of 2+1 dimensions
3
Why (2+1)-dimensional gravity? Sumary General relativity in 2+1 dimensions
4
Classical Gravity in (2+1)-dimensions Introduction The ADM decomposition The first-order formalism: The Chern-Simons formulation
5
Quantum Gravity in (2+1)-dimensions
6
The (2+1)-Dimensional Black Hole: BTZ BTZ Geometry
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
16 / 71
Why (2+1)-dimensional gravity?
General relativity in 2+1 dimensions
General relativity in 2+1 dimensions Theory of gravity obtained from the standard Einstein-Hilbert action
Theorem I=
1 16πG
Z M
√ d 3 x −g (R − 2Λ) + Imatter (1)
where M is differentiable manifold, g = det gµν , Λ is the cosmological constant and Imatter is the matter action. As in 3+1 dimensions, the resulting Euler-Lagrange equations are the standard Einstein field equations 1 Rµν − gµν R + Λgµν = −8πGTµν 2 (2)
Proof details
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
Skip proof
November, 2005
17 / 71
Why (2+1)-dimensional gravity?
General relativity in 2+1 dimensions
General relativity in 2+1 dimensions Relativity in 2+1 6= Relativity in 3+1 : curvature tensor ∼ Ricci tensor 1 Rµνρσ = gµρ Rνσ + gνσ Rµρ − gνρ Rµσ − gµσ Rνρ − (gµρ gνσ − gµσ gνρ )R. 2
(3)
Every solution of the vacuum Einstein equations with Λ = 0 is flat. Every solution with a nonvanishing cosmological constant has constant curvature: de Sitter (Λ > 0), or anti-de Sitter (Λ < 0). A (2+1)dimensional spacetime has no local degrees of freedom. General relativity in 2+1 dimensions has no Newtonian limit: d 2 xi 2 (n − 3) + ∂i Φ = 0 dt 2 n−2 (4)
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
18 / 71
Why (2+1)-dimensional gravity?
General relativity in 2+1 dimensions
General relativity in 2+1 dimensions Relativity in 2+1 6= Relativity in 3+1 : curvature tensor ∼ Ricci tensor 1 Rµνρσ = gµρ Rνσ + gνσ Rµρ − gνρ Rµσ − gµσ Rνρ − (gµρ gνσ − gµσ gνρ )R. 2
(3)
Every solution of the vacuum Einstein equations with Λ = 0 is flat. Every solution with a nonvanishing cosmological constant has constant curvature: de Sitter (Λ > 0), or anti-de Sitter (Λ < 0). A (2+1)dimensional spacetime has no local degrees of freedom. General relativity in 2+1 dimensions has no Newtonian limit: d 2 xi 2 (n − 3) + ∂i Φ = 0 dt 2 n−2 (4)
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
18 / 71
Why (2+1)-dimensional gravity?
General relativity in 2+1 dimensions
General relativity in 2+1 dimensions Relativity in 2+1 6= Relativity in 3+1 : curvature tensor ∼ Ricci tensor 1 Rµνρσ = gµρ Rνσ + gνσ Rµρ − gνρ Rµσ − gµσ Rνρ − (gµρ gνσ − gµσ gνρ )R. 2
(3)
Every solution of the vacuum Einstein equations with Λ = 0 is flat. Every solution with a nonvanishing cosmological constant has constant curvature: de Sitter (Λ > 0), or anti-de Sitter (Λ < 0). A (2+1)dimensional spacetime has no local degrees of freedom. General relativity in 2+1 dimensions has no Newtonian limit: d 2 xi 2 (n − 3) + ∂i Φ = 0 dt 2 n−2 (4)
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
18 / 71
Why (2+1)-dimensional gravity?
General relativity in 2+1 dimensions
General relativity in 2+1 dimensions Relativity in 2+1 6= Relativity in 3+1 : curvature tensor ∼ Ricci tensor 1 Rµνρσ = gµρ Rνσ + gνσ Rµρ − gνρ Rµσ − gµσ Rνρ − (gµρ gνσ − gµσ gνρ )R. 2
(3)
Every solution of the vacuum Einstein equations with Λ = 0 is flat. Every solution with a nonvanishing cosmological constant has constant curvature: de Sitter (Λ > 0), or anti-de Sitter (Λ < 0). A (2+1)dimensional spacetime has no local degrees of freedom. General relativity in 2+1 dimensions has no Newtonian limit: d 2 xi 2 (n − 3) + ∂i Φ = 0 dt 2 n−2 (4)
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
18 / 71
Classical Gravity in (2+1)-dimensions
Introduction
Outline 1
Abstract
2
Introduction Sumary History The pedagogical value of 2+1 dimensions
3
Why (2+1)-dimensional gravity? Sumary General relativity in 2+1 dimensions
4
Classical Gravity in (2+1)-dimensions Introduction The ADM decomposition The first-order formalism: The Chern-Simons formulation
5
Quantum Gravity in (2+1)-dimensions
6
The (2+1)-Dimensional Black Hole: BTZ BTZ Geometry
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
19 / 71
Classical Gravity in (2+1)-dimensions
Introduction
Classical Gravity in (2+1)-dimensions
Introduction Our goal is not to study the detailed characteristics of particular solutions, but rather to develop an understanding of the generic properties of the space of solutions. Two fundamental approaches to classical general relativity in 2+1 dimensions: I I
Arnowitt-Deser-Misner (ADM) decomposition of the metric. The first-order formalism.
In both cases, the goal is to set up the field equations in a manner that permits a complete characterization of the classical solutions.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
20 / 71
Classical Gravity in (2+1)-dimensions
Introduction
Classical Gravity in (2+1)-dimensions
Introduction Our goal is not to study the detailed characteristics of particular solutions, but rather to develop an understanding of the generic properties of the space of solutions. Two fundamental approaches to classical general relativity in 2+1 dimensions: I I
Arnowitt-Deser-Misner (ADM) decomposition of the metric. The first-order formalism.
In both cases, the goal is to set up the field equations in a manner that permits a complete characterization of the classical solutions.
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
20 / 71
Classical Gravity in (2+1)-dimensions
Introduction
Classical Gravity in (2+1)-dimensions
Introduction Our goal is not to study the detailed characteristics of particular solutions, but rather to develop an understanding of the generic properties of the space of solutions. Two fundamental approaches to classical general relativity in 2+1 dimensions: I I
Arnowitt-Deser-Misner (ADM) decomposition of the metric. The first-order formalism.
In both cases, the goal is to set up the field equations in a manner that permits a complete characterization of the classical solutions.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
20 / 71
Classical Gravity in (2+1)-dimensions
Introduction
Classical Gravity in (2+1)-dimensions
Introduction Our goal is not to study the detailed characteristics of particular solutions, but rather to develop an understanding of the generic properties of the space of solutions. Two fundamental approaches to classical general relativity in 2+1 dimensions: I I
Arnowitt-Deser-Misner (ADM) decomposition of the metric. The first-order formalism.
In both cases, the goal is to set up the field equations in a manner that permits a complete characterization of the classical solutions.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
20 / 71
Classical Gravity in (2+1)-dimensions
Introduction
Classical Gravity in (2+1)-dimensions
Introduction Our goal is not to study the detailed characteristics of particular solutions, but rather to develop an understanding of the generic properties of the space of solutions. Two fundamental approaches to classical general relativity in 2+1 dimensions: I I
Arnowitt-Deser-Misner (ADM) decomposition of the metric. The first-order formalism.
In both cases, the goal is to set up the field equations in a manner that permits a complete characterization of the classical solutions.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
20 / 71
Classical Gravity in (2+1)-dimensions
The ADM decomposition
Outline 1
Abstract
2
Introduction Sumary History The pedagogical value of 2+1 dimensions
3
Why (2+1)-dimensional gravity? Sumary General relativity in 2+1 dimensions
4
Classical Gravity in (2+1)-dimensions Introduction The ADM decomposition The first-order formalism: The Chern-Simons formulation
5
Quantum Gravity in (2+1)-dimensions
6
The (2+1)-Dimensional Black Hole: BTZ BTZ Geometry
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
21 / 71
Classical Gravity in (2+1)-dimensions
The ADM decomposition
Classical Gravity in (2+1)-dimensions The ADM decomposition The Hamiltonian formulation of a field theory involves a decomposition of spacetime into space and time. Geometrically, this corresponds to a foliation of spacetime into nonintersecting spacelike hypersurfaces Σ. In this 2+1 decomposition, the spacetime metric gµν is decomposed into an induced metric hµν , a shift vector N i , and a lapse scalar N; while the induced metric is concerned with displacements within Σ, the lapse and shift are concerned with displacements away from the hypersurface. The Hamiltonian is a functional of the field configuration and its conjugate momentum on Σ. In general relativity, the Hamiltonian is a functional of hµν and its conjugate momentum π i j , which is closely related to the extrinsic curvature of the hypersurface Σ; the lapse and shift are freely specifiable, and they do not appear in the Hamiltonian as dynamical variables. The gravitational Hamiltonian inherits boundary terms from the action functional; those are defined on the two-surface S formed by the intersection of ∂ M and Σ.
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
22 / 71
Classical Gravity in (2+1)-dimensions
The ADM decomposition
Classical Gravity in (2+1)-dimensions Characteristics of ADM formalism The ADM form of the metric: ds2 = −N 2 dt 2 + gi j dxi + N i dt
dx j + N i dt
(5)
The ADM decomposition Characteristics of ADM formalism The ADM decomposition ⇔ Lorentzian version of the Pythagoras theorem
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
23 / 71
Classical Gravity in (2+1)-dimensions
The ADM decomposition
Classical Gravity in (2+1)-dimensions Poisson brackets ADM formalism allows us apply standard Hamiltonian techniques to (2+1)-dimensional gravity. δ F [g, π] F [g, π] , gi j = − δ πi j
δ F [g, π] F [g, π] , π i j = δ gi j
for any functional F of the positions and momenta. The Hamilton equations are ġi j =
δH = − H, gi j δ πi j
π̇ i j =
δH = − H, π i j δ gi j
where Z
H= Σ
d 2 x NH + Ni H i
πi j =
∂L ∂ ∂t gi j
√ 1 H = √ πi j π i j − π 2 − g (R − 2Λ) g β
The classical Poisson brackets are: {mα , pβ } = δα ,
c 2005
H i = −2∇ j π i j
{mα , mβ } = {pα , pβ } = 0.
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
24 / 71
Classical Gravity in (2+1)-dimensions
The first-order formalism: The Chern-Simons formulation
Outline 1
Abstract
2
Introduction Sumary History The pedagogical value of 2+1 dimensions
3
Why (2+1)-dimensional gravity? Sumary General relativity in 2+1 dimensions
4
Classical Gravity in (2+1)-dimensions Introduction The ADM decomposition The first-order formalism: The Chern-Simons formulation
5
Quantum Gravity in (2+1)-dimensions
6
The (2+1)-Dimensional Black Hole: BTZ BTZ Geometry
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
25 / 71
Classical Gravity in (2+1)-dimensions
The first-order formalism: The Chern-Simons formulation
Classical Gravity in (2+1)-dimensions The first-order formalism: The Chern-Simons formulation Fundamental variables: eµ a → gµν = ηab eµ a eν b where ea = eµ a dxµ and a connection ωµ ab given for ω a = 21 ε abc ωµbc dxµ the first-order action takes the form Z 1 Λ I=2 ea ∧ dωa + εabc ω b ∧ ω c + εabc ea ∧ eb ∧ ec , 2 6 M
(6)
and the Euler-Lagrange equations are: Ta = dea + εabc ω b ∧ ec = 0,
1 Λ Ra = dωa + εabc ω b ∧ ω c = − εabc eb ∧ ec . 2 2
The basic Poisson brackets are: {ei a (x), ω j b (x0 )} = 21 η ab εi j δ 2 (x − x0 )
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
26 / 71
Classical Gravity in (2+1)-dimensions
The first-order formalism: The Chern-Simons formulation
Classical Gravity in (2+1)-dimensions The first-order formalism: The Chern-Simons formulation Fundamental variables: eµ a → gµν = ηab eµ a eν b where ea = eµ a dxµ and a connection ωµ ab given for ω a = 21 ε abc ωµbc dxµ the first-order action takes the form Z 1 Λ I=2 ea ∧ dωa + εabc ω b ∧ ω c + εabc ea ∧ eb ∧ ec , 2 6 M
(6)
and the Euler-Lagrange equations are: Ta = dea + εabc ω b ∧ ec = 0,
1 Λ Ra = dωa + εabc ω b ∧ ω c = − εabc eb ∧ ec . 2 2
The basic Poisson brackets are: {ei a (x), ω j b (x0 )} = 21 η ab εi j δ 2 (x − x0 )
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
26 / 71
Classical Gravity in (2+1)-dimensions
The first-order formalism: The Chern-Simons formulation
Classical Gravity in (2+1)-dimensions The first-order formalism: The Chern-Simons formulation Fundamental variables: eµ a → gµν = ηab eµ a eν b where ea = eµ a dxµ and a connection ωµ ab given for ω a = 21 ε abc ωµbc dxµ the first-order action takes the form Z 1 Λ I=2 ea ∧ dωa + εabc ω b ∧ ω c + εabc ea ∧ eb ∧ ec , 2 6 M
(6)
and the Euler-Lagrange equations are: Ta = dea + εabc ω b ∧ ec = 0,
1 Λ Ra = dωa + εabc ω b ∧ ω c = − εabc eb ∧ ec . 2 2
The basic Poisson brackets are: {ei a (x), ω j b (x0 )} = 21 η ab εi j δ 2 (x − x0 )
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
26 / 71
Quantum Gravity in (2+1)-dimensions
Quantum Gravity in (2+1)-dimensions Goal The main goal of studying general relativity in (2+1) dimensions is to gain insight into the problems of quantum gravity. In its most basic form, quantization is the procedure of replacing the Poisson brackets of classical theory with conmutators of operators acting on somee Hilber space {x, y} → i}[x̂, ŷ]
(7)
One of the main lessons of (2+1)-dimensional gravity seems to be that a thorough understanding of the classical solutions is crucial for the formulation of a quantum theory. Before starting in on the problem of quantization, it is worth recalling why quantum gravity is so hard. The difficulties are partly technical: general relativity is a complicated, and nonlinear theory.
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
27 / 71
Quantum Gravity in (2+1)-dimensions
Quantum Gravity in (2+1)-dimensions Problems In particular, general relativity is perturbatively nonrenormalizable, and while we know a few examples of nonrenormalizable theories that can be sensibly quantized, the general problem is poorly understood. Beyond these technical failures, however, lie the basic conceptual problems that plague quantum gravity. Conventional quantum theory starts with a fixed, passive spacetime background that provides a setting in which particles and fields interact. According to general relativity, however, spacetime is itself dynamical, and much of the conventional framework becomes, at best, ambiguous. Without a fixed definition of time, we do not know how to describe dynamics or interpret probabilities. Without an a priori distinction between past, present, and future, we do not how to impose causality.
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
28 / 71
Quantum Gravity in (2+1)-dimensions
Quantum Gravity in (2+1)-dimensions ...Problems Fundamentally, we do not understand what it means to quantize the structure of spacetime. The usefulness of (2+1)-dimensional gravity comes from the fact that it eliminates the technical problems while preserving the conceptual foundations. We have seen that for typical topologies, (2+1)-dimensional general relativity has only finitely many degrees of freedom. Quantum field theory is thus reduced to quantum mechanics, and the problem of nonrenormalizability disappears. On the other hand, (2+1)-dimensional gravity is still a diffeomorphism-invariant theory of spacetime geometry, and most of the basic conceptual issues of the full theory remain unchanged.
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
29 / 71
Quantum Gravity in (2+1)-dimensions
Quantum Gravity in (2+1)-dimensions Three main directions
The covariant line of research This schemes try to apply quantum ideas in a full spacetime context; is the attempt to build the theory as a quantum field theory of the fluctuations of the metric over a flat Minkowski space, or some other background metric space. The program was started by Rosenfeld, Fierz and Pauli in the thirties. The Feynman rules of general relativity (GR, from now on) were laboriously found by DeWitt and Feynman in the sixties. t’Hooft and Veltman, Deser and Van Nieuwenhuizen, and others, found firm evidence of non-renormalizability at the beginning of the seventies. Then, a search for an extension of GR giving a renormalizable or finite perturbation expansion started. Through high derivative theory and supergravity, the search converged successfully to string theory in the late eighties.
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
30 / 71
Quantum Gravity in (2+1)-dimensions
Quantum Gravity in (2+1)-dimensions Three main directions
The canonical line of research This schemes start with a pre-quantum division of three-dimensional spacetime into a two-dimensional space plus time; is the attempt to construct a quantum theory in which the Hilbert space carries a representation of the operators corresponding to the full metric, or some functions of the metric, without background metric to be fixed. The program was set by Bergmann and Dirac in the fifties. Unraveling the canonical structure of GR turned out to be laborious. Bergmann and his group, Dirac, Peres, Arnowit Deser and Misner completed the task in the late fifties and early sixties. The formal equations of the quantum theory were then written down by Wheeler and DeWitt in the middle sixties, but turned out to be too ill-defined. A well defined version of the same equations was successfully found only in the late eighties, with loop quantum gravity.
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
31 / 71
Quantum Gravity in (2+1)-dimensions
Quantum Gravity in (2+1)-dimensions Three main directions
The sum over histories line of research Is the attempt to use some version of Feynman’s functional integral quantization to define the theory. Hawking’s Euclidean quantum gravity, introduced in the seventies, most of the the discrete (lattice-like, posets...) approaches and the spin foam models, recently introduced, belong to this line.
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
32 / 71
Quantum Gravity in (2+1)-dimensions
Quantum Gravity in (2+1)-dimensions Other directions Others: There are of course other ideas that have been explored: Twistor theory has been more fruitful on the mathematical side than on the strictly physical side, but it is still actively developing. I Noncommutative geometry has been proposed as a key mathematical tool for describing Planck scale geometry, and has recently obtained very surprising results, particularly with the work of Connes and collaborators. I Finkelstein, Sorkin, and others, pursue courageous and intriguing independent paths. I Penrose idea of a gravity induced quantum state reduction have recently found new life with the perspective of a possible experimental test. I ... I
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
33 / 71
Quantum Gravity in (2+1)-dimensions
Quantum Gravity in (2+1)-dimensions
Models of Quantum Gravity in (2+1)-dimensions Reduced phase space quantization Chern–Simons quantization Covariant canonical quantization “Quantum geometry” Lattice methods The Wheeler–DeWitt equation Lorentzian path integrals Euclidean path integrals and quantum cosmology
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
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Quantum Gravity in (2+1)-dimensions
Schwarzschild Black Hole The Schwarzschild metric 2Gm 2Gm −1 2 2 ds = − 1 − 2 dt + 1 − 2 dr + r2 dΩ2 , c r c r 2
(8) is the unique solution to the Einstein field equations that describes the vacuum spacetime outside a spherically symmetric body of mass m. The difficulties of the Schwarzschild metric is in r = 2Gm/c2 and r = 0. This problem can be circumvented by introducing another coordinate system → Even Horizon: represents the boundary of all events which can be observed in principle by an external observer. The value r = rs = 2Gm/c2 is known as the Schwarzschild radius.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
35 / 71
Quantum Gravity in (2+1)-dimensions
Schwarzschild Black Hole Properties of Schwarzschild Black Hole Is stationary; Is sphericaly symmetric Is asymtoticaly flat Is static ↔ is time-symetric (t → t 0 = −t) and time translation invariant (t → t 0 = t + constant)
Thermodynamics
Hawking temperature: TH = }κ/2π Entropy of black hole: SBH = A/4}G
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
36 / 71
Quantum Gravity in (2+1)-dimensions
Kerr Black Hole Kerr Black Hole Metric A solution to the Einstein field equations describing a rotating black hole was discovered by Roy Kerr in 1963. As we shall see, the Kerr metric can be written in a number of different ways. In the standard Boyer-Lindquist coordinates, it is given by ds2
=
2Mr 4Mar sin2 θ Σ ρ2 2 − 1 − 2 dt 2 − dtdφ + 2 sin2 θ dφ 2 + dr + ρ 2 dθ 2 2 ρ ρ ρ ∆
=
ρ 2∆ 2 Σ ρ2 2 − dt + 2 sin2 θ (dφ − ω dt)2 + dr + ρ 2 dθ 2 , Σ ρ ∆
(9)
where ρ 2 = r2 + a2 cos2 θ ,
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∆ = r2 − 2Mr + a2
Σ = (r2 + a2 )2 − a2 ∆ sin2 θ ,
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
ω ≡−
gtφ 2Mar = gφ φ Σ
November, 2005
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Quantum Gravity in (2+1)-dimensions
Kerr Black Hole Properties is stationary, axisymmetric and asymtotically flat is invariant under transformations t → −t,
φ → −φ ,
and
t → −t,
a → −a
represents the field exterior to a spinning source where the spin of the field is related to a and the angular momentum to ma has a ring singularity at x2 + y2 = a,
z=0
has two surfaces of infinite red shift 1/2 r = m ± m2 − a2 cos2 θ in the case a2 < m2 , has two event horizons: r = m ± m2 − a2
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
1/2
November, 2005
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Outline 1
Abstract
2
Introduction Sumary History The pedagogical value of 2+1 dimensions
3
Why (2+1)-dimensional gravity? Sumary General relativity in 2+1 dimensions
4
Classical Gravity in (2+1)-dimensions Introduction The ADM decomposition The first-order formalism: The Chern-Simons formulation
5
Quantum Gravity in (2+1)-dimensions
6
The (2+1)-Dimensional Black Hole: BTZ BTZ Geometry
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
39 / 71
The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
The (2+1)-Dimensional Black Hole: The hydrogen atom of Quantum Gravity
BTZ Black Hole Geometry In 2+1 dimensions any solution of the Einstein field equations Gµν = 8πGTµν − Λgµν ,
(10)
with vanishing stress-energy tensor has constant curvature. Despite this limitation, Bañados et al. [5] have made the interesting observation that when Λ = −1/`2 < 0, the field equations have a black hole solution, characterized by the metric ds2 = −N 2 dt 2 + N −2 dr2 + r2 (N φ dt + dφ )2 , −∞ < t < ∞ ,
0 < r < ∞,
(11)
0 ≤ φ ≤ 2π,
with lapse and angular shift functions N 2 (r) = −M +
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J2 r2 + 2, 2 ` 4r
N φ (r) = −
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
J . 2r2
(12)
November, 2005
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
The (2+1)-Dimensional Black Hole: BTZ
...BTZ Geometry As a space of constant curvature, this geometry can be obtained directly from anti-de Sitter space by means of appropriate identifications, as discussed in Ref. [6]. When M > 0 and |J| ≤ M` , the solution has an outer event horizon at r = r+ , where " 2 #1/2 2 M` J 2 r+ = 1+ 1− , (13) 2 M` and an inner horizon at r− = J`/2r+ . The parameters J and M have been shown to be the quasilocal angular momentum and mass of the black hole [7]; alternatively they can be expressed in terms of Casimir invariants in a gauge-theoretic formulation of (2 + 1)-dimensional gravity [8]. The parameter M can also be expressed in terms of the initial energy density of a disk of collapsing dust in AdS space [9].
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps What Have We Learned? Existence and nonuniqueness
Singularities
(2+1)-dimensional gravity as a test bed
Sums over topologies Quantized geometry
Lorentzian dynamical triangulations
Euclidean vs. Lorentzian gravity
Observables and the “problem of time”
Which approaches are equivalent?
Singularities
Higher genus
Is length quantized?
Coupling matter
“Doubly special relativity”
The cosmological constant
Topology change
Again, (2+1)-dimensional gravity as a test bed
Sums over topologies
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What Can We Still Learn?
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
42 / 71
The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps What Have We Learned? Existence and nonuniqueness
Singularities
(2+1)-dimensional gravity as a test bed
Sums over topologies
Lorentzian dynamical triangulations
Euclidean vs. Lorentzian gravity
Observables and the “problem of time”
Which approaches are equivalent?
Singularities
Higher genus
Is length quantized?
Coupling matter
“Doubly special relativity”
The cosmological constant
Topology change
Again, (2+1)-dimensional gravity as a test bed
Sums over topologies
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What Can We Still Learn?
Quantized geometry
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
42 / 71
The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps What Have We Learned
Existence and nouniqueness Perhaps the most important lesson of (2+1)-dimensional quantum gravity is that general relativity can, in fact, be quantized. While additional ingredients—strings, for instance—may have their own attractions, they are evidently not necessary for the existence of quantum gravity. More than an “existence theorem,” though, the (2+1)-dimensional models also provide a “nonuniqueness theorem”: many approaches to the quantum theory are possible, and they are not all equivalent. This is perhaps a bit of a disappointment, since many in this field had hoped that once we found a self-consistent quantum theory of gravity, the consistency conditions might be stringent enough to make that theory unique. In retrospect, though, we should not be so surprised: quantum gravity is presumably more fundamental than classical general relativity, and it is not so strange to learn that more than one quantum theory can have the same classical limit.
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps ...What Have We Learned
(2+1)-dimensional gravity as a test bed General relativity in 2+1 dimensions has provided a valuable test bed for a number of specific proposals for quantum gravity. Some of these are “classics”—the Wheeler-DeWitt equation, for instance, and reduced phase space quantization—while others, like spin foams, Lorentzian dynamical triangulations, and covariant canonical quantization, are less well established. We have discovered some rather unexpected features, such as the difficulties caused by spatial diffeomorphism invariance and the consequent nonlocality in Wheeler-DeWitt quantization, and the necessity of understanding the representations of the group of large diffeomorphisms in almost all approaches. For particular quantization programs, (2+1)-dimensional models have also offered more specific guidance: special properties of the loop operators, methods for treating noncompact groups in spin foam models, and properties of the sums over topologies have all been generalized to 3+1 dimensions.
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps ...What Have We Learned
Lorentzian dynamical triangulations A particular application of (2+1)-dimensional gravity as a test bed is important enough to deserve special mention. The program of “Lorentzian dynamical triangulations” is a genuinely new approach to quantum gravity. Given the failures of ordinary “Euclidean dynamical triangulations,” one might normally be quite skeptical of such a method. But the success in reproducing semiclassical states in 2+1 dimensions, although still fairly limited, provides a strong argument that the approach should be taken seriously.
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps ...What Have We Learned
Observables and the “problem of time” One of the deepest conceptual difficulties in quantum gravity has been the problem of reconstructing local, dynamical spacetime from the nonlocal diffeomorphism-invariant observables required by quantum gravity. The notorious “problem of time” is a special case of this more general problem of observables. As we saw in section ??, (2+1)-dimensional quantum gravity points toward a solution, allowing the construction of families of “local” and “time-dependent” observables that nevertheless commute with all constraints. The idea that “frozen time” quantum gravity is a Heisenberg picture corresponding to a fixed-time-slicing Schrödinger picture is a central insight of (2+1)-dimensional gravity. In practice, though, we have also seen that the transformation between these pictures relies on our having a detailed description of the space of classical solutions of the field equations. We cannot expect such a fortunate circumstance to carry over to full (3+1)-dimensional quantum gravity; it is an open question, currently under investigation, whether one can use a perturbative analysis of classical solutions to find suitable approximate observables.
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps ...What Have We Learned
Singularities It has long been hoped that quantum gravity might smooth out the singularities of classical general relativity. Although the (2+1)-dimensional model has not yet provided a definitive test of this idea, some progress has been made. Puzio, for example, has shown that a wave packet initially concentrated away from the singular points in moduli space will remain nonsingular. On the other hand, Minassian has recently demonstrated that quantum fluctuations do not “push singularities off to infinity” , and that several classically singular (2+1)-dimensional quantum spacetimes also have singular “quantum b-boundaries.”
c 2005
::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps ...What Have We Learned
Is length quantized? Another long-standing expectation has been that quantum gravity will lead to discrete, quantized lengths, with a minimum length on the order of the Planck length. Partial results in quantum geometry and spin foam approaches to (2+1)-dimensional quantum gravity suggest that this may be true, but also that the problem is a bit subtle. The most recent result in this area,relates the spectrum of lengths to representations of the (2+1)-dimensional Lorentz group, which can be discrete or continuous. Freidel et al. argue that spacelike intervals are p continuous, while timelike intervals are discrete, with a spectrum of the form n(n − 1)` p . The analysis is a bit tricky, since the length “observables” do not, in general, commute with the Hamiltonian constraint. A first step towards defining truly invariant operators describing distances between point particles supports this picture, but the results are not yet conclusive.
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps ...What Have We Learned
“Doubly special relativity” Quantum gravity contains two fundamental dimensionful constants, the Planck length ` p and the speed of light c. This has suggested to some that special relativity might itself be altered so that both ` p and c are constants. This requires a nonlinear deformation of the Poincaré algebra, and leads to a set of theories collectively called “doubly special relativity”. It has recently been pointed out that (2+1)-dimensional gravity automatically displays such a deformation. A few attempts have been made to connect this picture to noncommutative spacetime, mainly in the context of point particles, but it seems too early to evaluate them.
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
November, 2005
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps ...What Have We Learned
Topology change Does consistent quantum gravity require spatial topology change? The answer in 2+1 dimensions is unequivocally no: canonical quantization gives a perfectly consistent description of a universe with a fixed spatial topology. On the other hand, the path integrals seem to allow the computation of amplitudes for tunneling from one topology to another. Problems with these topology-changing amplitudes remain, particularly in the regulation of divergent integrals over zero-modes. If these can be resolved, however, we will have to conclude that we have found genuinely and deeply inequivalent quantum theories of gravity.
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
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BTZ Geometry
Next steps ...What Have We Learned
Sums over topologies In conventional descriptions of the Hartle-Hawking wave function, and in other Euclidean path integral descriptions of quantum cosmology, it is usually assumed that a few simple contributions dominate the sum over topologies. The results of (2+1)-dimensional quantum gravity indicate that such claims should be treated with skepticism; as discussed in section, the sum over topologies is generally dominated by an infinite number of complicated topologies, each individually exponentially suppressed. This is a new and unexpected result, whose implications for realistic (3+1)-dimensional gravity are just starting to be explored.
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps What Can We Still Learn?
Singularities A key question in quantum gravity is whether quantized spacetime “resolves” the singularities of classical general relativity. This is a difficult question—already classically, it is highly nontrivial to even define a singularity, and the quantum extensions of the classical definitions are far from obvious. This is an area in which (2+1)-dimensional gravity provides a natural arena, but results so far are highly preliminary.
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps ...What Can We Still Learn?
Sums over topologies Another long-standing question in quantum gravity is whether spacetime topology can (or must) undergo quantum fluctuations. Some real progress has been made in 2+1 dimensions. Often, though, the results require saddle point approximations, and pick out particular classes of saddle points. The nonperturbative summation techniques promise much deeper results, and may point toward a measure on the space of topologies analogous to the measure on the space of geometries induced by the DeWitt metric.
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps ...What Can We Still Learn?
Quantized geometry We saw above that there is some evidence for quantization of timelike intervals in (2+1)-dimensional gravity. A systematic exploration of this issue might teach us a good deal about differences among approaches to quantization. In particular, it would be very interesting to see whether any corresponding result appears in reduced phase space quantization, Wheeler-DeWitt quantization, or path integral approaches. To address this problem properly, one must introduce genuine observables for quantities such as length and area, either by adding point particles or by looking at shortest geodesics around noncontractible cycles. Note that for the torus universe, the moduli can be considered as ratios of lengths, and there is no sign that these need be discrete. This does not contradict the claims since the lengths in question are spacelike, but it does suggest an interesting dilemma in Euclidean quantum gravity, where spacelike as well as timelike intervals might naturally be quantized.
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps ...What Can We Still Learn?
Euclidean Vs. Lorentzian gravity In the Chern-Simons formalism, “Euclidean” and “Lorentzian” quantum gravity seem to be dramatically inequivalent: they have different gauge groups, different holonomies, and very different behaviors under the actions of large diffeomorphisms. In the ADM approach, on the other hand, the differences are almost invisible. This suggests that further study might finally tell us whether Euclideanization is merely a technical trick, analogous to Wick rotation in ordinary quantum field theory, or whether it gives a genuinely different theory; and, if the latter, just how different the Euclidean and Lorentzian theories are. In canonical quantization, a key step would be to relate Chern-Simons and ADM amplitudes in the Euclidean theory. In spin foam and path integral approaches, it might be possible to explicitly compare amplitudes.
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps ...What Can We Still Learn?
Which approaches are equivalent? A more general problem is to understand which of the approaches described here are equivalent. In particular, it is not obvious how much of the difference among various methods of quantization can be attributed to operator ordering ambiguities, and how much reflects a deeper inequivalence, as reflected (for instance) in different length spectra or different possibilities for topology change. An answer might help us understand just how nonunique quantum gravity in higher dimensions will be.
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps ...What Can We Still Learn?
Higher genus Most of the detailed, explicit results in (2+1)-dimensional quantum gravity hold only for the torus universe R × T 2 . This topology has some exceptional features, and might not be completely representative. In particular, the relationship between the ADM and Chern-Simons quantizations relied on a particularly simple operator ordering; it is not obvious that such an ordering can be found for the higher genus case. An extension to arbitrary genus might be too difficult, but a full treatment of the genus two topology, using the relation to hyperelliptic curves or the sigma model description, may be possible. It could also be worthwhile to further explore the case of spatially nonorientable manifolds to see whether any important new features arise.
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BTZ Geometry
Next steps ...What Can We Still Learn?
Coupling matter This review has dealt almost exclusively with vacuum quantum gravity. We know remarkably little about how to couple matter to this theory. Some limited progress has been made: for example, there is some evidence that (2+1)-dimensional gravity is renormalizable in the 1/N expansion when coupled to scalar fields. This is apparently no longer the case when gravity is coupled to fermions and a U(1) Chern-Simons gauge theory, although Anselmi has argued that if coupling constants are tuned to exact values, renormalizability can be restored, and in fact the theory can be made finite. Certain matter couplings in supergravity have been studied, and work on circularly symmetric “midi-superspace models” has led to some surprising results, including unexpected bounds on the Hamiltonian. But the general problem of coupling matter remains very difficult, not least because—except in the special case of “topological matter” —we lose the ability to represent diffeomorphisms as ISO(2, 1) gauge transformations. Difficult as it is, however, an understanding of matter couplings may be the key to many of the conceptual issues of quantum gravity. One can explore the properties of a singularity, for example, by investigating the reaction of nearby matter, and one can look for quantization of time by examining the behavior of physical clocks. Moreover, some of the deep questions of quantum gravity can be answered only in the presence of matter.
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps ...What Can We Still Learn?
The cosmological constant Undoubtedly, the biggest embarrassment in quantum gravity today is the apparent prediction, at least in effective field theory, that the cosmological constant should be some 120 orders of magnitude larger than the observed limit. Several attempts have been made to address this problem in the context of (2+1)-dimensional quantum gravity. First, Witten has suggested a novel mechanism by which supersymmetry in 2+1 dimensions might cancel radiative corrections to Λ without requiring the equality of superpartner masses, essentially because even if the vacuum is supersymmetric, the asymptotics forbid the existence of unbroken supercharges for massive states. This argument requires special features of 2+1 dimensions, though, and it is not at all clear that it can be generalized to 3+1 dimensions (although some attempts have been made in the context of “deconstruction”). Second, the discovery that the sum over topologies can lead to a divergent partition function has been extended to 3+1 dimensions, at least for Λ < 0, and it has been argued that this behavior might signal a phase transition that could prohibit a conventional cosmology with a negative cosmological constant. The crucial case of a positive cosmological constant is not yet understood, however, and if a phase change does indeed occur, its nature is still highly obscure. It may be that the nonperturbative summation over topologies could cast light on this question.
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Next steps ...What Can We Still Learn?
Again, (2+1)-dimensional gravity as a test bed As new approaches to quantum gravity are developed, the (2+1)-dimensional model will undoubtedly remain important as a simplified test bed. For example, a bit of work has been done on the null surface formulation of classical gravity in 2+1 dimensions; a quantum treatment might be possible, and could tell us more about the utility of this approach in 3+1 dimensions. Similarly, (2+1)-dimensional gravity has recently been examined as an arena in which to test for a new partially discrete, constraint-free formulation of quantum gravity.
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The (2+1)-Dimensional Black Hole: BTZ
BTZ Geometry
Acknowledgments
Thanks! The author would like to thank to God and Professors J. M. Tejeiro, J. R. Arenas, and all professors of the Observatorio Astronómico Nacional for their collaboration and their attendance and very especially to my wife and children. Start
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Appendix
Appendix Outline
7
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Appendix A note on units Derivation of Einstein Field Equations
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Appendix
A note on units
A note on units
It is customary in quantum gravity to express masses in terms of the Planck mass and lengths in terms of the the Planck length. In 2+1 dimensions the gravitational constant G has units of an inverse momentum, and the Planck mass (in units with c = 1, } = 1) is
the Planck length LP := (G}/c3 )1/2 ' 10−33 cm; the Planck time TP := LP /c = (}G/c5 ) ' 10−44 s; −5 the Planck mass MP := }/cLP = }c G ' 10 g;
the Planck energy EP = MP c2 ' 1018 GeV ; If a cosmological constant is present, |Λ|−1/2 has units of length. The theory then has a dimensionless length scale,
`=
1
(14)
16π}G |Λ|1/2
Roughly speaking, this scale measures the radius of curvature of the universe. Throughout this work, I will use units such that 16πG = 1 and } = 1, unless otherwise stated. This choice simplifies a number of equations. In concrete applications, of course it is important to restore factors of G and }.
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Appendix
Derivation of Einstein Field Equations
General relativity in 2+1 dimensions Theory of gravity obtained from the standard Einstein-Hilbert action
Theorem I=
1 16πG
Z M
√ d 3 x −g (R − 2Λ) + Imatter (1)
where M is differentiable manifold, g = det gµν , Λ is the cosmological constant and Imatter is the matter action. As in 3+1 dimensions, the resulting Euler-Lagrange equations are the standard Einstein field equations 1 Rµν − gµν R + Λgµν = −8πGTµν 2 (2)
Proof. See Appendix Derivation of Einstein Field Equations Return
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Appendix
Derivation of Einstein Field Equations
Derivation of Einstein Field Equations
We are going to define [13] the energy-momentum tensor for a material system described by an action Imatter as the “functional derivate” of Imatter with respect to gµν 1 δ Imatter = 2
Z M
√ d 3 x −gTµν (x) δ gµν
(15)
The coeffitient Tµν (x) is defined to be the energy-momentum tensor of this system. The total action is I = IG + Imatter
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
(16)
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Appendix
Derivation of Einstein Field Equations
...Derivation of Einstein Field Equations where
√ d 3 x −g (R − 2Λ)
(17)
1 d x −g Rµν − gµν R + Λgµν δ gµν 2 M
(18)
IG =
1 16πG
Z M
In [13] it is demonstrated that 1 δ IG = 16πG
Z
3 √
Combining (15) with (18), we see that the total action I is stationary with respect to arbitrary variation in gµν if and only if 1 Rµν − gµν R + Λgµν = −8πGTµν 2 which, of course, is the Einstein field equation.
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::..Darío Alberto Castro Castro::.. (Universidad Gravity in Nacional (2+1)-dimensions de Colombia)
Return
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Appendix
Derivation of Einstein Field Equations
For Further Reading
[1] Achúcarro, A. and Townsend, P. K. (1986). A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories. Phys. Lett. B180, 89-92. [2] Barrow, J. D., Burd, A. B., and Lancaster, D. (1986). Three-dimensional classical spacetimes. Class. Quant. Grav. 3 551-567. [3] Brown, J. D. (1988). Lower dimensional gravity. World Scientific, Singapure.
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Appendix
Derivation of Einstein Field Equations
For Further Reading
[1] Achúcarro, A. and Townsend, P. K. (1986). A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories. Phys. Lett. B180, 89-92. [2] Barrow, J. D., Burd, A. B., and Lancaster, D. (1986). Three-dimensional classical spacetimes. Class. Quant. Grav. 3 551-567. [3] Brown, J. D. (1988). Lower dimensional gravity. World Scientific, Singapure.
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Appendix
Derivation of Einstein Field Equations
For Further Reading
[1] Achúcarro, A. and Townsend, P. K. (1986). A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories. Phys. Lett. B180, 89-92. [2] Barrow, J. D., Burd, A. B., and Lancaster, D. (1986). Three-dimensional classical spacetimes. Class. Quant. Grav. 3 551-567. [3] Brown, J. D. (1988). Lower dimensional gravity. World Scientific, Singapure.
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Appendix
Derivation of Einstein Field Equations
[4] Carlip, S. (1998). Quantum Gravity in 2+1 dimensions. Cambridge University Press, Cambridge, UK. (2003) [5] M. Bañados, C. Teitelboim and J. Zanelli. The Black Hole in Three Dimensional Spacetime. Phys. Rev. Lett. 69, 1849 (1992). [6] M. Bañados, M. Henneaux, C. Teitelboim, and J. Zanelli. Geometry of the 2+1 Black Hole. Phys. Rev. D48, 1506 (1992).
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Appendix
Derivation of Einstein Field Equations
[4] Carlip, S. (1998). Quantum Gravity in 2+1 dimensions. Cambridge University Press, Cambridge, UK. (2003) [5] M. Bañados, C. Teitelboim and J. Zanelli. The Black Hole in Three Dimensional Spacetime. Phys. Rev. Lett. 69, 1849 (1992). [6] M. Bañados, M. Henneaux, C. Teitelboim, and J. Zanelli. Geometry of the 2+1 Black Hole. Phys. Rev. D48, 1506 (1992).
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Appendix
Derivation of Einstein Field Equations
[4] Carlip, S. (1998). Quantum Gravity in 2+1 dimensions. Cambridge University Press, Cambridge, UK. (2003) [5] M. Bañados, C. Teitelboim and J. Zanelli. The Black Hole in Three Dimensional Spacetime. Phys. Rev. Lett. 69, 1849 (1992). [6] M. Bañados, M. Henneaux, C. Teitelboim, and J. Zanelli. Geometry of the 2+1 Black Hole. Phys. Rev. D48, 1506 (1992).
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Appendix
Derivation of Einstein Field Equations
[7] J. D. Brown, J. Creighton and R. B. Mann. Phys. Rev. D, accepted for publication, gr-qc/9405007. Un. of Waterloo preprint WATPHYS-TH-94-02. [8] D. Cangemi, M. Leblanc and R. B. Mann. Gauge formulation of the spinning black hole in 2+1 dimesnional anti-de Sitter space. Phys. Rev. D48, 3606 (1993). [9] R. B. Mann and S. F. Ross. Phys. Rev. D47. 3319 (1993).
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Appendix
Derivation of Einstein Field Equations
[7] J. D. Brown, J. Creighton and R. B. Mann. Phys. Rev. D, accepted for publication, gr-qc/9405007. Un. of Waterloo preprint WATPHYS-TH-94-02. [8] D. Cangemi, M. Leblanc and R. B. Mann. Gauge formulation of the spinning black hole in 2+1 dimesnional anti-de Sitter space. Phys. Rev. D48, 3606 (1993). [9] R. B. Mann and S. F. Ross. Phys. Rev. D47. 3319 (1993).
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Appendix
Derivation of Einstein Field Equations
[7] J. D. Brown, J. Creighton and R. B. Mann. Phys. Rev. D, accepted for publication, gr-qc/9405007. Un. of Waterloo preprint WATPHYS-TH-94-02. [8] D. Cangemi, M. Leblanc and R. B. Mann. Gauge formulation of the spinning black hole in 2+1 dimesnional anti-de Sitter space. Phys. Rev. D48, 3606 (1993). [9] R. B. Mann and S. F. Ross. Phys. Rev. D47. 3319 (1993).
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Appendix
Derivation of Einstein Field Equations
[10] R. M. Wald Quantum Field Theory in Curved Sapcetime and Black Hole Thermodynamics. The university of Chicago Press (1994). [11] F. de Felice and C. J. S. Clark. Relativity on Curved Manifolds. Cambridge University Press, Cambridge, UK (1995). [12] R. De Castro. El Universo LATEX, Segunda Edición. Universidad Nacional de Colombia, Dpto. Matemáticas, (2003).
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Appendix
Derivation of Einstein Field Equations
[10] R. M. Wald Quantum Field Theory in Curved Sapcetime and Black Hole Thermodynamics. The university of Chicago Press (1994). [11] F. de Felice and C. J. S. Clark. Relativity on Curved Manifolds. Cambridge University Press, Cambridge, UK (1995). [12] R. De Castro. El Universo LATEX, Segunda Edición. Universidad Nacional de Colombia, Dpto. Matemáticas, (2003).
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Appendix
Derivation of Einstein Field Equations
[10] R. M. Wald Quantum Field Theory in Curved Sapcetime and Black Hole Thermodynamics. The university of Chicago Press (1994). [11] F. de Felice and C. J. S. Clark. Relativity on Curved Manifolds. Cambridge University Press, Cambridge, UK (1995). [12] R. De Castro. El Universo LATEX, Segunda Edición. Universidad Nacional de Colombia, Dpto. Matemáticas, (2003).
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Appendix
Derivation of Einstein Field Equations
[12] Wagoner, R. V. (1970). Scalar-tensor theory and gravitational waves. Phys. Rev. D1, 3209-3216. [13] Weinberg, S. (1972). Gravitation and Cosmology: Principles and Applications of General Theory of Relativity. John Wiley & Sons, USA.
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Appendix
Derivation of Einstein Field Equations
[12] Wagoner, R. V. (1970). Scalar-tensor theory and gravitational waves. Phys. Rev. D1, 3209-3216. [13] Weinberg, S. (1972). Gravitation and Cosmology: Principles and Applications of General Theory of Relativity. John Wiley & Sons, USA.
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