Gravity in Three Dimensions - How General Is Holography?

„

Das Gehirn ist der wichtigste Muskel beim Klettern.

(Your brain is the most important muscle when climbing.)

– Wolfgang Güllich German climber

G

eneral Relativity in three dimensions is very special in many regards and, as already mentioned in the introduction, there are a lot of reasons why it is beneficial to study gravity in this setup, especially if one is interested in general features of holography.

First and foremost, gravity in three dimensions is technically much simpler than in four or higher dimensions. For example the Riemann tensorR_abcd can be expressed in terms of the Ricci tensorRab, the Ricci scalarRand the metricgabas

R_abcd=g_acR_bd+g_bdR_ac−g_adR_bc−g_bcR_ad−1

2R(g_acg_bd−g_adg_bc). (1.1) Now taking also into account Einstein’s equations

R_µν+

Λ−R 2

g_µν = 8πG_NT_µν, (1.2) whereG_N is Newton’s constant in three dimensions,Λis the cosmological constant andTµν the energy-momentum tensor which encodes the local energy-momentum distribution. This implies that the curvature of spacetime in three dimensions is completely determined in terms of the local energy-momentum distribution and the value of the cosmological constant. Thus, if there are no matter sources the curvature of spacetime is completely determined by the value of the cosmological constant. This in turn also means that there are no local propagating (bulk-) degrees of freedom i.e. massless gravitons¹.

At first sight this sounds like bad news since a theory with no local propagating degrees of freedom seems to be trivial. Luckily, both local and global effects play an important role in (three dimensional) gravity so that the theory is physically non-trivial. It is also noteworthy that Einstein gravity in three dimensions is a topological theory.

1This is true for Einstein-Hilbert gravity in three dimensions. One could, however, also consider other gravity theories in three dimensions which allow for (typically massive) gravitons.

Probably the most famous example illustrating this feature is the BTZ black hole solution found by Bañados, Teitelboim and Zanelli [12]. This black hole solution is locally AdS, but at the boundary of the AdS spacetime is characterized by canonical charges which differ from the usual AdS vacuum. In addition the BTZ black hole has a horizon, singularity and exhibits an ergoregion in general.

In [11] Brown and Henneaux presented boundary conditions for three dimensional gravity, whose corresponding canonical charges generate two copies of the Virasoro algebra. This ultimately lead to the (holographic) conjecture that AdS in three dimensions can equivalently be described by a two-dimensional conformal field theory located at the boundary of AdS [15].

Since gravity in three dimensions is a purely topological theory one might expect that this theory can also be formulated in a way that makes its topological character explicit i.e. a Chern-Simons formulation. I will review Chern-Simons formulations and its properties in Chapter2. Before doing so it will be instructive to explain how one has to formulate gravity in three dimensions in order to be able to rewrite the Einstein-Hilbert action

I_EH= 1 16πG_N

d³x√

−g(R−2Λ), (1.3)

whereg≡detgµν, as a Chern-Simons action.

The action (1.3) takes as the fundamental dynamic field the symmetric tensorg_µν which acts as a symmetric bilinear form on the tangent space of the manifoldM.

Writing the metric in a given basis thus does not necessarily mean that this basis is orthonormal at each given point of spacetime. For many purposes it is, however, advantageous to have a notion of a local orthonormal laboratory frame i.e. a family of ideal observers embedded in a given spacetime. Such a family of ideal observers can be introduced in General Relativity via frame fields e^a = e^aµdx^µ, which are often also calledvielbein. This frame field is a function of the spacetime coordinates x^µand carries spacetime indices, which will be denoted by Greek lettersµ, ν, . . .and internal local Lorentz indices denoted by Latin lettersa, b, . . .. The frame fieldse^a and the metricg_µν are related by

g_µν =e^a_µe^b_νη_ab, (1.4)

whereη_abis the 2+1 dimensional Minkowski metric with signature (−,+,+). In this formulation local Lorentz indices can be raised and lowered using the Minkowski metric η_ab, while spacetime indices are raised and lowered using the spacetime metricg_µν.

The big advantage of using a formulation in terms of frame fields is that one now can very easily promote objects from a flat, Lorentz invariant setting to a description in a coordinate invariant and curved background². Take for example some objectV^a

2One example would be a formulation of the Dirac equation in curved backgrounds.

which transforms under local Lorentz transformationsΛ (x^µ)^a_b like the components of a vector,

V˜^a= Λ (x^µ)^a_bV^b. (1.5) Then one can easily describe this object in a curved background using the frame field³as

V^µ=e_a^µV^a. (1.6)

Local Lorentz invariance of the frame fields also means that there should be a gauge field associated to that local Lorentz invariance. This gauge field is the spin connectionω^ab = ω^ab_µdx^µ withω^ab_µ = −ω^ba_µ which also allows one to define a covariant derivative acting on generalized tensors i.e. tensors which have both spacetime and Lorentz indices as

D_µV^a_ν =∂_µV^a_ν +ω^a_bµV^b_ν−Γ^σ_νµV^a_σ, (1.7) whereΓ^σ_νµdenotes the affine connection associated to the metricg_µν

Γ^σ_νµ= 1

2g^σδ(∂νgδµ+∂µgνδ−∂δgνµ). (1.8) One particular convenient feature in three dimensions is that one can (Hodge) dualize the spin connection in such a way that it has the same index structure as the vielbein. In terms of Lorentz indices this can be achieved by using the 3d Levi-Civita symbol in order to obtain

ω^a= 1

2^abcω_bc ⇔ ω_ab=−_abcω^c, (1.9) where⁰¹²= 1. It is exactly this dualization of the spin connection which makes it possible to combine the vielbein and the spin connection into a single gauge field as I will review later in Chapter2.

Using the dualized spin connection one can write the associated curvature two-form R^aas

R^a= dω^a+1

2^a_bcω^b∧ω^c, (1.10) and consequently the Einstein-Hilbert-Palatini action (1.3) in terms of these new (first order) variables as

IEHP = 1 8πGN

ea∧R^a−Λ

6abce^a∧e^b∧e^c

. (1.11)

The equations of motion of the second order action (1.3) which are obtained by varying the action with respect to the metricg_µν are given by the Einstein equations (1.2). Since in the frame-like formalism one has two independent fieldse^aandω^a

3To be more precise this is the inverse of the frame fielde^aµdefined bye^aµeaν

=δµ^ν.

one has to vary (1.11) with respect to both of these fields and subsequently also obtains two equations which encode curvature and torsion respectively as

R^a= dω^a+1

2^abcω^b∧ω^c= Λ

2^abc∧e^b∧e^c, (1.12a) T^a= de^a+^a_bcω^b∧e^c= 0. (1.12b) This basic knowledge of frame fields, spin connections and how to use those two fields to cast the second order Einstein-Hilbert action (1.3) into a first order form (1.11) is already sufficient to be able to move on to the the next chapter, in which I will describe how to rewrite the Einstein-Hilbert-Palatini action (1.11) as a Chern-Simons action.

2 Chern-Simons Formulation of

Im Dokument How General Is Holography? (Seite 29-33)