Symmetries and Three-Dimensional Gravity

Symmetries and

Three-Dimensional Gravity

Friedrich Schöller

TU Wien (Austria)

66th Yearly Meeting of the Austrian Physical Society September 28, 2016

What Can You Expect?

I Why three-dimensional gravity

I Background structure

I Symmetries and charges

I Distinct Minkowski vacua

Why Three-Dimensional Gravity?

I Technically simple: No local gravitational degrees of freedom

I Shares conceptual features with four dimensional gravity

I Similar symmetries and conserved charges

I Symmetry reduction

Background Structure

I Provides the stage

I ⇒ Symmetries

I ⇒ Conservation laws

I For example Minkowski space

I Link different physical systems

I General relativity is “background independent”

Asymptotically Flat

I Isolated systems

I Approaches Minkowski space at infinity

I Geometric formulation: Add boundary at lightlike infinity[Penrose, 1963]

I Asymptotic structure: Induced

degenerate metric g_ij and normal vector nⁱ on the boundary

I Works in even spacetime dimensions with N ≥4[Hollands and Ishibashi, 2005]

I And inN = 3 dimensions[Ashtekar, Bičák, and Schmidt, 1997]

I t

r

Asymptotic Symmetries

I BMS Transformations= diffeomorphisms keeping (g_ij, nⁱ) invariant

I Infinite dimensional group

I Semidirect product between supertranslations and the conformal group ofS^N−2

I Supertranslations generated by: ξ=T(Ω)∂t

I N >3: Conf(S^N−2) = SO(N −1,1) = Lorentz group

I N = 3: Conf(S¹) = Diff(S¹) =superrotations

I Many Poincaré subgroups

Generalized Charges

I Not conserved if radiation present

I Covariant definition[Wald and Zoupas, 2000]

I Charges are Hamiltonians that generate symmetries

H_ξ(Σ) =H_ξ(∂Σ) =^Z

∂Σ

q

I t

r Σ ∂Σ

Coordinate Expressions

Using “retarded time” coordinateu=t−r:

g_µνdx^µdx^ν =M du²−2e^2βdu dr+r²dφ²+ 2N du dφ BMS Transformations:

ξ =T(φ) +uY⁰(φ)∂u+Y(φ)∂_φ Charges:

Hξ= 1

16πG_r→∞lim ^{Z 2π}

0

M(T+uY⁰) + 2N Y + 2rβY⁰dφ Same as[Barnich and Troessaert, 2010]

Supertranslations of Minkowski Space

Minkowski space:

gµνdx^µdx^ν =−du²−2du dr+r²dφ Supertranslation:

ξ =α(φ)∂u+ subleading Outcome:

gµνdx^µdx^ν =−du²−2du dr+r²dφ²+ 2N du dφ N =−α⁰⁰⁰(φ)−α⁰(φ)

H_ξ = 1

8πG_r→∞lim ^{Z 2π}

0 α(φ)Y⁰⁰⁰(φ) +Y⁰(φ)dφ

Summary & Outlook

Summary:

I Charges as Hamiltonian generators of symmetries

I Distinct Minkowski vacua

Work in progress[Stefan Prohazka, Jakob Salzer, FS]:

I BMS transformations as symmetries of an S-matrix

I Additional scalar field

I Gravitational memory effect

Thank you!