Abstract
Expanding upon [arXiv:1404.4472, 1511.06079], we provide further detailed analysis of Ba\~nados geometries, the most general solutions to the AdS3 Einstein gravity with Brown-Henneaux boundary conditions. We analyze in some detail the causal, horizon and boundary structure, and geodesic motion on these geometries, as well as the two class of symplectic charges one can associate with these geometries: charges associated with the exact symmetries and the Virasoro charges. We elaborate further the one-to-one relation between the coadjoint orbits of two copies of Virasoro group and Ba\~nados geometries. We discuss that the information about the Ba\~nados goemetries fall into two categories: "orbit invariant" information and "Virasoro hairs". The former are geometric quantities while the latter are specified by the non-local surface integrals. We elaborate on multi-BTZ geometries which have some number of disconnected pieces at the horizon bifurcation curve. We study multi-BTZ black hole thermodynamics and discuss that the thermodynamic quantities are orbit invariants. We also comment on the implications of our analysis for a 2d CFT dual which could possibly be dual to AdS3 Einstein gravity.
Highlights
To fix the normalization of the horizon generating Killing vectors, we focus on the regions where ψ1ψ2 and φ1φ2 are both positive, and the event horizon is generated by ζH+
Irrespective of from which patch at the boundary we look into the bulk, one would see exactly the same geometry, with the same mass and angular momentum
In this work we elaborated on the Bañados geometries
Summary
With Brown–Henneaux boundary conditions [9]. These solutions are all locally AdS3 with local sl(2, R) × sl(2, R) isometries and their causal boundary is a 2d cylinder (in orthonormal coordinates). − r2dx+dx−, and without loss of generality one may choose x± = τ ± φ, where φ ∈ [0, 2π ] is a space-like circle, while τ is a time-like coordinate This is in accord with the fact that the causal boundary of the geometry is (part of) the cylinder parametrized by x±. With this choice, to avoid the appearance of closed time-like curves (CTCs), |r 2| cannot take a large negative value. We shall cut the r 2 range from a negative value, rC2TC1, where CTC develops and we take rC2TC1 < r 2. For illustrative purposes we discuss the special case of a BTZ black hole [7,8], corresponding to constant positive L± in the appendix and compare the relation between Bañados and the more standard BTZ-coordinate systems
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