Abstract
We provide boundary conditions for three-dimensional gravity including boosted Rindler spacetimes, representing the near-horizon geometry of non-extremal black holes or flat space cosmologies. These boundary conditions force us to make some unusual choices, like integrating the canonical boundary currents over retarded time and periodically identifying the latter. The asymptotic symmetry algebra turns out to be a Witt algebra plus a twisted u(1) current algebra with vanishing level, corresponding to a twisted warped CFT that is qualitatively different from the ones studied so far in the literature. We show that this symmetry algebra is related to BMS by a twisted Sugawara construction and exhibit relevant features of our theory, including matching micro- and macroscopic calculations of the entropy of zero-mode solutions. We confirm this match in a generalization to boosted Rindler-AdS. Finally, we show how Rindler entropy emerges in a suitable limit.
Highlights
Rindler space arises generically as the near horizon approximation of non-extremal black holes or cosmological spacetimes
We provide boundary conditions for three-dimensional gravity including boosted Rindler spacetimes, representing the near-horizon geometry of non-extremal black holes or flat space cosmologies
The result (5.25) coincides with the microscopic one (5.16). In this final section we highlight some of the unusual features that we unraveled in our quest for near horizon holography
Summary
Rindler space arises generically as the near horizon approximation of non-extremal black holes or cosmological spacetimes. The key ingredient to their discovery that AdS3 Einstein gravity must be dual to a CFT2 was the imposition of precise (asymptotically AdS) boundary conditions This led to the realization that some of the bulk first class constraints become second class at the boundary, so that boundary states emerge and the physical Hilbert space forms a representation of the 2dimensional conformal algebra with specific values for the central charges determined by the gravitational theory. Upon writing surface charges as integrals over u and taking time to be periodic, the asymptotic symmetry algebra turns out to describe a warped CFT of a type not encountered before: there is no Virasoro central charge nor a u(1)-level; instead, there is a non-trivial cocycle in the mixed commutator Based on these results we determine the entropy microscopically and find that it does not coincide with the naive Rindler entropy, as a consequence of the different roles that u and x play in quasi-Rindler, versus Rindler, holography. Questions related to standard Rindler thermodynamics are relegated to appendix B
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