Abstract
In anti-de Sitter space a highly accelerating observer perceives a Rindler horizon. The two Rindler wedges in AdSd+1 are holographically dual to an entangled conformal field theory that lives on two boundaries with geometry ℝ × Hd−1. For AdS3, the holographic duality is especially tractable, allowing quantum-gravitational aspects of Rindler horizons to be probed. We recover the thermodynamics of Rindler-AdS space directly from the boundary conformal field theory. We derive the temperature from the two-point function and obtain the Rindler entropy density precisely, including numerical factors, using the Cardy formula. We also probe the causal structure of the spacetime, and find from the behavior of the one-point function that the CFT “knows” when a source has fallen across the Rindler horizon. This is so even though, from the bulk point of view, there are no local signifiers of the presence of the horizon. Finally, we discuss an alternate foliation of Rindler-AdS which is dual to a CFT living in de Sitter space.
Highlights
Emphasizing that Rindler-AdS space is a advantageous spacetime for studying the quantum gravity of horizons
We find that the boundary theory “knows” when a source has fallen past the Rindler horizon even though, from a bulk point of view, there are no local invariants that mark the presence of the event horizon
Specializing to AdS3, we show that the Cardy formula precisely reproduces the Bekenstein-Hawking entropy density, including the numerical coefficient, both for nonrotating and rotating Rindler-AdS space
Summary
We would like to cover anti-de Sitter space in the Rindler coordinates natural to an accelerating observer. Consider a Rindler observer in d + 2-dimensional Minkowski space (with two time directions) uniformly accelerating in the X1 direction: X0 = ξ sinh(t/L) X1 = ξ cosh(t/L). Instead of choosing an arbitrary acceleration parameter g (as one does in flat Rindler space), we have used the existence of the AdS scale L to rescale the time coordinate such that g is replaced by 1/L; since g is unphysical, there is no loss of generality. As ξ → ∞, the transformation ξ → γξ and (χ, t) → γ−1(χ, t) is the usual scale-radius duality, and is manifestly an isometry of the asymptotic metric Another feature unique to three-dimensional anti-de Sitter space is the existence of a kind of rotating Rindler space [10]: ds2 = −.
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