Abstract
We prove a positive volume theorem for asymptotically AdS spacetimes: the maximal volume slice has nonnegative vacuum-subtracted volume, and the vacuum-subtracted volume vanishes if and only if the spacetime is identically pure AdS. Under the Complexity=Volume proposal, this constitutes a positive holographic complexity theorem. The result features a number of parallels with the positive energy theorem, including the assumption of an energy condition that excludes false vacuum decay (the AdS weak energy condition). Our proof is rigorously established in broad generality in four bulk dimensions, and we provide strong evidence in favor of a generalization to arbitrary dimensions. Our techniques also yield a holographic proof of Lloyd’s bound for a class of bulk spacetimes. We further establish a partial rigidity result for wormholes: wormholes with a given throat size are more complex than AdS-Schwarzschild with the same throat size.
Highlights
Where is a reference length scale which we take to be the AdS radius L, vol[Στ ] is the cutoff-regulated volume of the hypersurface Στ, and the maximum is taken over all bulk timeslices anchored at τ in the dual spacetime (M, g)
We prove a positive volume theorem for asymptotically AdS spacetimes: the maximal volume slice has nonnegative vacuum-subtracted volume, and the vacuumsubtracted volume vanishes if and only if the spacetime is identically pure AdS
With equality if and only if (M, g) is identical to n copies of pure AdSd+1 on Σ. The upshot of this theorem is that any asymptotically AdS (AAdS) spacetime with spherical or planar symmetry that satisfies the WCC has a positive complexity of formation unless it is exactly pure AdS on the maximal volume slice, and in its domain of dependence
Summary
We prove the positive volume (complexity) theorem for AdS4, state the necessary conjecture for the proof of the statement in AdSd+1=4, and prove the general d case for certain classes of bulk spacetimes. We state our Penrose inequality for spatially symmetric spacetimes.
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