A hole-ographic spacetime (1310.4204v3)

Abstract: We embed spherical Rindler space -- a geometry with a spherical hole in its center -- in asymptotically AdS spacetime and show that it carries a gravitational entropy proportional to the area of the hole. Spherical AdS-Rindler space is holographically dual to an ultraviolet sector of the boundary field theory given by restriction to a strip of finite duration in time. Because measurements have finite durations, local observers in the field theory can only access information about bounded spatial regions. We propose a notion of Residual Entropy that captures uncertainty about the state of a system left by the collection of local, finite-time observables. For two-dimensional conformal field theories we use holography and the strong subadditivity of entanglement to propose a formula for Residual Entropy and show that it precisely reproduces the areas of circular holes in AdS3. Extending the notion to field theories on strips with variable durations in time, we show more generally that Residual Entropy computes the areas of all closed, inhomogenous curves on a spatial slice of AdS3. We discuss the extension to higher dimensional field theories, the relation of Residual Entropy to entanglement between scales, and some implications for the emergence of space from the RG flow of entangled field theories.

• The paper formulates spherical AdS-Rindler space as dual to a boundary sector by restricting the field theory to finite time intervals.
• It introduces differential entropy to quantify residual entanglement uncertainty and reproduces circular hole areas in AdS₃.
• The study challenges conventional views on spacetime emergence and offers new insights into gravitational entropy within holographic frameworks.

An Examination of Hole-ographic Spacetime Concepts in Holography and Entropy

This essay explores a research paper titled "A hole-ographic spacetime," which explores the concept of spherical Rindler space within the framework of holography and gravitational entropy. The paper embeds a spherical Rindler space—characterized by a spherical hole situated at the center—within the asymptotic boundary of Anti-de Sitter (AdS) spacetime. This geometric formulation enables a novel holographic interpretation and highlights a unique aspect of gravitational entropy.

The crux of the paper involves the formulation of spherical AdS-Rindler space as dual to a sector of boundary field theory. Specifically, the boundary field theory is restricted to a strip defined by finite time intervals, thereby illustrating the intrinsic limitations imposed on local observers in accessing information from bounded spatial regions. The remarkable feature of this setup is the introduction of a concept termed "differential entropy," which quantifies the residual uncertainty stemming from entanglement.

Using holographic methods and leveraging the strong subadditivity property of entanglement, the authors propose a formula for differential entropy in the context of two-dimensional conformal field theories (CFTs). This formulation reproduces the areas of circular holes within AdS$_3$, establishing a compelling correlation between geometric areas and entropic measures derived from holography. Additionally, they extend the idea of differential entropy to field theories defined on strips with variable durations, suggesting a general applicability in computing the areas of closed, inhomogeneous curves on spatial slices of AdS$_3$.

The implications of these results are noteworthy, both theoretically and practically, as they challenge conventional understandings of space emergence from RG flows in entangled field theories. One significant implication is the potential impact on how gravitational entropy is interpreted within the holographic paradigm, particularly concerning the entropic nature of information encoded in spacetime geometries. The findings evoke further exploration into higher dimensional field theories, extending the notion of differential entropy beyond the vacuum state entanglement in AdS$_3$.

This research prompts inquiries into the existence of a universal entropic measure applicable to diverse holographic systems, especially concerning discrepancies between bulk locality and boundary observables. It invites speculation on future advancements in quantum gravity frameworks, particularly where holographic dualities might elucidate broader entropy-related phenomena across different geometries.

In summary, the paper presents a meticulous exploration into the embedding of spherical Rindler spaces in holography. Its contributions toward understanding gravitational entropy provide a platform for further discussions and potential breakthroughs in the theoretical underpinnings of quantum gravity, holography, and spacetime emergence.