# Sensing the shape of spacetime: detector response and entanglement harvesting in curved space

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## Date

2019-01-24

## Authors

Ng, Keith

## Advisor

Mann, Robert

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## Publisher

University of Waterloo

## Abstract

This thesis seeks to understand what information Unruh-DeWitt detectors can gain about the structure of spacetime. By coupling to the quantum state of a field on a curved space, the detectors can gain information about their surroundings, beyond their proper acceleration. They can then receive more information than a na\"{i}ve application of the equivalence principle might suggest. Therefore, we seek to characterize the response of detector(s) in curved space, and find the limits of their abilities.
We first consider the transition probability of a single detector. We show that the transition probability is sensitive to the quantum state of the field.
Suppose a detector is enclosed by a shell of transparent matter: then, the gravitational field is completely flat inside. However, the detector can still determine it is not in flat space, and can do so faster than the signalling time to the shell. Thus, some information about the global structure of spacetime is stored locally in the vacuum state, and the detector can extract this information.
Next, we consider the question of the (3+1)-dimensional geon, a Schwarzschild black hole which has a topological identification behind its horizon. Even though it is identical to the usual black hole outside the event horizon, the detector can still distinguish the geon from an ordinary black hole. We compare our results in (3+1) dimensions to those already found for the BTZ geon in (2+1) dimensions. Thus, the vacuum state contains information about the global structure of spacetime, and this information can be extracted by a single detector.
We then consider two detectors, and the entanglement they gain by being coupled to the field. The quantity of entanglement harvested is, once again, sensitive to the structure of the spacetime in which the detectors live; we thus seek to characterize the dependence of entanglement on spacetime structure. We find a new formula for calculating the harvested entanglement, and apply it to find a new analytic expression in flat space. This new method is optimized for calculating the entanglement harvested from a state where only the mode expansion is known, as in most spacetimes; and in the special case where the detectors are spacelike separated, the formula simplifies dramatically.
Lastly, we apply our new formula to (3+1)-dimensional Anti-de Sitter spacetime, a highly symmetric space of constant negative curvature. We use our new expression to find the entanglement harvested by two detectors, both in geodesic and static configurations. We find that the entanglement is indeed affected by global structures and choices in AdS$_4$. However, we also find many new and novel phenomena in the static case, where two detectors remain static at different redshifts; these phenomena merit further investigation.

## Description

## Keywords

General Relativity, Quantum Information, Relativistic Quantum Information, Unruh-DeWitt detector, entanglement