Quantum Aspects of Space and Time

Abstract

In this thesis, we explore different ways in which spacetime exhibits peculiar properties when subjected to the rules of quantum mechanics. These rules are naturally implemented at the level of semiclassical physics, where the dynamical nature of the spacetime metric is neglected. In particular, we explore how quantum effects modify some of the fundamental statements of General Relativity, ranging from different possible solutions, such as traversable wormholes and time machines, to some of the more foundational conjectures, with an emphasis to the one of cosmic censorship. Chapter One takes a deeper look into the connection between geometry and entropy. We revisit the original reasoning leading to their entwinement, and we clarify the different notions of entropy that play a role in it. We emphasize the recurring theme and the pattern in such a relationship: how the union between area and entropy makes sense when put together on the same footing, hinting towards a deeper meaning in a complete theory of quantum gravity. This seemingly simple unification is then shown to lead to incredible results, ranging from improved conjectures about quantum gravity, to illuminating one of the most critical problems of modern theoretical physics - the black hole information paradox. In particular, we mainly focus on one example of semiclassical statements, the (quantum) Penrose inequality, and we show in detail the difficulties one has to overcome for a meaningful conjecture to hold. Furthermore, we revise the basic arguments underlying the recent progress regarding the black hole interior and lay out the possible paths to the interpretation of these striking results. Chapter Two explores different solutions that classical General Relativity forbade, but quantum physics advanced. A number of no-go theorems get circumvented, and configurations previously thought of as impossible become available, and even natural. This is especially clear for solutions such as traversable wormholes and their inherent use in studies of entanglement structures. Indeed, such connections will be relevant in gauge/gravity duality for a fuller understanding of the holographic dictionary. But we can also see the way in which other no-go theorems become easier to infer. In essence, the creation of closed causal curves was understood as a problem of quantum gravity due to the incredibly high energies one seems to need for their demise. However, we show how simple, low-energy arguments are enough to shatter the fiction of time machines. The final Chapter Three perhaps comes closer to the study of quantum gravity than the previous ones. We undertake the problem of naked singularities in gravity, and we see how including quantum effects solidifies some foundational statements while completely fragmenting other ones. In a nutshell, the strong cosmic censorship conjecture is shown to be on much firmer ground than previously thought. Quantum physics is used to destabilize the relevant Cauchy horizon once and for all. However, including quantum effects necessarily means we must abandon our na¨ıve understanding of the weak cosmic censorship and embark on a much stranger path towards a meaningful statement about naked singularities. In doing so, we discuss the purpose of cosmic censorship and its interpretation in the realm of quantum gravity. We finish the dissertation with a summary and a further discussion on the nature of naked sin- gularities, providing a framework in which these questions can be meaningfully posed. After a brief overview of recent developments in this research line, we discuss the possible ways in which we can tackle such a perplexing problem. Namely, the role of critical phenomena in gravitational collapse is emphasized, and a proposal for a future study is outlined.

Como es propio de toda teoría clásica, la Relatividad General no puede aspirar a ser más que una teoría efectiva, cuyo campo de estudio se reduce al de fenómenos emergentes de estructuras más elementales. Sin embargo, se trata de una teoría dificil de tratar al poseer propiedades no compartidas por el resto de teorías clásicas: una descripción holográfica. A pesar de no haber proporcionado todas las respuestas que buscábamos acerca de la naturaleza del espacio y del tiempo, la holografía ha jugado un papel fundamental; en especial mostrándonos una conexión entre nociones tan dispares como la información cuántica y la geometría, similar a la conexión que Gibbons y Hawking [1] dieron a conocer entre el área y la entropía. Esta tesis tiene como objetivo el estudio de casos en los que esta relación se vuelve manifiesta, usando el régimen semiclásico de gravedad.

El primer capítulo profundiza en la conexión entre área y entropía y algunas de las consecuencias que esta implica: la formulación semiclásica de la Desigualdad de Penrose y las posibles intepretaciones relativas al interior de los agujeros negros. El segundo capítulo se adentra en el estudio de escenarios prohibidos por la Relatividad General pero que resultan accesibles, y naturales, al considerar efectos cuánticos. Se centra en los agujeros de gusano y su relación con el entrelazamiento cuántico (a través de la dualidad “gauge/gravity”), así como en la imposibilidad de transformarse en máquinas del tiempo. El capítulo tercero es el que más avanza hacia el régimen cuántico de la gravedad, explorando el problema de las singularidades desnudas y la Hipótesis de la Censura Cósmica. Se muestra cómo la versión fuerte sale reforzada tras un análisis semiclásico, mientras que la versión débil requiere de nuevas reinterpretaciones para su adaptación a la nueva realidad cuántica. Finalmente se ofrece un resumen junto con una discusión adicional sobre la naturaleza de las singularidades desnudas, con un pequeño repaso sobre los avances en este campo y las posibles rutas que tomar, haciendo hincapié en el papel del colapso crítico gravitatorio y proponiendo una línea de investigación más allá de esta tesis.

Bibliografía: [1] G. W. Gibbons and S. W. Hawking, “Action integrals and partition functions in quantum gravity,” Phys. Rev. D 15 (May, 1977) 2752–2756. https://link.aps.org/doi/10.1103/PhysRevD.15.2752.