Causality as a guiding principle for physics beyond General Relativity

TL;DR

This thesis argues that causality can serve as a guiding principle for physics beyond General Relativity by exploring emergent gravity scenarios inspired by analogue systems. It develops a two-part program: first, foundational analyses of background structures (including unimodular/WTDiff formulations) and their role in self-consistent graviton dynamics; second, a horizonless, non-singular-object program examining no-hair theorems and compactness bounds in horizonless spacetimes. Key findings include: (i) chronology protection emerges naturally in analogue setups and is tied to the underlying background causality; (ii) the bootstrapping of linear spin-2 theories can yield background-dependent nonlinear completions such as Unimodular Gravity, showing GR is not the unique nonlinear extension and that background structures can persist; (iii) in horizonless regimes, no-hair results and toroidal black-hole constructions illustrate how external fields constrain horizons and interior structure. Collectively, the work clarifies how background structures and emergent causality shape viable beyond-GR theories and informs the broader program of quantum gravity, including potential connections to string theory and semiclassical gravity.

Abstract

This thesis is situated within the context of quantum gravity, broadly understood as any effort to explore the interplay between gravitation and the quantum realm, without necessarily requiring the quantization of the gravitational field itself. We focus on emergent theories, particularly those in which the causal structure and geometric concepts underlying the gravitational field in General Relativity are not fundamental but instead arise from more basic underlying degrees of freedom. Our attention is directed toward emergent approaches inspired by condensed matter physics. Rather than developing a full-fledged emergent theory and analyzing its detailed consequences, this work offers a concise roadmap of analyses and reflections relevant to emergent frameworks, without committing to any specific model. The thesis is divided into two parts, reflecting the distinct tools and analyses employed in each. The first addresses fundamental and conceptual aspects of emergent theories, focusing on the role of background structures, both regarding their implications for the theory and their constructive relevance. The second assumes the absence of singularities and horizons, a feature often expected in emergent frameworks, though the analyses remain agnostic and independent of specific theoretical commitments.

Figures (45)

• Figure 1: The three figures above illustrate two inertial frames, $S$ (blue) and $S^{\prime}$ (red), with their respective origins at $\text{O}$ at $(t=0,x=0)$ and $\text{O}^{\prime}$ at $(t^{\prime}=0,x^{\prime}=0)$. The frame $S^{\prime}$ is translated and boosted with respect to $S$, as the tilt of their axes due to a hyperbolic transformation represents. The dotted lines at 45 degrees represent the light cones. If superluminal signals are allowed, the only restriction is that they must be future-oriented from the emitter’s perspective, even if they travel outside the light cones. This is shown in the upper left panel, where the shaded angular region represents the range of superluminal signals that can be sent from $\text{O}$ to the right. Notably, a signal can be sent to an observer passing through $\text{O}^{\prime}$, despite $\text{O}^{\prime}$ being spacelike separated from $\text{O}$. Similarly, in the upper right panel, the shaded region illustrates the possible superluminal signals that can be sent from $\text{O}^{\prime}$ to the left. The brown arrow represents a specific signal that ultimately reaches the past of $\text{O}$. The lower panel demonstrates how these two superluminal signals, combined with a third timelike signal depicted in orange, can create a causal paradox. In this setup, a signal originating from $\text{O}$ can be arranged to return to the past of $\text{O}$, i.e., it can reach $x = 0$ for $t<0$. In other words, an observer can receive a reply to a message before even sending the original message. This construction highlights how, in Lorentz-invariant theories, tachyons (superluminal particles) can be used to transmit signals into the past, leading to violations of causality.
• Figure 2: The figure represents a warp drive tube that starts in a location A and ends in location B. The underlying spacetime can be considered $(D+1)$-dimensional with the warp-drive bubble moving in a straight line (along the $x$-coordinate, in the picture). This is the simplest construction of a warp drive and it can be simulated in an analogue gravity model without further problems.
• Figure 3: The figure represents a warp drive tube that starts in a location $\text{A}^{\prime}$ and ends in location $B^{\prime}$. The underlying spacetime can be considered $(D+1)$-dimensional with the warp-drive bubble moving in a straight line (along the $x$-coordinate, in the picture).
• Figure 4: We represent here the setup of two warp drives engineered to allow the presence of CTCs. The purple curve represents a generic CTC on this background. The problem with this configuration $1+1$ spacetime dimensions is that the metric in the region where the two warp-drive bubbles cross is ill-defined. In higher spacetime dimensions this problem is absent, we just need to engineer the two warp drives bubbles in different parallel planes to avoid the crossing.
• Figure 5: We represent here the setup described in the text that already shows the difficulty present when trying to build CTCs in a spacetime with a trivial topology. The shaded region represents the region of abnormal behavior of the light cones. It is impossible to regularize the light cones without removing points from the spacetime, otherwise the metric would need to vanish at some point.