[Paper Review] Analogue spacetimes: Toy models for "quantum gravity''

Matt Visser, Silke Weinfurtner|ArXiv.org|Dec 4, 2007

Quantum Electrodynamics and Casimir Effect47 references4 citations

TL;DR

This paper proposes analogue spacetimes—particularly acoustic models in fluids and Bose-Einstein condensates—as controlled, physically grounded toy models for exploring quantum gravity phenomenology. By studying dispersion relations and effective geometries, it demonstrates that Lorentz violation can be systematically tuned and that emergent spacetime structures like rainbow metrics and pseudo-Finsler geometries naturally arise, offering testable hints for quantum gravity beyond standard approaches.

ABSTRACT

Why are "analogue spacetimes'' interesting? For the purposes of this workshop the answer is simple: Analogue spacetimes provide one with physically well-defined and physically well-understood concrete models of many of the phenomena that seem to be part of the yet incomplete theory of "quantum gravity'', or more accessibly, "quantum gravity phenomenology''. Indeed "analogue spacetimes'' provide one with concrete models of "emergence'' (whereby the effective low-energy theory can be radically different from the high-energy microphysics). They also provide many concrete and controlled models of "Lorentz symmetry breaking'', and extensions of the usual notions of pseudo-Riemannian geometry such as "rainbow spacetimes'', and pseudo-Finsler geometries, and more. I will provide an overview of the key items of "unusual physics'' that arise in analogue spacetimes, and argue that they provide us with hints of what we should be looking for in any putative theory of "quantum gravity''. For example: The dispersion relations that naturally arise in the known emergent/analogue spacetimes typically violate analogue Lorentz invariance at high energy, but do not do so in completely arbitrary manner. This suggests that a search for arbitrary violations of Lorentz invariance is possibly overkill: There are a number of natural and physically well-motivated restrictions one can put on emergent/ analogue dispersion relations, considerably reducing the plausible parameter space.

Motivation & Objective

• To explore whether spacetime and gravity could emerge from more fundamental microscopic physics, inspired by fluid dynamics as a historical precedent.
• To investigate how effective spacetime geometries and modified dispersion relations arise in systems with known microphysics, such as water waves and BECs.
• To identify physically motivated constraints on Lorentz symmetry breaking in quantum gravity phenomenology, avoiding arbitrary modifications.
• To demonstrate that key features of quantum gravity—such as rainbow metrics, pseudo-Finsler structures, and signal velocity behavior—can be realized in concrete analogue models.
• To argue that emergent gravity and effective field theories may be viable even without full knowledge of the underlying quantum gravity degrees of freedom.

Proposed method

• Use of acoustic spacetime models in hydrodynamic systems (e.g., shallow water, BECs) to simulate effective Lorentzian geometries.
• Analysis of dispersion relations derived from the hydrodynamic and quantum mechanical equations of motion, particularly the Bogoliubov spectrum.
• Derivation of phase and group velocities in terms of wavenumber and depth, using the relation $ c_{ ext{phase}}^2 = c_0^2 \left\{1 + \epsilon (kd)^2 \right\} \frac{\tanh(kd)}{kd} $, where $ \epsilon = \frac{3\sigma}{\rho g d} $.
• Expansion of the phase velocity in power series of $ kd $ to identify the leading-order Lorentz-violating terms.
• Identification of the critical value $ \epsilon = 1/3 $, which cancels the first-order Lorentz-violating term in the dispersion relation.
• Comparison of signal velocity with phase and group velocities to assess causality and the onset of non-standard behavior.

Experimental results

Research questions

• RQ1 Can effective spacetime geometries and modified dispersion relations emerge from known microscopic physics in condensed matter systems?
• RQ2 How can Lorentz symmetry breaking be systematically controlled and tuned in analogue models, and what are the physical constraints on such violations?
• RQ3 To what extent can analogue models reproduce complex geometric structures like rainbow metrics or pseudo-Finsler geometries?
• RQ4 What are the implications of tunable dispersion relations for quantum gravity phenomenology, particularly in delaying or eliminating observable Lorentz violation?
• RQ5 Can the emergence of Einstein gravity be derived from a class of analogue models, and what does this imply for the fundamental nature of spacetime?

Key findings

• The dispersion relation in shallow water waves yields a Bogoliubov-type spectrum with a tunable parameter $ \epsilon $, which governs the degree of Lorentz violation.
• At $ \epsilon = 1/3 $, the leading-order Lorentz-violating term in the phase velocity expansion is canceled, corresponding to a water depth of approximately 0.47 cm for standard conditions.
• Both subluminal and superluminal behavior can coexist in different wavenumber regimes, challenging the simplistic classification of dispersion relations as purely superluminal or subluminal.
• The signal velocity is infinite in the models studied, indicating that causality is not necessarily violated in the same way as in standard field theory with finite signal speeds.
• Analogue models naturally generate rainbow metrics and pseudo-Finsler geometries, suggesting that such structures are not merely mathematical curiosities but physically realizable in effective theories.
• The existence of a hydrodynamic limit with finite group velocity and controlled dispersion supports the viability of effective field theory approaches even in the absence of full knowledge of the underlying microphysics.