7.12 Quantum gravity – fundamental models

When it comes to dealing with “fundamental” theories of quantum gravity, the analogue models play an interesting role which is complementary to the more standard approaches. The search for a quantum theory of gravity is fundamentally a search for an appropriate mathematical structure in which to simultaneously phrase both quantum questions and gravitational questions. More precisely, one is searching for a mathematical framework in which to develop an abstract quantum theory which then itself encompasses classical Einstein gravity (the general relativity), and reduces to it in an appropriate limit [113Jump To The Next Citation Point, 580Jump To The Next Citation Point, 221Jump To The Next Citation Point].

The three main approaches to quantum gravity currently in vogue, “string models” (also known as “M-models”), “loop space” (and the related “spin foams”), and “lattice models” (Euclidean or Lorentzian) all share one feature: They attempt to develop a “pre-geometry” as a replacement for classical differential geometry (which is the natural and very successful mathematical language used to describe Einstein gravity) [113, 580Jump To The Next Citation Point, 221, 534, 533, 83, 507]. The basic idea is that the mooted replacement for differential geometry would be relevant at extremely small distances (where the quantum aspects of the theory would be expected to dominate), while at larger distances (where the classical aspects dominate) one would hope to recover both ordinary differential geometry and specifically Einstein gravity or possibly some generalization of it. The “string”, “loop”, and “lattice” approaches to quantum gravity differ in detail in that they emphasise different features of the long-distance model, and so obtain rather different short-distance replacements for classical differential geometry. Because the relevant mathematics is extremely difficult, and by and large not particularly well understood, it is far from clear which, if any, of these three standard approaches will be preferable in the long run [580].

A recent (Jan. 2009) development is the appearance of Hořava gravity [287Jump To The Next Citation Point, 288Jump To The Next Citation Point, 289Jump To The Next Citation Point]. This model is partially motivated by condensed matter notions such as (deeply non-perturbative) anomalous scaling and the existence of a “Lifshitz point”, and additionally shares with most of the analogue spacetimes the presence of modified dispersion relations and high-energy deviations from Lorentz invariance [287, 288, 289, 634, 635, 585, 584, 681]. Though Hořava gravity is not directly an analogue model per se, there are deep connections – with some steps toward an explicit connection being presented in [694].

We feel it likely that analogue models can shed new light on this very confusing field by providing a concrete specific situation in which the transition from the short-distance “discrete” or “quantum” theory to the long-distance “continuum” theory is both well understood and non-controversial. Here we are specifically referring to fluid mechanics, where, at short distances, the system must be treated using discrete atoms or molecules as the basic building blocks, while, at large distances, there is a well-defined continuum limit that leads to the Euler and continuity equations. Furthermore, once one is in the continuum limit, there is a well-defined manner in which a notion of “Lorentzian differential geometry”, and in particular a “Lorentzian effective spacetime metric” can be assigned to any particular fluid flow [607, 624Jump To The Next Citation Point, 470Jump To The Next Citation Point]. Indeed, the “analogue gravity programme” is extremely successful in this regard, providing a specific and explicit example of a “discrete” → “continuum” → “differential geometry” chain of development. What the “analogue gravity programme” does not seem to do as easily is to provide a natural direct route to the Einstein equations of general relativity, but that merely indicates that current analogies have their limits and therefore, one should not take them too literally [624, 470]. Fluid mechanics is a guide to the mathematical possibilities, not an end in itself. The parts of the analogy that do work well are precisely the steps where the standard approaches to quantum gravity have the most difficulty, and so it would seem useful to develop an abstract mathematical theory of the “discrete” → “continuum” → “differential geometry” chain using this fluid mechanical analogy (and related analogies) as inspiration.

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