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) [65, 356, 129, 327, 326, 40]. 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 [356].
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 [376, 389, 284
].
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 [389, 284]. 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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