Analogue gravity from field theory normal modes?

We demonstrate that the emergence of a curved spacetime `effective Lorentzian geometry' is a common and generic result of linearizing a classical scalar field theory around some non-trivial background configuration. This investigation is motivated by considering the large number of `analogue models' of general relativity that have recently been developed based on condensed matter physics, and asking whether there is something more fundamental going on. Indeed, linearization of a classical field theory (that is, a field-theoretic `normal-mode analysis') results in fluctuations whose propagation is governed by a Lorentzian-signature curved spacetime `effective metric'. In the simple situation considered in this paper (a single classical scalar field), this procedure results in a unique effective metric, which is quite sufficient for simulating kinematic aspects of general relativity (up to and including Hawking radiation). Upon quantizing the linearized fluctuations around this background geometry, the one-loop effective action is guaranteed to contain a term proportional to the Einstein-Hilbert action of general relativity, suggesting that while classical physics is responsible for generating an `effective geometry', quantum physics can be argued to induce an `effective dynamics'. The situation is strongly reminiscent of, though not identical to, Sakharov's `induced-gravity' scenario, and suggests that Einstein gravity is an emergent low-energy long-distance phenomenon that is insensitive to the details of the high-energy short-distance physics. (We mean this in the same sense that hydrodynamics is a long-distance emergent phenomenon, many of whose predictions are insensitive to the short-distance cut-off implicit in molecular dynamics.)