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We revisit the theoretical analysis of an expanding ring-shapedBose-Einstein condensate. Starting from the action and integrating overdimensions orthogonal to the phonon’s direction of travel, we derive aneffective one-dimensional wave equation for azimuthally-travellingphonons. This wave equation shows that expansion redshifts the phononfrequency at a rate determined by the effective azimuthal sound speed,and damps the amplitude of the phonons at a rate given by \dot{\mathcal{V}}/{\mathcal{V}} 𝒱 ̇ / 𝒱 ,where \mathcal{V} 𝒱 is the volume of the background condensate. This behavior is analogousto the redshifting and ``Hubble friction’’ for quantum fields in theexpanding universe and, given the scalings with radius determined by theshape of the ring potential, is consistent with recent experimental andtheoretical results. The action-based dimensional reduction methods usedhere should be applicable in a variety of settings, and are well suitedfor systematic perturbation expansions. 

We derive the Einstein equation from the condition that every small causal diamond is a variation of a flat empty diamond with the same free conformal energy, as would be expected for a near-equilibrium state. The attractiveness of gravity hinges on the negativity of the absolute temperature of these diamonds, a property we infer from the generalized entropy. 

The static patch of de Sitter spacetime and the Rindler wedge of Minkowski spacetime are causal diamonds admitting a true Killing field, and they behave as thermodynamic equilibrium states under gravitational perturbations. We explore the extension of this gravitational thermodynamics to all causal diamonds in maximally symmetric spacetimes. Although such diamonds generally admit only a conformal Killing vector, that seems in all respects to be sufficient. We establish a Smarr formula for such diamonds and a ``first law" for variations to nearby solutions. The latter relates the variations of the bounding area, spatial volume of the maximal slice, cosmological constant, and matter Hamiltonian. The total Hamiltonian is the generator of evolution along the conformal Killing vector that preserves the diamond. To interpret the first law as a thermodynamic relation,it appears necessary to attribute a negative temperature to the diamond, as has been previously suggested for the special case of the static patch of de Sitter spacetime. With quantum corrections included, for small diamonds we recover the ``entanglement equilibrium'' result that the generalized entropy is stationary at the maximally symmetric vacuum at fixed volume, and we reformulate this as the stationarity of free conformal energy with the volume not fixed.