Relativistic fluctuating hydrodynamics

By applying the fluctuation dissipation theorem [1] to the Navier Stokes equation, Landau and Lifshitz developed long ago the standard theory of fluctuating hydrodynamics [2] [3] [4]. This theory is both appealing and successful, and it is consistent with the fluctuating hydrodynamics which is derived if fluctuation dissipation theory is first applied to the Boltzmann equation [5], and then the hydrodynamic limit is taken by conventional means [6]. It has also been applied to black hole fluctuations and vacuum decay [7]. The simplest relativistic generalization of Navier Stokes theory being the Eckart theory of dissipative fluids [8] (by simplicity, in this paper we shall not discuss the related Landau Lifhsitz theory [3], nor other ‘first order’ theories [9]), it is only natural to provide a relativistic version of fluctuating hydrodynamics by applying fluctuation dissipation theory to it; this step has been taken, and the results applied to such fields as cosmology [10]. Trouble is, Eckart theory has strong drawbacks as a relativistic theory, among them lack of stable solutions and acausal propagation of perturbations [9] [11]. This paper asks what changes must be introduced in Landau Lifshitz fluctuating hydrodynamics if the Eckart theory is replaced by a suitable relativistic theory. Unfortunately, there is not a single causal version of relativistic real fluid dynamics as compelling as the Eckart and Landau Lifshitz theories [12] [13], the main contenders