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While at the macroscopic level (where we live) it is easy to tell the direction of time, on the microscopic level we can run a process forwards and backwards and not be able to tell which way is which. This time reversal covariance is an important property of microscopic theories, and only disappears for us because of the arrow of time (entropy). Image borrowed from wikimedia.
Time reversal covariance, which corresponds to Onsager reciprocity, is a fundamental constraint that is often applied to hydrodynamics. Its meaning is that in a microscopic interaction one can run time forwards and backwards without being able to tell which direction it is moving in. This manifests in (quasi-)hydrodynamics as a constraint on correlators.
In this formal paper I considered linearised, general relaxation terms for a relativistic, U(1) charged fluid and determined what was consistent with time reversal covariance. In particular, I showed that it was possible to have a linearised quasihydrodynamic theory that was consistent with positivity of entropy production. I then showed how to embed this in a spacetime that was slightly curved. Doing so is extremely useful for obtaining the full suite of correlation functions as the other standard method - Martin-Kadanoff - only allows one to calculate some of the current-current correlation functions. The upshot was that any linearised fluid with the relaxation terms we were considering, must necessarily break background independence.
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