Abstract
The equilibrium conditions of a relativistic fluid with nonzero viscosity and heat conduction coefficients are known to reduce to Einstein’s equations for a barotropic perfect fluid in rigid motion. We consider here the linearization of these equations on a static spherically symmetric background and show that the solution space is three-dimensional (parametrized by the angular velocity vector, for example), provided the exterior vacuum region is asymptotically Euclidean and the equation of state ρ=ρ (p) (satisfying ρ⩾p⩾0 and dp/dρ≳0) is fixed, as well as the central values of the pressure and the gravitational potential. In the exterior region this solution agrees with the Kerr solution, linearized on the Schwarzschild background. This result is the first step towards proving a certain uniqueness of the possible equilibrium configurations of slowly rotating relativistic fluids. It is obtained using invariantly defined global conditions, without assuming the existence of particular coordinate systems.
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