The stress-energy tensor is decomposed as T = S + n j + n j + n n ρ μν μν μ ν ν μ μ ν or, equivalently, S ≡⊥ T ij ij, j ≡ − ⊥ nμT i iμ, and ρ ≡ nμnνT μν where ⊥ denotes the operation of projection onto the spatial hypersurface and it acts on all free indices. Note that the matter terms in Eqs. (12View Equation), (14View Equation), and (15View Equation) are defined with respect to a normal observer. This is in contrast to the usual definition of ρ with respect to the rest frame of the fluid in hydrodynamics.