2.3 The rotating fluid

When sources of non-isotropic stresses (such as a magnetic field or a solid state of parts of the star), viscous stresses, and heat transport are neglected in constructing an equilibrium model of a relativistic star, then the matter can be modeled as a perfect fluid, described by the stress-energy tensor
ab a b ab T = (𝜀 + P )u u + Pg , (11 )
where ua is the fluid’s 4-velocity. In terms of the two Killing vectors ta and ϕa, the 4-velocity can be written as
a e− ν a a u = √------2(t + Ωϕ ), (12 ) 1 − v
where v is the 3-velocity of the fluid with respect to a local ZAMO, given by
ψ− ν v = (Ω − ω)e , (13 )
and Ω ≡ u ϕ∕ut = dϕ∕dt is the angular velocity of the fluid in the coordinate frame, which is equivalent to the angular velocity of the fluid as seen by an observer at rest at infinity. Stationary configurations can be differentially rotating, while uniform rotation (Ω = const.) is a special case (see Section 2.5).
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