Equations of structure

2.4 Equations of structure

Having specified an equation of state of the form $𝜀 = 𝜀(P )$ , the structure of the star is determined by solving four components of Einstein’s gravitational field equations

(where $R ab$ is the Ricci tensor and $T = T a a$ ) and the equation of hydrostationary equilibrium. Setting $ζ = μ + ν$ , one common choice for the gravitational field equations is [55

]

1- 2 2 3 −4ν ∇ ⋅ (B ∇ ν) = 2 r sin 𝜃B e ∇ ω ⋅ ∇ ω (15 ) [ (𝜀 + P )(1 + v2) ] +4 πBe2 ζ−2ν ---------2-----+ 2P , (16 ) 1 − v 2 2 3 −4ν 2 2ζ−4ν(𝜀 + P)v ∇ ⋅ (r sin 𝜃B e ∇ ω) = − 16πr sin𝜃B e ------2--, (17 ) 2ζ−2ν 1 − v ∇ ⋅ (rsin 𝜃∇B ) = 16πr sin𝜃Be P, (18 )

supplemented by a first order differential equation for $ζ$ (see [55

]). Above, $∇$ is the 3-dimensional derivative operator in a flat 3-space with spherical polar coordinates $r$ , $𝜃$ , $ϕ$ .

Thus, three of the four gravitational field equations are elliptic, while the fourth equation is a first order partial differential equation, relating only metric functions. The remaining nonzero components of the gravitational field equations yield two more elliptic equations and one first order partial differential equation, which are consistent with the above set of four equations.

The equation of hydrostationary equilibrium follows from the projection of the conservation of the stress-energy tensor normal to the 4-velocity $(δcb + ucub)∇aT ab = 0$ , and is written as

where a comma denotes partial differentiation and $i = 1,...,3$ . When the equation of state is barotropic then the hydrostationary equilibrium equation has a first integral of motion

where $F(Ω ) = u ut ϕ$ is some specifiable function of $Ω$ only, and $Ω c$ is the angular velocity on the symmetry axis. In the Newtonian limit, the assumption of a barotropic equation of state implies that the differential rotation is necessarily constant on cylinders, and the existence of the integral of motion (20

) is a direct consequence of the Poincaré–Wavre theorem (which implies that when the rotation is constant on cylinders, the effective gravity can be derived from a potential; see [301 ]).