Rotation law and equilibrium quantities

2.5 Rotation law and equilibrium quantities

A special case of rotation law is uniform rotation (uniform angular velocity in the coordinate frame), which minimizes the total mass-energy of a configuration for a given baryon number and total angular momentum [49 , 147 ]. In this case, the term involving $F(Ω )$ in (20

) vanishes.

More generally, a simple choice of a differential-rotation law is

where $A$ is a constant [183

, 184

]. When $A → ∞$ , the above rotation law reduces to the uniform rotation case. In the Newtonian limit and when $A → 0$ , the rotation law becomes a so-called $j$ -constant rotation law (specific angular momentum constant in space), which satisfies the Rayleigh criterion for local dynamical stability against axisymmetric disturbances ( $j$ should not decrease outwards, $dj∕d Ω < 0$ ). The same criterion is also satisfied in the relativistic case [184

]. It should be noted that differentially rotating stars may also be subject to a shear instability that tends to suppress differential rotation [335 ].

The above rotation law is a simple choice that has proven to be computationally convenient. More physically plausible choices must be obtained through numerical simulations of the formation of relativistic stars.

Table 1: Equilibrium properties.

circumferential radius	$ψ R = e$
gravitational mass	$M = ∫(T − 1g T )taˆnbdV ab 2 ab$
baryon mass	$∫ a M0 = ρua ˆn dV$
internal energy	$U = ∫ uu ˆnadV a$
proper mass	$M = M + U p 0$
gravitational binding energy	$W = M − Mp − T$
angular momentum	$J = ∫ T ϕaˆnbdV ab$
moment of inertia	$I = J∕Ω$
kinetic energy	$T = 1 JΩ 2$

Equilibrium quantities for rotating stars, such as gravitational mass, baryon mass, or angular momentum, for example, can be obtained as integrals over the source of the gravitational field. A list of the most important equilibrium quantities that can be computed for axisymmetric models, along with the equations that define them, is displayed in Table 1. There, $ρ$ is the rest-mass density, $2 u = 𝜀 − ρc$ is the internal energy density, $a b 1∕2 nˆ = ∇at∕ |∇bt ∇ t|$ is the unit normal vector field to the t = const. spacelike hypersurfaces, and $∘ ---- dV = |3g| d3x$ is the proper 3-volume element (with $3g$ being the determinant of the 3-metric). It should be noted that the moment of inertia cannot be computed directly as an integral quantity over the source of the gravitational field. In addition, there exists no unique generalization of the Newtonian definition of the moment of inertia in general relativity and $I = J∕Ω$ is a common choice.