### 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 in (20) vanishes.
More generally, a simple choice of a differential-rotation law is

where is a constant [183, 184]. When , the above rotation law reduces to the uniform
rotation case. In the Newtonian limit and when , the rotation law becomes a so-called -constant
rotation law (specific angular momentum constant in space), which satisfies the Rayleigh criterion for
local dynamical stability against axisymmetric disturbances ( should not decrease outwards,
). 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.

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, is the
rest-mass density, is the internal energy density, is the
unit normal vector field to the t = const. spacelike hypersurfaces, and is
the proper 3-volume element (with 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 is a common
choice.