2.6 Equations of state2 The Equilibrium Structure of 2.4 Equations of structure

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 tex2html_wrap_inline3848 in (20Popup Equation) vanishes.

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

equation189

where A is a constant [184Jump To The Next Citation Point In The Article, 185Jump To The Next Citation Point In The Article]. When tex2html_wrap_inline3852, the above rotation law reduces to the uniform rotation case. In the Newtonian limit and when tex2html_wrap_inline3854, 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, tex2html_wrap_inline3860). The same criterion is also satisfied in the relativistic case [185Jump To The Next Citation Point In The Article]. 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.

   table197
Table 1: Equilibrium properties.

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, tex2html_wrap_inline3686 is the rest-mass density, tex2html_wrap_inline3882 is the internal energy density, tex2html_wrap_inline3884 is the unit normal vector field to the tex2html_wrap_inline3782 spacelike hypersurfaces, and tex2html_wrap_inline3888 is the proper 3-volume element (with tex2html_wrap_inline3890 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 tex2html_wrap_inline3892 is a common choice.



2.6 Equations of state2 The Equilibrium Structure of 2.4 Equations of structure

image Rotating Stars in Relativity
Nikolaos Stergioulas
http://www.livingreviews.org/lrr-2003-3
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei.mpg.de