More generally, a simple choice of a differential-rotation law is
where
A
is a constant [184,
185
]. When
, the above rotation law reduces to the uniform rotation case. In
the Newtonian limit and when
, 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,
). The same criterion is also satisfied in the relativistic
case [185
]. 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.
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
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.
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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 |