(where
is the Ricci tensor and
) and the equation of hydrostationary equilibrium. Setting
, one common choice for the gravitational field equations
is [55
]
supplemented by a first order differential equation for
(see [55
]). Above,
is the 3-dimensional derivative operator in a flat 3-space with
spherical polar coordinates
r,
,
.
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
, and is written as
where a comma denotes partial differentiation and
. When the equation of state is barotropic then the
hydrostationary equilibrium equation has a first integral of
motion
where
is some specifiable function of
only, and
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 [302]).
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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 |