Here,
and
are functions of
r
and
only.
is the angular velocity of the matter as measured at
infinity.
It is common to define v as the relative velocity between the matter and a normal observer (often called a zero angular momentum observer) so that
The velocity
v
is then fixed by the normalization condition
.
If we assume that the matter source is a perfect fluid, then the stress-energy tensor is given by
where
and
P
are the total energy density and pressure, respectively, as
measured in the rest frame of the fluid. The vanishing of the
divergence of the stress-energy tensor yields the equation of
hydrostatic equilibrium (often referred to as the relativistic
Bernoulli equation). In differential form, this is
If the fluid is barytropic
, then we can define the relativistic
enthalpy
as
and rewrite the relativistic Bernoulli equation as
The constants
,
, and
are the values their respective quantities have at some
reference point, often taken to be the surface of the neutron
star at the axis of rotation. When uniform rotation is assumed (
), Eq. (118
) is rather easy to solve. The case of differential rotation is
somewhat more complicated. An integrability condition of (116
) requires that
be expressible as a function of
, so
is a specifiable function of
which determines the rotation law that the neutron star must
obey [35
].
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Initial Data for Numerical Relativity
Gregory B. Cook http://www.livingreviews.org/lrr-2000-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |