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If rapidly rotating neutron stars were nonaxisymmetric,
they would emit gravitational waves in a very short time scale and
settle down to axisymmetric configurations. Moreover, the gravity of
typical neutron star is strong because
 
(1) 
where G is the gravitational constant, and and are the mass and the radius of a typical
neutron star. Therefore, we need to solve for rotating and axisymmetric
configurations in the framework of general relativity.
For the matter and the spacetime we make the following assumptions:
 1.
 The matter distribution and the spacetime are axisymmetric.
 2.
 The matter and the spacetime are in a stationary state.
 3.
 The matter has no meridional motions. The only motion of
the matter is a circular one that is represented by the angular velocity.
 4.
 The angular velocity is constant, as seen by a distant observer at rest.
 5.
 The matter can be described as a perfect fluid.
Under these assumptions, the metric can be expressed as follows
(Papapetrou 1966; Carter 1969):
 

 (2) 
 
 (3) 
 (4) 
where t is the time coordinate and the polar coordinates are used. The metric depends on four metric functions (or
metric potentials) which are functions of r and only. Different
authors have used the set of functions , or , which are related
to each other through (2). In (2) and throughout the
text we use gravitational units (c=G=1), unless otherwise stated.
The energy momentum tensor is expressed as
 
(5) 
where , p and are the energy density, the pressure
and the fourvelocity, respectively. In the coordinate basis defined
by (2), the components of the fourvelocity are
 
(6) 
where the proper velocity v with respect to a local zero angular momentum
observer is defined by
 
(7) 
Using the metric and the energy momentum tensor mentioned above,
we can write down the Einstein equations for the metric components.
Although we omit detailed expressions for the Einstein equations here,
one can easily derive them by straightforward calculations or consult
the papers by Butterworth & Ipser (1976), Komatsu et al. (1989a)
and Bonazzola et al. (1993).
The equation of hydrostationary equilibrium can be derived from the
equations of motion and takes the following form:
 
(8) 
This equation can be integrated, if we specify the equation of state
which relates the energy density to the pressure. For a given EOS, a
model is uniquely specified by fixing two parameters, such as the
central energy density and the ratio of the polar to the
equatorial coordinate radius, . Then, four Einstein
field equations and the equation of hydrostationary equilibrium must be
solved with appropriate boundary conditions, to yield the four metric
functions and the density distribution. The available codes for
obtaining relativistic rotating neutron star models differ basically
in the choice and method of integration of the four field equations
and in the finite grid and finite difference scheme used for the
integration.
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