A binary configuration obviously lacks the azimuthal symmetry
that was assumed in the discussions of stationary solutions of
Einstein's equations in Section
2.4
and hydrostatic equilibrium in Section
4.1
. Fortunately, the condition of hydrostatic equilibrium requires
only the presence of a single, timelike Killing vector. With the
assumption that gravitational radiation is negligible, we can
assume that the matter is in some equilibrium state as viewed
from the reference frame that is rotating along with the binary.
That is, if the binary has a constant orbital angular velocity of
, then the time vector in the rotating frame
is a Killing vector and it is related to the time vector in the
rest frame of the binary
by
where
is a generator of rotations about the rotation axis
. Bonazzola
et al
. [18
] refer to
as a
helicoidal
Killing vector.
Two equilibrium states for the matter have been explored in
the literature. The simplest case is that of
co
-rotation, where the 4-velocity of the matter is proportional to
. In this case, the matter is at rest in the frame of reference
rotating with the binary system, the
corotating reference frame
. The second equilibrium state is that of
counter
-rotation, where there is no rotation in the
rest frame of the binary
. We will explore these two cases further below.
Stationarity of the gravitational field, unlike hydrostatic
equilibrium, requires the presence of separate timelike and
azimuthal Killing vectors. For the case of a binary system, there
is no unique definition of quasiequilibrium. The earliest results
on constructing quasiequilibrium solutions of Einstein's
equations stem from work by Wilson and Mathews [104,
105
,
106
], and others have explored similar schemes [11
,
18
,
12
]. Although written in slightly different forms, the system of
equations for the gravitational fields in all of these schemes
are fundamentally identical. While they were developed before the
conformal thin-sandwich decomposition of Section
2.3, the conformal thin-sandwich decomposition (see Eq. (51
)) offers the easiest way to interpret this approach. We consider
ourselves to be in the corotating reference frame so that our
time vector is
. To make the transition back to the rest frame of the binary as
easy as possible, we write the shift vector of our
decomposition as
so that
remains as the shift vector of the
decomposition made with respect to the rest frame of the binary
system.
The primary assumptions are that the conformal 3-metric
is flat, the initial-data slice is maximal so that
K
=0, and
. We see from (42
) that the last assumption implies that the conformal 3-geometry
is
instantaneously
stationary as seen in the corotating reference frame. The final
choice that must be made is for the conformally rescaled lapse
. An elliptic equation for the lapse can be obtained by
demanding that the trace of the extrinsic curvature
K
also be instantaneously stationary in the corotating reference
frame. This is the so-called
maximal slicing
condition on the lapse. For the particular assumptions we have
made here, this equation can be written
It is interesting to note that, for a conformally flat 3-geometry,
so
does not appear in the equations for the gravitational fields
except in the matter terms and possibly in boundary conditions.
Equations (51
) and (124
) can be solved for the gravitational fields on an initial-data
hypersurface, given values for the matter terms and appropriate
boundary conditions.
If
we choose the matter so that it is in hydrostatic equilibrium
with respect to the pseudo-Killing vector
, then these equations for the gravitational fields will yield
data that are in quasiequilibrium in the sense that
and
K
are both
instantaneously
stationary.
For corotating binaries, the matter is at rest in the
corotating reference frame of the binary. It is
rigidly
rotating and hydrostatic equilibrium is specified by solving the
relativistic Bernoulli equation (118), with
, self-consistently with the equations for the gravitational
fields.
For counterrotating binaries, the matter is not rotating in
the rest frame of the binary. Counterrotating equilibrium
configurations can be obtained by assuming the matter to have
irrotational flow
[99,
92
]. As long as the flow is
isentropic, we can express the enthalpy (117
) as
where
is the rest-mass density. For irrotational flow, the vorticity
of the fluid (cf. Ref. [99
]) is zero. Combining this with the Euler equation, we find that
the 4-velocity of the fluid can be expressed as
where
is the velocity potential, or flow field. Equation (127
), together with the normalization condition
, automatically satisfies the Euler equation and we are left
with the continuity equation which must be satisfied,
The continuity equation (128), the normalization condition, and Eq. (127
) yield
Stationarity (or quasistationarity) in the corotating reference frame requires that
where
C
is a positive constant. Now, in terms of the
decomposition with
as the shift vector (see Eq. (123
)), we find that the Bernoulli equation is written as
The flow field
must satisfy
subject to the boundary condition
at the surface of the flow,
Solving Eqs. (131) and (132
) self-consistently with the equations for the gravitational
fields yields a counterrotating binary in hydrostatic
equilibrium.
As mentioned above, the earliest work on neutron-star binaries was carried out by Wilson and Mathews [104, 105]. Wilson et al . [106] describe their approach for generating initial data for equilibrium neutron-star binaries. In these early works, the equation of hydrostatic equilibrium was not used. Rather, an initial guess for the density profile was chosen and the full hydrodynamic system was evolved with viscous damping until equilibrium was reached. During each step of the hydrodynamic evolution, the equations for the gravitational fields were resolved. The resulting data represented neither strictly co- nor counter-rotating binary neutron stars. This work led to the controversial result [75] that each neutron star in the binary may become radially unstable and collapse prior to the merger of the pair of stars. While an error was found in this work [49, 76] with the result that the signature of collapse is significantly weaker, the controversy has not yet been completely resolved.
The first use of corotating hydrostatic equilibrium with the Wilson-Mathews approach for specifying the gravitational fields was by Cook et al . [46] for the test case of an isolated neutron star. This approach was then used to study corotating neutron-star binaries by Baumgarte et al . [11, 12] and by Marronetti et al . [71]. Interestingly, turning-point methods for detecting secular instabilities [95, 96, 53] can be applied to the case of corotating binaries [13].
Corotating binary configurations are relatively easy to
construct. However, it is believed that the viscosity of
neutron-star matter is not large enough to allow for
synchronization of the spin with the orbit [61,
14]. But if the initial spins of the neutron stars are not too
large, close binaries should be well approximated by irrotational
models. Bonazzola
et al
. [18] (as corrected by Asada [7]) developed the first approach for constructing counterrotating
binary configurations. However, simpler formulations of
irrotational flow were developed independently by
Teukolsky [99] and Shibata [92], and Gourgoulhon [55] showed that all three approaches were equivalent. Numerical
solutions of the equations for irrotational flow coupled to the
equations for the gravitational fields are more difficult to
construct than those for corotation because of the boundary
condition (133) on the flow field that must be applied on the surface of each
neutron star. This boundary condition is particularly difficult
to implement because the location of the surface of the star is
not known
a priori, and will move as the equations are being solved. The first
models of irrotational binary neutron stars were constructed by
Bonazzola
et al
. [20], Marronetti
et al
. [72], and Uryu and collaborators [101,
102]. A description of the numerical methods used by Bonazzola
et al
. can be found in Refs. [57,
58].
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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 |