where
,
,
and
are four metric functions that depend on the coordinates
r
and
only (see
e.g.
Bardeen and Wagoner [26]). Unless otherwise noted, we will assume
c
=
G
=1. In the exterior vacuum, it is possible to reduce the number
of metric functions to three, but as long as one is interested in
describing the whole spacetime (including the source-region of
nonzero pressure), four different metric functions are required.
It is convenient to write
in the the form
where
B
is again a function of
r
and
only [24
].
One arrives at the above form of the metric assuming that
i)
the spacetime has a timelike Killing vector field
and a second Killing vector field
corresponding to axial symmetry,
ii)
the spacetime is asymptotically flat,
i.e.
,
and
at spatial infinity. According to a theorem by Carter [57], the two Killing vectors commute and one can choose coordinates
and
(where
,
are the coordinates of the spacetime), such that
and
are coordinate vector fields. If, furthermore, the source of the
gravitational field satisfies the circularity condition (absence
of meridional convective currents), then another theorem [58] shows that the 2-surfaces orthogonal to
and
can be described by the remaining two coordinates
and
. A common choice for
and
are
quasi-isotropic coordinates, for which
and
(in spherical polar coordinates), or
and
(in cylindrical coordinates). In the slow rotation formalism by
Hartle [143
], a different form of the metric is used, requiring
.
The three metric functions
,
and
can be written as invariant combinations of the two Killing
vectors
and
, through the relations
while the fourth metric function
determines the conformal factor
that characterizes the geometry of the orthogonal
2-surfaces.
There are two main effects that distinguish a rotating
relativistic star from its nonrotating counterpart: The shape of
the star is flattened by centrifugal forces (an effect that first
appears at second order in the rotation rate), and the local
inertial frames are dragged by the rotation of the source of the
gravitational field. While the former effect is also present in
the Newtonian limit, the latter is a purely relativistic effect.
The study of the dragging of inertial frames in the spacetime of
a rotating star is assisted by the introduction of the local
Zero-Angular-Momentum-Observers (ZAMO) [23,
24]. These are observers whose worldlines are normal to the
hypersurfaces, and they are also called
Eulerian
observers. Then, the metric function
is the angular velocity of the local ZAMO with respect to an
observer at rest at infinity. Also,
is the time dilation factor between the proper time of the local
ZAMO and coordinate time
t
(proper time at infinity) along a radial coordinate line. The
metric function
has a geometrical meaning:
is the
proper circumferential radius
of a circle around the axis of symmetry. In the nonrotating
limit, the metric (5
) reduces to the metric of a nonrotating relativistic star in
isotropic coordinates
(see [321] for the definition of these coordinates).
In rapidly rotating models, an
ergosphere
can appear, where
. In this region, the rotational frame-dragging is strong enough
to prohibit counter-rotating time-like or null geodesics to
exist, and particles can have negative energy with respect to a
stationary observer at infinity. Radiation fields (scalar,
electromagnetic, or gravitational) can become unstable in the
ergosphere [108], but the associated growth time is comparable to the age of the
universe [68].
The asymptotic behaviour of the metric functions
and
is
where
M,
J
and
Q
are the gravitational mass, angular momentum and quadrupole
moment of the source of the gravitational field (see
Section
2.5
for definitions). The asymptotic expansion of the dragging
potential
shows that it decays rapidly far from the star, so that its
effect will be significant mainly in the vicinity of the
star.
![]() |
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 |