2.3 The rotating fluid2 The Equilibrium Structure of 2.1 Assumptions

2.2 Geometry of spacetime

In general relativity, the spacetime geometry of a rotating star in equilibrium can be described by a stationary and axisymmetric metric tex2html_wrap_inline3700 of the form

  equation63

where tex2html_wrap_inline3702, tex2html_wrap_inline3704, tex2html_wrap_inline3706 and tex2html_wrap_inline3708 are four metric functions that depend on the coordinates r and tex2html_wrap_inline3712 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 tex2html_wrap_inline3716 in the the form

equation71

where B is again a function of r and tex2html_wrap_inline3712 only [24Jump To The Next Citation Point In The Article].

One arrives at the above form of the metric assuming that i) the spacetime has a timelike Killing vector field tex2html_wrap_inline3724 and a second Killing vector field tex2html_wrap_inline3726 corresponding to axial symmetry, ii) the spacetime is asymptotically flat, i.e. tex2html_wrap_inline3728, tex2html_wrap_inline3730 and tex2html_wrap_inline3732 at spatial infinity. According to a theorem by Carter [57], the two Killing vectors commute and one can choose coordinates tex2html_wrap_inline3734 and tex2html_wrap_inline3736 (where tex2html_wrap_inline3738, tex2html_wrap_inline3740 are the coordinates of the spacetime), such that tex2html_wrap_inline3724 and tex2html_wrap_inline3726 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 tex2html_wrap_inline3724 and tex2html_wrap_inline3726 can be described by the remaining two coordinates tex2html_wrap_inline3750 and tex2html_wrap_inline3752 . A common choice for tex2html_wrap_inline3750 and tex2html_wrap_inline3752 are quasi-isotropic coordinates, for which tex2html_wrap_inline3758 and tex2html_wrap_inline3760 (in spherical polar coordinates), or tex2html_wrap_inline3762 and tex2html_wrap_inline3764 (in cylindrical coordinates). In the slow rotation formalism by Hartle [143Jump To The Next Citation Point In The Article], a different form of the metric is used, requiring tex2html_wrap_inline3766 .

The three metric functions tex2html_wrap_inline3702, tex2html_wrap_inline3704 and tex2html_wrap_inline3706 can be written as invariant combinations of the two Killing vectors tex2html_wrap_inline3724 and tex2html_wrap_inline3726, through the relations

eqnarray90

while the fourth metric function tex2html_wrap_inline3708 determines the conformal factor tex2html_wrap_inline3780 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 tex2html_wrap_inline3782 hypersurfaces, and they are also called Eulerian observers. Then, the metric function tex2html_wrap_inline3706 is the angular velocity of the local ZAMO with respect to an observer at rest at infinity. Also, tex2html_wrap_inline3786 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 tex2html_wrap_inline3704 has a geometrical meaning: tex2html_wrap_inline3716 is the proper circumferential radius of a circle around the axis of symmetry. In the nonrotating limit, the metric (5Popup Equation) 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 tex2html_wrap_inline3794 . 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 tex2html_wrap_inline3702 and tex2html_wrap_inline3706 is

  equation108

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 tex2html_wrap_inline3706 shows that it decays rapidly far from the star, so that its effect will be significant mainly in the vicinity of the star.



2.3 The rotating fluid2 The Equilibrium Structure of 2.1 Assumptions

image Rotating Stars in Relativity
Nikolaos Stergioulas
http://www.livingreviews.org/lrr-2003-3
© Max-Planck-Gesellschaft. ISSN 1433-8351
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