Geometry of spacetime

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 $gab$ of the form

where $ν$ , $ψ$ , $ω$ and $μ$ are four metric functions that depend on the coordinates $r$ and $𝜃$ 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 $ψ e$ in the the form

where $B$ is again a function of $r$ and $𝜃$ only [24

One arrives at the above form of the metric assuming that i) the spacetime has a timelike Killing vector field $ta$ and a second Killing vector field $ϕa$ corresponding to axial symmetry, ii) the spacetime is asymptotically flat, i.e., $tata = − 1$ , $ϕaϕa = +∞$ and $taϕa = 0$ at spatial infinity. According to a theorem by Carter [57 ], the two Killing vectors commute and one can choose coordinates $0 x = t$ and $x3 = ϕ$ (where $xa$ , $a = 0,...,3$ are the coordinates of the spacetime), such that $ta$ and $ϕa$ 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 $ta$ and $ϕa$ can be described by the remaining two coordinates $x1$ and $x2$ . A common choice for $x1$ and $x2$ are quasi-isotropic coordinates, for which $gr𝜃 = 0$ and $g𝜃𝜃 = r2grr$ (in spherical polar coordinates), or $gϖz = 0$ and $gzz = r2gϖ ϖ$ (in cylindrical coordinates). In the slow rotation formalism by Hartle [143 ], a different form of the metric is used, requiring $2 g𝜃𝜃 = gϕϕ∕ sin 𝜃$ .

The three metric functions $ν$ , $ψ$ and $ω$ can be written as invariant combinations of the two Killing vectors $ta$ and $ϕa$ , through the relations

t ta = g , (7 ) a tt ϕaϕa = gϕϕ, (8 ) t ϕa = g , (9 ) a tϕ

while the fourth metric function $μ$ determines the conformal factor $e2μ$ 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 t = const. hypersurfaces, and they are also called Eulerian observers. Then, the metric function $ω$ is the angular velocity of the local ZAMO with respect to an observer at rest at infinity. Also, $e−ν$ 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 $ψ$ has a geometrical meaning: $eψ$ 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 $gtt > 0$ . 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 $ν$ and $ω$ is

ν ∼ − M--+ Q- P2(cos𝜃), r r3 (10 ) 2J- ω ∼ r3 ,

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