Spherically symmetric spacetime


In physics, spherically symmetric spacetimes are commonly used to obtain analytic and numerical solutions to Einstein's field equations in the presence of radially moving dust, compressible or incompressible fluids, or baryons. Because spherically symmetric spacetimes are by definition irrotational, they are not realistic models of black holes in nature. However, their metrics are considerably simpler than those of rotating spacetimes, making them much easier to analyze.
Spherically symmetric models are not entirely inappropriate: many of them have Penrose diagrams similar to those of rotating spacetimes, and these typically have qualitative features that are unaffected by rotation. One such application is the study of mass inflation due to counter-moving streams of infalling matter in the interior of a black hole.

Formal definition

A spherically symmetric spacetime is a spacetime whose isometry group contains a subgroup which is isomorphic to the rotation group SO and the orbits of this group are 2-spheres. The isometries are then interpreted as rotations and a spherically symmetric spacetime is often described as one whose metric is "invariant under rotations". The spacetime metric induces a metric on each orbit 2-sphere. Conventionally, the metric on the 2-sphere is written in polar coordinates as
and so the full metric includes a term proportional to this.
Spherical symmetry is a characteristic feature of many solutions of Einstein's field equations of general relativity, especially the Schwarzschild solution and the Reissner–Nordström solution. A spherically symmetric spacetime can be characterised in another way, namely, by using the notion of Killing vector fields, which, in a very precise sense, preserve the metric. The isometries referred to above are actually local flow diffeomorphisms of Killing vector fields and thus generate these vector fields. For a spherically symmetric spacetime, there are precisely 3 rotational Killing vector fields. Stated in another way, the dimension of the Killing algebra is 3; that is,. In general, none of these are time-like, as that would imply a static spacetime.
It is known that any spherically symmetric solution of the vacuum field equations is necessarily isometric to a subset of the maximally extended Schwarzschild solution. This means that the exterior region around a spherically symmetric gravitating object must be static and asymptotically flat.

Spherically symmetric metrics

Conventionally, one uses spherical coordinates, to write the metric. Several coordinate charts are possible; these include:
One popular metric, used in the study of mass inflation, is
Here, is the standard metric on the unit radius 2-sphere. The radial coordinate is defined so that it is the circumferential radius, that is, so that the proper circumference at radius
is. In this coordinate choice, the parameter is defined so that is the proper rate of change of the circumferential radius. The parameter can be interpreted as the radial derivative of the circumferential radius in a freely-falling frame; this becomes explicit in the tetrad formalism.

Orthonormal tetrad formalism

Note that the above metric is written as a sum of squares, and therefore it can be understood as explicitly encoding a vierbein, and, in particular, an orthonormal tetrad. That is, the metric tensor can be written as a pullback of the Minkowski metric :
where the is the inverse vierbein. The convention here and in what follows is that the roman indexes refer to the flat orthonormal tetrad frame, while the greek indexes refer to the coordinate frame. The inverse vierbein can be directly read off of the above metric as
where the signature was take to be. Written as a matrix, the inverse vierbein is
The vierbein itself is the inverse of the inverse vierbein
That is, is the identity matrix.
The particularly simple form of the above is a prime motivating factor for working with the given metric.
The vierbein relates vector fields in the coordinate frame to vector fields in the tetrad frame, as
The most interesting of these two are which is the proper time in the rest frame, and which is the radial derivative in the rest frame. By construction, as noted earlier, was the proper
rate of change of the circumferential radius; this can now be explicitly written as
Similarly, one has
which describes the gradient of the circumferential radius along the radial direction. This is not in general unity; compare, for example, to the standard Swarschild solution, or the Reissner–Nordström solution. The sign of effectively determines "which way is down"; the sign of distinguishes incoming and outgoing frames, so that is an ingoing frame, and is an outgoing frame.
These two relations on the circumferential radius provide another reason why this particular parameterization of the metric is convenient: it has a simple intuitive characterization.

Connection form

The connection form in the tetrad frame can be written in terms of the Christoffel symbols in the tetrad frame, which are given by
and all others zero.

Einstein equations

A complete set of expressions for the Riemann tensor, the Einstein tensor and th Weyl curvature scalar can be found in Hamilton & Avelino. The Einstein equations become
where is the covariant time derivative, the radial pressure, and the radial energy flux. The mass is the Misner-Thorne mass or interior mass, given by
As these equations are effectively two-dimensional, they can be solved without overwhelming difficulty for a variety of assumptions about the nature of the infalling material