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 matter or energy. 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 (such as Cauchy horizons) 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.
A spherically symmetric spacetime is a spacetime whose isometry group contains a subgroup which is isomorphic to the rotation group SO(3) and the orbits of this group are 2-spheres (ordinary 2-dimensional spheres in 3-dimensional Euclidean space). 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 (and this induced metric must be a multiple of the metric of a 2-sphere). Conventionally, the metric on the 2-sphere is written in polar coordinates as
g\Omega=d\theta2+\sin2\thetad\varphi2
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
M
K(M)
\dimK(M)=3
It is known (see Birkhoff's theorem) 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.
x\mu=(t,r,\theta,\phi)
One popular metric,[1] used in the study of mass inflation, is
ds2=g\mu\nudx\mudx\nu=-
dt2 | + | |
\alpha2 |
1 | ||||||
|
\left(dr-\beta | ||||
|
\right)2+r2g(\Omega).
Here,
g(\Omega)
\Omega=(\theta,\phi)
r
r
2\pir
\betat
\betat=dr/d\tau
\tau
\betar
ηij
g\mu\nu=ηij
i | |
e | |
\mu |
j | |
e | |
\nu |
where the
i | |
e | |
\mu |
t | |
e | |
\mu |
| ||||
dx |
r | |
e | |
\mu |
| ||||
dx |
\left(dr-\beta | ||||
|
\right)
\theta | |
e | |
\mu |
dx\mu=rd\theta
\phi | |
e | |
\mu |
dx\mu=r\sin\thetad\phi
where the signature was take to be
(-+++)
i | |
e | |
\mu |
=\begin{bmatrix}
1 | |
\alpha |
&0&0&0\ -
\betat | |
\alpha\betar |
&
1 | |
\betar |
&0&0\\ 0&0&r&0\\ 0&0&0&r\sin\theta\\ \end{bmatrix}
The vierbein itself is the inverse(-transpose) of the inverse vierbein
\mu | |
e | |
i |
=\begin{bmatrix}\alpha&\betat&0&0\ 0&\betar&0&0\\ 0&0&
1 | |
r |
&0\\ 0&0&0&
1 | |
r\sin\theta |
\\ \end{bmatrix}
That is,
i | |
(e | |
\mu |
)T
\nu | |
e | |
i |
i | |
=e | |
\mu |
\nu | |
e | |
i |
\nu | |
=\delta | |
\mu |
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
\partiali=e
\mu | |
i |
\partial | |
\partialx\mu |
\partialt
\partialr
\betat
\betat=\partialtr
Similarly, one has
\betar=\partialrr
which describes the gradient (in the free-falling tetrad frame) 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
\betar
\betar
\betar>0
\betar<0
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.
\Gammaijk
\Gammartt=-\partialrln\alpha
\Gammartr=
-\beta | ||||
|
+
\partial\betat | |
\partialr |
-\partialtln\betar
\Gamma\theta=\Gamma\phi=
\betat | |
r |
\Gamma\theta=\Gamma\phi=
\betar | |
r |
\Gamma\phi\theta\phi=
\cot\theta | |
r |
and all others zero.
A complete set of expressions for the Riemann tensor, the Einstein tensor and the Weyl curvature scalar can be found in Hamilton & Avelino.[1] The Einstein equations become
\nablat\beta
|
-4\pirp
\nablat\betar=4\pirf
where
\nablat
\nabla
p
f
M(r)
2M | |
r |
2 | |
-1=\beta | |
r |
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 (that is, for the assumption of a spherically symmetric black hole that is accreting charged or neutral dust, gas, plasma or dark matter, of high or low temperature, i.e. material with various equations of state.)