Ferrando & Saez invariant characterization of LTB dusts
Just noticed a new eprint (11 May 2010)
(glance at the notation on p. 13 first!) which gives a
local characterization of spherically symmetric spacetimes given entirely in terms of tensorial quantities, and as an application gives a tensorial test for whether or not a Lorentzian spacetime is locally isometric to an LTB dust, with an appropriate dust congruence, in much the same sense as the well known tensorial test for whether or not a Lorentzian manifold is locally isometric to Minkowksi vacuum (namely: iff the Riemann tensor vanishes).
The authors, Ferrando and Saez, have given several such invariant characterizations in the past decade which leverage Cartan's solution of the very imporant problem of local equivalence: given two Riemannian (or Lorentzian) manifolds of the same dimension, when is there a local isometry between them? (Global equivalence is much harder and AFAIK little is known about the general problem.) In special cases it is possible to give "tests" which construct various special tensorial quantities and give necessary and sufficient conditions in terms of these whether or note your Lorentzian manifold is locally isometric to an example of known family of exact solutions, and if so tells you the physical data (e.g. the distinguished timelike congruence representing the world lines of the dust particles, from which you can obtain the dust density, as a specific function on your manifold, by computing the Einstein tensor, for example).
The ingredients include
- the simplest quadratic curvature invariant constructed algebraically from the Ricci tensor
R_{mn} \, R^{mn}
(no, this is not the Ricci scalar, which is the trace of the Ricci tensor and thus a linear invariant; this is the trace of the square of the Ricci tensor, and thus a quadratic invariant, i.e. a sum of squares of components),
- two second rank tensors constructed algebraically from the metric tensor and the Ricci tensor
- a scalar constructed algebraically from Weyl curvature tensor (basically, "cube root of the trace of the cube of the Weyl tensor")
<br />
\rho = -\left( \frac{1}{12} \, {C_{ab}}^{pq} \, {C_{pq}}^{rs} \, {C_{rs}}^{ab} \right)^{1/3}<br />
(in case of a Petrov D Weyl tensor, this scalar together with "eigendirections" completely characterizes the Weyl tensor and is essentially the Coulomb type Weyl spinor component, in the NP formalism)
- a fourth rank tensor constructed algebraically from the metric tensor:
<br />
2 \, G_{abmn} = g_{am} \, g_{bn} + g_{bn} \, g_{am} - g_{an} \, g_{bm} - g_{bm} \, g_{an}<br />
(compare inner product on bivectors W^{ab} at an event, induced from the given metric tensor, treated as giving an inner product on vectors at the same event)
- a fourth rank tensor constructed algebraically from the metric tensor and Weyl curvature tensor:
<br />
S_{abcd} = \frac{1}{3 \, \rho} \; \left( C_{abcd} - \rho \, G_{abcd} \right)<br />
- a second rank tensor constructed using same plus divergence of S_{abcd}
As another application, an obvious test for the Schwarzschild vacuum combines the authors test for whether or not a Lorentzian manifold is locally isometric to a spherically symmetric with the condition that the Ricci tensor vanishes. This can be simplified to
- the (second rank) Ricci tensor R_{ab} vanishes,
- the scalar \rho is nonzero,
- the fourth rank tensor {S_{ab}}^{mn} \, {S_{mn}}^{pq} + {S_{ab}}^{pq} vanishes,
- the scalar \kappa is positive,
- the second rank tensor \tilde{S}_{ab} vanishes,
- the scalar S_{ambn} \, U^a \, \rho^{;m} \, U^b \, \rho^{;n} - \rho_{;p} \, \rho^{;p} is positive,
where \vec{U} is any timelike unit vector,
<br />
\kappa = \frac{1}{9 \, \rho^2} \; \rho_{;m} \, \rho^{;m} - 2 \rho<br />
is the Gaussian curvature of a two-manifold (think of its tangent spaces as analogous to eigenvectors), and where
<br />
\tilde{S}_{ab} = {{}^\ast S}_{ambn} \rho^{;m} \, \rho^{;n}<br />
(note the similarity to the electroriemann tensor, and also note that the left and right Hodge duals agree here).
Exercise: verify that the geometric units for these quantities are consistent with their interpretations (e.g. a Gaussian curvature).
Here, only the fourth and sixth conditions involve differentiation, and only first order derivatives are needed. So the metric tensor, the curvature tensor (second derivatives of the metric) and its first derivatives, i.e. metric tensor and at most third order derivatives, suffice to determine whether or not a Lorentzian manifold is locally isometric to the Schwarzschild vacuum.
It is known that in general, classification up to local isometry may require many succesive covariant differentiations, although the upper bound on how many which was given by Cartan is too generous. So this is one way of understanding in precise terms what we mean by saying that the Schwarzschild vacuum is an unusually "simple" Lorentzian manifold!
If you've ever tried to transform a hairy chart to some canonical form (e.g. Weyl canonical form for a static vacuum you suspect is axisymmetric), you'll probably appreciate that the kind of tensorial test offered by Ferrando and Saez is much easier to apply in practice! Indeed, their test is easily coded under GRtensorII.
Exercise: Spot check whether I parsed their notation correctly, by coding the above test using GRTensorII and immediately applying it to some suitable test cases, including the Schwarzschild vacuum written in a chart in which the spherical symmetry is somewhat disguised, such as the exponential chart obtained by put Z = \log(r) in the standard exterior chart (this chart appears at first glance to reflect cylindrical symmetry rather than spherical symmetry) and the Weyl canonical chart (the canonical chart for the exterior of the Schwarzschild vacuum respects the obvious axial symmetry but well disguises the spherical symmetry). What happens in the future interior region? Check that \rho really does give the Coulomb tidal field (m/r^3 in the standard chart). Can you find a local criterion for identifying r? (Then you can obtain a local measure of m.)
(Heh, I see some new crank misconceptions in our future!)
Excercise: which of these conditions fail for the Vaidya null dust (with nonzero energy density)? The Kerr vacuum (with nonzero angular momentum)?
To all you SA/Ms who are coders ambitious to contribute to some worthy open source project, I say again: many here at PF and elsewhere would no doubt welcome a serious effort to coax Maxima into performing such tests! (Far less important that coaxing it into computing the kinematic decompositions for timelike congruences and null geodesic congruences, however.)