Solutions with 7 Killing vectors: two explicit examples, plus caveat
Hi, Stingray,
Stingray said:
In 4 dimensions, a spacetime can have a maximum of 10 linearly independent Killing vectors. Are there known examples of spacetimes (satisfying Einstein's equation) with 7, 8, or 9 Killing vectors? I know FRW cosmologies have 6 Killing vectors, but I'm looking for something a bit more symmetric that still has varying curvature.
The book by Stephani et al., Exact Solutions of the Einstein Field Equations, 2n Ed., Cambridge University Press, 2001, is a gold mine of information for this classical topic, although it will require some work to extract all the relevant information. (This book offers some nifty tables which can help to quickly get the general idea, however.)
A complete answer would be very complex, but the very short answer is that there are many solutions with 0-4 Killing vectors, and of course an certain explicit vacuum solution has the maximal number of 10, but not very many exact solutions possesses "intermediate" dimensional Lie algebras of Killing vectors. Indeed, there are various results to the effect that such and such a class is the only spacetime model with various properties and 5 or 6 Killing vectors (one of these refers to the famous Goedel dust, which has 5 Killing vectors but which escapes being isotropic as well as homogeneous). Similarly if you allow homotheties, affine collineations, or other generalizations of Killing vectors.
I seem to recall that there are some dimensions in the range 7-9 which do not occur at all, at least with some restrictions on the Ricci curvature (i.e. on the stress-energy tensor).
On a more positive note, I can offer a few explicit examples of exact solutions with 7 Killing vectors . The generic plane wave (EK9, the ninth class in the Ehlers & Kundt classification of vacuum plane waves, also SG10, the tenth class in the Sipple and Goenner classification of all plane waves) has a 5 dimensional Lie algebra of Killing vectors, but there are some interesting special cases which have one or two extra ones. In particular, SG16 and SG17 possesses seven dimensional Lie algebras of Killing vector fields.
A specific example: the line element of SG16 can be written (in the harmonic or Brinkmann chart)
[itex]ds^2 = -a^2 \, \left( X^2+Y^2 \right) \, dU^2 - 2 \, dU \, dV + dX^2 + dY^2,[/itex]
[itex]-\infty < U, V, X, Y < \infty[/itex]
If you compute the Einstein tensor you find this is a "null dust solution" modeling something like "incoherent EM radiation" unaccompanied by gravitational radiation, since this exact plane wave solution happens to conformally flat! A simple choise of seven linearly independent Killing vector fields is:
[itex]\partial_U, \; \partial_V, \; \partial_\Theta = -Y \, \partial_X + X \, \partial_Y[/itex]
[itex]a X \, \cos(a U) \, \partial_V + \sin(a U) \partial_X[/itex]
[itex]a X \, \sin(a U) \, \partial_V - \cos(a U) \partial_X[/itex]
[itex]a Y \, \cos(a U) \, \partial_V + \sin(a U) \partial_Y[/itex]
[itex]a Y \, \sin(a U) \, \partial_V - \cos(a U) \partial_Y[/itex]
where the first and third in this list are "extras".
You mentioned "varying curvature"; you'd probably consider this example to fail that test. For example, using the standard NP tetrad constructed from the Brinkmann chart,
[tex]\vec{\ell} = \partial_U - a^2/2 \, \left( X^2 + Y^2 \right) \, \partial_V,<br />
\; \vec{n} = \partial_V, \; \vec{m} = \frac{1}{\sqrt{2}} \left( \partial_X + i \, \partial_Y \right)[/tex]
the Weyl scalars all vanish and the only nonvanishing Ricci scalar (other than the NP Lambda) is [tex]\Phi_{00} = a^2[/tex].
But you would probably admit SG17 as an example with "time-varying curvature". In a harmonic or Brinkmann chart, the line element can be written
[itex]ds^2 = -\frac{m \, \left( X^2+Y^2 \right)}{U^2} \; dU^2 - 2 \, dU \, dV + dX^2 + dY^2,[/itex]
[itex]0 < U < \infty, \; -\infty < V, X, Y < \infty[/itex]
The two extra Killing vectors here are
[tex]U \partial_U - V \partial_V, \; \partial_\Theta[/tex]
(SG17 also admits an affine collineation which is not a Killing vector, by the way, [tex]U \, \partial_V[/tex].) This has [tex]\Phi_{00} = m/U^2[/tex] with respect to the standard NP tetrad.
I should stress that "invariant characterizations of curvature" can be quite tricky when radiation is present. That is, different observers, even different classes of inertial observers, might observe very different behavior and might even disagree on whether or not any curvature components diverge on some locus. So we naturally reach for curvature invariants, but these are no help at all, since ALL the curvature invariants of plane waves vanish identically, yet these are curved spacetimes. (This is analogous to the fact that in Lorentzian manifolds, the "length" of a nonzero null vector field vanishes identically.) This observation (due to Penrose) gives rise to the provisional rough classification of curvature singularities as scalar or nonscalar and strong or weak in various senses. Some of the other EK and SG classes in fact provide classic examples of plane waves exhibiting some of the heirarchy of strength (where "weaker" singularities are more survivable by small objects).
Chris Hillman