In GR exercises I've often seen metrics of the form $g_{ab} = \eta_{ab} + k_ak_b$ where $k_a$ is null with respect to $g$ (or equivalently $\eta$). I'm happy doing calculations with such metrics, but I wondered what the physical interpretation of such a metric is?

If $k$ is in some sense small then this is a linearised limit, but I don't think this always has to be the case.

  • 2
    $\begingroup$ That's probably an induced metric on a hypersurface. $\endgroup$
    – Javier
    May 18 '16 at 21:24
  • $\begingroup$ In the Kerr-schild case, do you know how we interpret this hyper surface? $\endgroup$
    – Wooster
    May 25 '16 at 18:47

The reason to consider such metrics was not a particular interpretation, but that for this ansatz one could hope to find some exact solutions. It is known as the Kerr-Schild metric, and its role in the process of finding exact solutions has been described by Kerr in Wiltshire, Visser, Scott (eds.), The Kerr Spacetime.

One can, of course, try to develop some sort of physical interpretations of special solutions, A.J.S. Hamilton, J.P. Lisle, The river model of black holes, Am.J.Phys.76:519-532 (2008), arxiv:0411060v2 would be an example for the slightly different ansatz $ds^2 = -dt^2 + (dr+\beta dt)^2 + r^2(d\theta^2 + \sin^2(\theta)d\varphi^2)$ . But I doubt it makes sense, a physical interpretation should be applicable to more than a few exceptional solutions, see my article http://ilja-schmelzer.de/papers/river.pdf for details.

  • $\begingroup$ Dear Schmelzer: For your information, Physics.SE has a policy that it is OK to cite oneself, but it should be stated clearly and explicitly in the answer itself, not in attached links. $\endgroup$
    – Qmechanic
    May 19 '16 at 8:50

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