# Metric that is Minkowski plus sum of null vectors

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.

• That's probably an induced metric on a hypersurface. Commented May 18, 2016 at 21:24
• In the Kerr-schild case, do you know how we interpret this hyper surface? Commented May 25, 2016 at 18:47

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.