Do gravitational lenses work on gravitational waves?

Do gravitational lenses work on gravitational waves? Could we get an Einstein cross of gravity?

Answer

Summary

This is a partial answer which cannot deal with general gravitational waves, but there is a simple answer, namely, yes gravitational lenses do work on gravitational waves, for a certain class of waves, i.e. those with small wavelengths and small amplitudes. Specifically, the components of a gravitational wave of small enough amplitude that it can be considered a perturbation field on the background metric will locally fulfil D'Alembert's equation. If, further the wavelength is small enough that the Eikonal equation is a good approximation to the propagation of these perturbation components, then the rays, i.e. tangents to the motions of points on the phasefronts of these waves, define tangents to lightlike geodesics: it doesn't matter whether the underlying field is an EM field or a gravitational wave perturbation, the geodesics will be the same. However, gravitational waves are often of much lower frequency than EM radiation, so the short wavelength condition is harder to fulfil.

Although only a partial answer, I believe this answer is important to the undertext of your question, which seems to be, "can we use this effect for increasing the likelihood of detecting gravitational wave experimentally?" In the light of the following, the answer to this latter question would seem to be "yes, we can".

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To get to this answer we use the central ideas of the linearized gravity described in e.g. Chapter 18 of Misner, Thorne and Wheeler "Gravitation" or §8.3 "Einstein's Equations for Weak Gravitational Fields" in Bernard Schutz, "A First Course in General Relativity".

Here one splits the metric $\mathbf{g}$ as

$$\mathbf{g}=\mathbf{\tilde{g}}+\mathbf{h}\tag{1}$$

where $\mathbf{\tilde{g}}$ is some background spacetime defined by all the galaxies, stars and other stress-energy-tensor sources whose lensing effects one wishes to study. $\mathbf{h}$ is a perturbation to the background metric, and it is small enough that its nonlinear coupling (through the $-\frac{1}{2}\,\mathbf{h}\,h$ term in the Einstein tensor, where $h = \mathrm{tr}(\mathbf{h})$) into the whole metric can be neglected when the form (1) is put back into the Einstein field equations. So we have exactly the same situation as we do for light or fluid mechanics on background spacetime: i.e. the perturbation $\mathbf{h}$ behaves like any old tensor field separate from and on top of the underlying gravitational background $\mathbf{\tilde{g}}$. We study it with whatever field equations apply to it cast in a covariant form so that the effect of the background $\mathbf{\tilde{g}}$ can be worked out, just as we would with Maxwell's equations on a curved background.

In weak field theory, the background metric $\mathbf{\tilde{g}}$ is simply taken to be flat Minkowski spacetime. But the simple perturbation idea is general. Moreover, having decided that we can neglect the $-\frac{1}{2}\,\mathbf{h}\,h$ nonlinearity, let's zoom in on a small region of spacetime, with small enough extent that curvature effects (from the background $\mathbf{\tilde{g}}$ arising from all the stars and lensing stuff) can be neglected over the extent of this spacetime chunk. Within this chunk, in some Riemann normal co-ordinates (so that we get a locally inertial frame) the perturbation on the curved background ideas above approximate to the perturbation on Minkowski spacetime theory studied in MTW Chapter 18 or Schutz Chapter 8. Therefore, locally, the components of the trace-reversed $\mathbf{\bar{h}} =\mathbf{h} - \frac{1}{2}\, \boldsymbol{\eta}\, h$ "field" fulfil D'Alembert's wave equation. So, if, further, the wavelength of these waves is short enough to approximate the propagation of their phasefronts by the Eikonal equation, then it cannot matter whether these fields be a scalar light field or the $\mathbf{\bar{h}}$ components: the "rays", i.e. normals to the phasefronts, will define tangents to lightlike geodesics, and exponentiate to the same system of geodesics.

In other words, the gravitational wave paths are the same as light paths as long as (1) the amplitudes of the waves are small enough that $-\frac{1}{2}\,\mathbf{h}\,h$ can be neglected from the Einstein field equations and the linearizing perturbation approach is sound (2) the wavelengths of the waves are short compared with the reciprocals of the sectional curvatures computed from the curvature tensor arising from the lensing background $\mathbf{\tilde{g}}$. The second approximation also applies to light: if it is not met and the Eikonal equation does not hold, one will get diffraction effects from the lensing background.

So if the light from a region arriving at your instruments is increased through the gravitational effects of something between you and the scrutinized region, so too will the gravitational waves be increased through the same partial focussing.

In particular, you can no more burn space ants with gravitational waves through a gravitational lens than you can with light through a gravitational lens!