Since there are gravitational lenses, are there gravitational mirrors?

Question

Gravitational lensing is a physical observed effect. Can one have gravitational mirror?

A slightly unrelated question: Can gravitational waves be reflected?

Answer 1

Can one have gravitational mirror?

Gravitational lensing phenomena are due to light deflection, i.e. the change in direction of a light beam, analogous to refraction by ordinary materials but, for building a decent "gravitational mirror" you would need a different phenomena: reflection.

Reflection in an ordinary mirror happens because electrons in the atoms of a metal layer on the back surface of the mirror, absorb and re-emit the photons back. There is no analogous phenomenon due to gravity, but you can think nevertheless of the possibility of a very strong deflection that turns a light beam back to you.

The formula Eddington used during the famous solar eclipse of 1919 for testing GR was first derived by Einstein. He departed from the assumption of having a nearly flat metric, a slightly perturbed version of the flat metric $\eta_{\mu\nu}$ of special relativity:

$$g_{\mu\nu}\approx \eta_{\mu\nu}+h_{\mu\nu}$$ $$\lvert h_{\mu\nu} \rvert \ll 1$$

He obtained a formula for the deflection angle, in terms of the impact parameter $d$, or minimum distance to the deflecting point mass:

$$\alpha=\frac{4GM}{c^{2}d}$$

Now you can see that the only way to have a 180 degrees refraction, i.e. a quite big $\alpha$ value, is by means of sending a light beam with a very small impact parameter $d$, quite close to the deflecting point mass. That is a problem, not only because then the nearly-flat metric will no longer hold, but also because usual astrophysical objects (stars) are extensive. You need an object that is both very small and very dense, that is, a black hole.

In summary, there is the theoretical possibility of a extreme light deflection, quite close to a black hole, so that a light beam you send happens to come back to you after a 180 degrees deflection. You can name that "mirror" in some sense, but it would not be enough for having a decent, spatially extended image of you.

Can gravitational waves be reflected?

In General Relativity, gravitational waves are supposed to propagate along null geodesics, exactly as light does. That is, you can have deflection of gravitational waves, but again no reflection.

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EDIT: You could theoretically send a light beam as close as you want to a sort of point-like black hole without an event horizon, a so-called naked singularity, see here

Answer 2

Gravitational mirrors are possible, but it would be a very unstable mirror. Nonetheless, they are possible.

Just before the event horizon of a black hole, there exists a place called a photon sphere. This orbit is the closest you can get to a black hole without having to accelerate. At this strange place, photons can orbit the black hole. So, theoretically, you can enter this sphere, turn your head to the side, and see the back of your head, because light from the back of your head would go around the black hole and go back to your face.

As I said, though, it would be a very unstable orbit. The slightest perturbation can cause the photons to escape the black hole, or to fall into it.

So there should be a very small chance, a chance close to nothing, where something happens: a perturbation that causes a significant part of the photons in this sphere to escape the black hole, but in a non-diffused way. If you're lucky enough to live to see this happen, the light coming from you could orbit in the photon sphere, turn around, and be released by a perturbation. It could then travel all the way back to you, as if you were looking into a mirror.

That's just science fiction for now, though.

As for mirroring gravitational waves, I'm not sure. I can't say it's impossible, because I don't think anybody has ever tried doing that before. If you were able to reflect gravitational waves, that would be something like gravitational shielding -- that is, antigravity. Antigravity is, as far as I know, not forbidden by the current laws of physics.

Edit - I realized with some thought, it is possible to "reflect" light around a curved spacetime. However, the curvature must be large over a large distance. This will not tend to distort light -- the downside being, light has to travel a large distance first, and so it will take a long time for you to be able to see your "reflection."

Consider a system of many Sun-sized stars. If you shine a light on it at just the right angle, it will deflect. If there is another faraway star to "receive" this deflected light, the light will deflect again. If you compoud this process many times, you will be able to get a full 180 degree "reflection" of light, without distorting the image. I'd assume it might take years for the original beam to make its way back to you, though.