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Gravitational lensing is an observed phenomenon. Can one have a gravitational mirror?

A slightly unrelated question: Can gravitational waves be reflected?

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    $\begingroup$ As far as I know gravitational lensing refers to the lensing of light waves, so in an analogous way a gravitational mirror would reflect light waves, and not gravitational waves. $\endgroup$
    – Joe
    Commented Jan 25, 2013 at 12:10

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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.

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

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    $\begingroup$ Totally speculative here: Light reflects through interaction with conductor electrons, so is there any hint from "heuristic quantum gravity" that light might interact with a graviton in a like way? $\endgroup$
    – Selene Routley
    Commented Oct 20, 2013 at 23:57
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The question has been addressed by a physicist (a respected specialist of quantum optics) Raymond Chiao and his coworkers in several articles, particularly in http://arxiv.org/abs/0903.0661 entitled Do Mirrors for Gravitational Waves Exist? The first sentence of the abstract claims :

Thin superconducting films are predicted to be highly reflective mirrors for gravitational waves at microwave frequencies.

The authors argue specifically that :

The existence of strong mass supercurrents within a superconducting film in the presence of a gravitational wave, dubbed the "Heisenberg-Coulomb effect," implies the specular reflection of a gravitational microwave from a film whose thickness is much less than the London penetration depth of the material, in close analogy with the electromagnetic case.

The least one can say is the idea is controversial and (paraphrasing a remark one can find in the Editor's note of another article by R.Chiao : https://arxiv.org/abs/0904.3956) if some people find the hypothesis that gravitational waves could be reflected by some superconducting films to be reasonable, others (the concensus?) think it is highly questionable and possibly inconsistent with the current theory of charged superfluids. Unfortunately I regret I have not enough expertise to point out to any specific conceptual flaw and haven't found any attempt to possibly debunk this theory.

Nevertheless it would be very interesting to know if it is possible to make any connection between the former speculations and the already experimentally demonstrated nonclassical rotational inertia of superfluids, superconductors and Bose Einstein condensates ...

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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.

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  • $\begingroup$ Since gravitational lense bends light by a gravitational field, by gravitational mirror I meant reflecting light by a gravitational field $\endgroup$
    – Revo
    Commented Jan 25, 2013 at 12:42
  • $\begingroup$ I think reading your comment I get the answer to the gravitational mirror question. I now realize that this would require spacetime with high degree of curvature to reflect light which exists only inside a blackhole $\endgroup$
    – Revo
    Commented Jan 25, 2013 at 12:44
  • $\begingroup$ That's not exactly right, either. Inside a black hole, you cannot "mirror" light. The spacetime curvature you're looking for is such that if you shine a light, it will follow a curved path that will U-turn back to you. But that kind of curvature would only exist if you were standing at the singularity. But, by the very nature of that same curvature, light cannot leave where it is. It can't, therefore, reach the U-turn point, and cannot be mirrored back. $\endgroup$
    – markovchain
    Commented Jan 25, 2013 at 12:52
  • $\begingroup$ @markovchain, no, see the final edit in my answer. There are naked singularities without the problems of an event horizon, and thus the U-shaped path is possible. $\endgroup$
    – Eduardo Guerras Valera
    Commented Jan 25, 2013 at 14:00
  • $\begingroup$ As long as light doesn't reach an event horizon, the U is possible or, more precisely, something like an open O (a fat U, if you prefer so) $\endgroup$
    – Eduardo Guerras Valera
    Commented Jan 25, 2013 at 21:42
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Your question can be rephrased in the following way: are there metrics which satisfy the Einstein field equations such that if you existed in it and looked around you, you can see your mirror image?

The answer to this is yes. A simple set of solutions that fits what you are looking for are conical angle deficit solutions, which describe the gravitational field of an infinitely long string. By picking a particularly massive string, you could either have one where the total angle around the string is 180 degrees or 90 degrees (as opposed to 360 degrees if it had mass zero). One half will act like a mirror, except it is split in two halves and your mirror image is in the half opposite to where you are standing. With 90 degrees you can have a "corner reflector" which would effectively look like two mirrors at right angles to each other.

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