Is it possible to bend light so that it forms a circle and goes round and round indefinitely without losing energy?
Answer
How could one manipulate light? It does not have mass, it does not have electric charge. For that matter, it also does not have any color or weak charge. There seems no way to change its direction of motion.
General relativity describes how masses can create curvature in spacetime. If you have enough mass, it will get curved significantly. Light will follow this curvature, because light will go “straight” which will become curved in curved spacetime. Right at the Schwarzschild radius of a black hole, the escape velocity is the speed of light. That means that a photon there trying to go straight away from the black hole will not get any further, although it moves with the speed of light.
That is not a closed orbit, of course. As Jerry Schirmer pointed out in the comments, a closed orbit happens at $r = 3M$ where $M$ is the mass of the black hole. The problem with this orbit is that it is unstable. Any perturbation will either send the photon away from the black hole or lets it spiral into the singularity. Either way it breaks from the closed orbit.
Since a photon has an energy, it also creates spacetime curvature. A moving photon will therefore radiate gravitational waves, although they will be minuscule. However, they are sufficient perturbation to prevent the orbit from being closed forever. This could be prevented by using a solid ring of light such that the mass density along the orbit is constant. Then no gravitational waves would be emitted.
If the Hawking temperature of the black hole does not exactly match the temperature of the ambient universe (think of the cosmic microwave background), the black hole will grow or shrink. This will change the radius of the orbit and also prevents an orbiting photon for eternity.
All in all this is very unstable and will not work out.
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Another possibility is to use refraction of light. If you have an optical medium with different optical densities (different index of refraction $n$), light will also bend. This is how a lens works. With the right setup of lenses one can refract light to go around a path. You could even set up three mirrors and let the light go round and round in a triangle!
The optical fiber is a bit more sophisticated, it has a gradient of the optical density and can therefore smoothly direct the light around a curve.
With quantum electrodynamics, there is the tiny interaction of light rays with other light rays. Although light has no charge in itself, it can couple to virtual charged fermions and create a closed loop that couples four photons in total. If you have enough light around in a particular configuration, one could bend light rays with that. However, I fear that this is not realizable in any experiment.
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Another valid issue was raised in the comments: If you would have this situation successfully set up, how would you know that it is working? If you try to observe the photon, you would change it. If it radiates something to the outside (scattered light, gravitational waves), it would lose energy over time and leave the orbit.
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