Could you view yourself in high gravity situations?

Question

I'm trying to understand what effects gravity can have on light. First of all, I don't understand how gravity can even affect it, since it doesn't have mass, right? That is probably a separate question though.

When gravity is strong enough, it bends light towards the source of the gravity. So if you were on a small planet and gravity were to gradually increase, would the horizon rise as well, allowing you to see further? If so, at some point, could you look up at some angle and have the light go all the way around the planet and back to yourself, in which case you would essentially be looking up at yourself? Also, as the gravity increases, is there a point at which light could orbit the planet indefinitely.

Is this the right place to ask questions like this?

Answer 1

When gravity is strong enough, it bends light towards the source of the gravity.

Roughly true

So if you were on a small planet and gravity were to gradually increase, would the horizon rise as well, allowing you to see further?

Yes!

If so, at some point, could you look up at some angle and have the light go all the way around the planet and back to yourself, in which case you would essentially be looking up at yourself? Also, as the gravity increases, is there a point at which light could orbit the planet indefinitely.

These two are essentially the same question, and the answer to both is yes.

Here is a nice picture illustrating a few different light-like geodesics; the lines indicate paths that light might take near a massive, compact spherical body:

(Image credit)

For a non-rotating spherical object, there is a sphere of space in which light has a stable orbit; i.e. ideally light could orbit forever if it were emitted tangentially at that specific radius. This sphere is known as the photon sphere, and it has a radius of

$$R_p = \frac 3 2 r_s = \frac{3 G M}{c^2}$$

Interesting point of reference: For the mass of the Earth, this radius is about $1.3 ~\rm cm$. Thus if the mass of the Earth were compressed into a marble with a radius of $1.3 ~\rm cm$, there would be a stable orbit for photons directly on its surface. If you were an ant standing on this Earth-marble, you would (ignoring other inconvenient realities of such a situation) be able to see the back of your head. Or thorax, or whatever you have.

For further research, there's a nice overview of Schwartzchild geodesics, the name for paths that free bodies (including light) take in the vicinity of a non-rotating spherical object. Of particular interest to this question is the "Bending of light by gravity" section.

Answer 2

Disclaimer for those who know more: I only talk about spherical black holes for simplicity; rotating black holes are more complicated.

Indeed, a planet's gravity bends light and allows you to see a little bit farther; this is an observed effect (though not on planets), and is called gravitational lensing. If you've seen the movie Interstellar, you might remember that the background of stars around the black hole is distorted. That's gravitational lensing at work, although of course that's a simulation and not a real black hole. A noteworthy fact is that the Earth's atmosphere also bends light (though this doesn't really have anything to do with gravity). When you see the sun about to set, it has actually already set! It is below the horizon but its light is bent towards you by the atmosphere.

The question of whether light can go the whole way around is a little bit deeper. Associated with a spherical body of mass $M$ is a number called its Schwarzschild radius, given by $r_S = 2GM/c^2$, where $G$ is the gravitational constant and $c$ is the speed of light. This is usually rather small; the Earth has a Schwarzschild radius of a bit under $1\ \text{cm}$, and its value for the Sun is around $3\ \text{km}$. Ordinarily the Schwarzschild radius is much smaller than the object's actual radius, and it doesn't play an important role.

Things change if you have an extremely dense object. General relativity predicts that any body that becomes smaller than its Schwarzschild radius (so for example if you were to compress the Sun into a star under $3\ \text{km}$ in radius) will inevitably collapse into a black hole. The "surface" of the black hole, also called its event horizon, will be at $r_S$.

Near the event horizon, the gravity is so strong that strange things can happen. Here comes the part relevant to your question: light can orbit a black hole. The orbit's radius is $\frac32 r_S$, so technically an object smaller than that but bigger than its Schwarzschild radius (so not quite a black hole) can have light going around it. Sadly, though, this orbit is unstable. This means that you would have to launch your photon at a very specific distance and in a very specific direction, and the smallest perturbation (such as a planet's gravity) would cause it to either spiral into the black hole or escape the orbit.

So I guess if you found a black hole and managed to float at exactly $\frac32 r_S$, you would see your back in front of you.