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I was asked about the question today:

suppose that you are observing from afar a spherically symmetric star of mass $M$. Its radius $R$ is $\textbf{not}$ much larger than its Schwarzschild radius. Due to the gravitational bending of light, you can see not only the front side of the star but also the part of its back side? what's it like and how much of the back side can you see?

I don't have any idea about how can one see the back side of the star in this question. If I can see the back side, then there must some light emit from the back side to me. But this confuses me. Any suggestion?

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    $\begingroup$ Did you mean Its radius R is not much larger than its Schwarzschild radius? If the star is much larger that the Schwarzschild radius, e.g. our Sun, then the bending of light is negligible. $\endgroup$ – John Rennie Jan 10 '18 at 16:10
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Because of the gravitational bending of light, some of the light emitted towards the back of the star is visible from the front. For a large star this is a very small effect, but as $R$ gets closer to the Schwarzschild radius more and more of the backside becomes visible.

This effect is of practical importance when it comes to the study of the $x$-ray pulse profiles of a rotating neutron star. Time variability of the $x$ ray signal is believed to be due to the fact that the star has a hot spot that is rotating around the star. Because of gravitational bending we can see the hot spot more than half of the time (even at the equator). There is now an experiment (the Nicer mission https://heasarc.gsfc.nasa.gov/docs/nicer/) which is trying to use this effect to measure the radius of a neutron star.

The fraction of the back side that is visible can be worked out using the form of the null geodesics in the Schwarzschild geometry. Simple approximations are discussed, for example, by Beloborodov https://arxiv.org/abs/astro-ph/0201117 .

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If the radius of the star is much larger than the Schwarzschild radius the gravitational bending is extremely negligible. The part of the back side of the star that theoretically would be visible to a far away observer, due to the bending, is extremely negligible as well.

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If space surrounding the observable universe wasn't filled with matter (which would be a universe that very fast resulted in a big crunch, but let's assume this for the sake of the argument), our visible universe would be a black hole with a radius much greater than the Schwarzschild radius. The light inside the radius would be destined to travel forever inside it, but light emerging from matter outside the radius could travel away from the universe (or from the star you mentioned in your question). Whatever the direction of the emerging light (between tangent and perpendicular) on the opposite side of where you're "standing", it can never reach you because it's bent not enough to make such a path to reach your eyes and make you see the backside of the universe (or star).

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