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This question obviously could have been improved in many ways, and I'd appreciate if you answer it and pinpoint some holes/inconsistencies.

Assume that there is a black hole (or just a very very massive body) in a flat universe which bends the spacetime around it. There is a light source which is on the opposite side of the black hole some distance away from it. If we trace the geodesic of light's travel, then we find some line around the blackhole (not straight, right?).

Could when traveling on the curvature takes less time than making a "detour" on flatter spacetime be described as black hole’s horizon (Fermat's principle of least time)?

Possible mistake I thought: wouldn't each heavy star have a small black dot in the centre, but it's not observed (as traveling at the steepest curvature is longer than just a bit around it); big stars would imitate blackholes (or do they do so inside?)

Research: this post says that Fermat's principle does hold, but my question extends to a question about black hole (/massive body). Can its radius be calculated through some very complicated derivative w.r.t. proper time? :) I appreciate QMechanic's refs, I'll take a look.

Kyrylo Lyskov
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We can use the Euler-Lagrange equations, which we can think of as a generalization to the Fermat's principle, to trace the path of the light rays, getting what we call null geodesics. Whether it can be used to determine the boundary of the black hole, the easy answer is no, but the actual answer is, it's complicated.

If I understand correctly, you're asking whether an observer can identify the boundary of a static black hole as the region through which, apparently, no light rays pass toward them. If that's the case, you are talking about the black hole shadow, which is larger than the actual event horizon. If we're being technical, then it would be the faint light ring, which is smaller than the shadow, but still larger than the actual event horizon. These can be obtained through Euler-Lagrange equations.

Defining the boundary of the black hole is much more complicated, because we need to take a tight bundle of null geodesics called a null congruence. The boundary of the black hole, or at least the "quasi-local" description of it, is where this null congruence stops expanding.

Christoffel
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