Usually, the theory of relativity is not studied much at school, and all the familiarity of students with the theory of gravity is limited by the Newtonian law of universal gravitation. As we know from this law, the force of attraction is proportional to the masses and inversely proportional to the distance between the bodies.
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Based on this, photons do not have mass. If we substitute zero in the place of one of the masses, we get zero. Therefore, the force of attraction will be zero. It turns out gravity should not affect the light in any way.
But the fact is that the theory of gravity in Newton’s formulation is empirical. Over time, inaccuracies were found in the results obtained using this formula, in particular, a shift in the perihelion of Mercury was discovered.
For a long time, they had been trying to explain it by the presence of a hypothetical planet Vulcan between the Sun and Mercury, but it could not be found.
Finally, at the beginning of the 20th century, a new theory of gravity was created: General Theory of Relativity. This theory not only gave a much more accurate mathematical apparatus but also gave an explanation of the nature of gravity. Einstein explained gravity by the curvature of space.
A good visual illustration of the curvature of space is the classic example with a stretched canvas with a ball lying in the middle. Canvas is a two-dimensional space, sagging under the weight of the ball it bends. If, before throwing the ball on the canvas, we draw trajectories of photons moving in two-dimensional space on it, after placing the ball, we will see that the trajectories of photons passing near the place where the ball lies will become curved.
In exactly the same way, a massive object (a planet, a star, or a black hole) creates the curvature in the fabric of space-time. Photons move along straight trajectories. However, since there is a curvature of space-time itself, the trajectories of photons are also curved. This is a well-known effect called “gravitational lensing.”
Due to this effect, we can see objects hidden from us by other massive objects. A good example is Einstein’s cross, i.e. the ZW 2237 + 030 galaxy-lens bends its mass from the QSO 2237 + 0305 quasar located behind it, and we see immediately four images of the same quasar on opposite sides of the galaxy.
Photons that fall under the black hole event horizon are at the bottom of its infinite gravity well. To break out, they need to actually overcome an infinite distance.
Thus, the gravity of very massive objects bends space-time in the vicinity of these objects, and this leads to a curvature of the trajectories of photons flying nearby. And of course, it’s worth adding that if we look at Einstein’s equation, we won’t find the mass of the word there at all, only the energy-momentum tensor.