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It is said that photons have zero rest mass so how can gravitational force of a black hole affect light?

And if photons do have some effective mass while traveling at speed of light then only can a black hole's gravitational force act upon them and if force acts upon them they will surely be accelerated.

Does this mean that near a black hole light itself crosses its own speed that is 3 X 10^8 m/s ? , which is not theoretically possible.

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Indeed photons are massless particles, so they follow "quickest paths" during their propagation in space-time; these are called "geodesics". However, general relativity doesn't simply says how the way masses attract each other is modified, it most of all says that mass (and energy density) curve space-time; this also curves the geodesics, which photons (in particular) follow.

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It is said that photons have zero rest mass so how can gravitational force of a black hole affect light?

Photons have zero rest mass so when they are at rest they have no mass. They are never at rest so this is a little misleading.

And if photons do have some effective mass while traveling at speed of light then only can a black hole's gravitational force act upon them and if force acts upon them they will surely be accelerated.

Photons have a fixed speed so when near a strong gravitational field (like a black hole) it will be bent towards it (but keep its speed constant). The extra energy or acceleration still goes into the photon, just as an increase in frequency not speed.

Does this mean that near a black hole light itself crosses its own speed that is 3 X 10^8 m/s ? , which is not theoretically possible.

No (see above).

Another way to look at this is via the Equivalence Principle. Have a look at the image of two lifts here.

(a) A beam of light emerges from a flashlight in an upward-accelerating elevator. Since the elevator moves up during the time the light takes to reach the wall, the beam strikes lower than it would if the elevator were not accelerated. (b) Gravity has the same effect on light, since it is not possible to tell whether the elevator is accelerating upward or acted upon by gravity.

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