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General theory of relativity proved that gravity does attract light ( photons ), what if I project a beam of light radially towards a black holes will it increases it's speed or not?

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A gravitational source will bend the path of the photons, but it doesn't change its speed. So even if you shine it right at a black hole, its speed will remain c.

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    $\begingroup$ This isn't true. See for example the two questions linked as duplicates that explain how and why gravitational fields do change the speed of light. $\endgroup$ – John Rennie Aug 5 '18 at 17:44