I have a dilemma. I wanted to find the time it would take for light to travel from a distance of S to R in a gravitational field, travelling from a large distance directly toward the center of gravity. The speed of light in a gravitational field according to a distant observer is c sqrt[1 - 2GM/xc^2], where x is the distance from the center of gravity for the instantaneous position of the photon. The time to travel a distance of dx at that instantaneous position, then, is dt = dx / (c sqrt(1 - 2GM/xc^2) ). If we set r = 2GM/c^2, then it is dt = dx / (c sqrt(1 - r/x) ). Integrating for that, we finddt = dx / (c sqrt(1 - r/x) )t = Int dx / (c sqrt[1 - r/x] )= [x sqrt(1 - r/x) + (r/2) log(2 x (sqrt(1 - r/x) + 1) - r)] / c from S to R= [S sqrt(1 - r/S) - R sqrt(1 - r/R) + (r/2) log((2 S (sqrt(1 - r/S) + 1) - r) / (2 R (sqrt(1 - r/R) + 1) - r))] / cOkay, so that should be the time it takes for light to travel from S to R in a gravitational field directly along a path to the center of gravity according to a distant observer. So let's find what the time would take for light to travel from some distance S all the way to the Schwarzschild radius. Now, at the Schwarzschild radius, light travels at a speed of c sqrt(1 - r/r) = 0, so it should never get there, taking an eternity of time according to the distant observer, so that is what I would expect for the result, infinity, but that's not what I get. If we set R=r, then we gett = [S sqrt(1 - r/S) - r sqrt(1 - r/r) + (r/2) log((2 S (sqrt(1 - r/S) + 1) - r) / (2 r (sqrt(1 - r/r) + 1) - r))] / c= [S sqrt(1 - r/S) + (r/2) log((2 S (sqrt(1 - r/S) + 1) - r) / r] / cwhich is far from infinity. Using S = 10 r, for example, the time according to this should be t = 11.30528 r / c. What did I do wrong?