I think it's high time we do this explicitly and calculate the actual

**trajectory** of our light ray, instead of arguing for pages and pages about speeds and times and velocity. Light satisfies the geodesic equation

with some affine parameter

. Because for light we have ds=0, it also must satisfy the additional parametrisation condition

Starting with the Schwarzschild metric, we can now calculate the connection coefficients, insert these into the above conditions, and solve the resulting set of partial differential equations. Typing all of that out is way too tedious, so I will skip that and simply state the end result ( the dot denotes differentiation with respect to

) :

Herein

, and the constants of integration have to be determined from the initial conditions of the trajectory. Having said that, due to the spherical symmetry of Schwarzschild space-time, we are free to fix one more constant at random; if we pick F=1, and fix E=0 for light, the equations of motion are solved by

wherein A and B are the coefficients of the Schwarzschild metric. As is immediately obvious without even having to evaluate the integral, the trajectory of light in Schwarzschild space-time is

**not** a straight radial line under any circumstance. If you shine a beam of light "straight up", regardless of where you are, the trajectory of the beam is given by the above integral, and it is not a straight line.