Does it take work to bend light? We all know that light always travels in a straight line.  Would it not then stand to reason that changing the path of light requires energy?  If so, would this not violate Newton's laws of motion if bending the light did not exchange energy thus changing the color of the light in the process?
 A: The light travels in a straight line. But what is a "straight line" in a curved space? The concept that replaces the striaght line in a curved spacetime is the geodesic https://en.wikipedia.org/wiki/Geodesic . And, as it turns out, light, as well as anything else, if not affected by a force (excluding gravity, which in Einstein theory of gravity is not a force) travels along a geodesic in a spacetime. To make something travel along a different curve, a force needs to be used.
Now, the shapes that geodesics take can be sometimes surprising. The curve that Earth creates in spacetime as it travels around the Sun is a geodesic. A bended ray of light passing next to a gravity source is a geodesic. As such in neither of these cases any force is required to keep them on their paths. On the other hand, just staying at the surface of Earth, without either orbiting or falling, isn't travelling along the geodesics. In this case case the force that makes us deviate from geodesics is the push of the surface of Earth on our feet - if we get deep into it, it's the electromagnetic force. And it's this force that we actually feel as our weight, not the gravity. As it was repeatedly shown, bodies that are free falling (and are traveling along a geodesic) do not feel their weight.
It can be also mentioned that even in a Newtonian dynamics if a force is orthogonal to the path it does not transfer any energy to the object.
A: You can use Newton mechanics in the limit to m=0 and will see that work is simply $\int \mathbf{f} \cdot \mathbf{ds}$. No net work is performed when integrated over the full hyperbolic orbital. In summary, it does not take work to bend light in Newton mechanics. I am not sure if the concept of work exists in General Relativity.
