The bending of light ray and unphysical solutions in general relativity

In general relativity, according to Einstein, a gauge is related to the time dilation and the space contractions, and thus a physically realizable gauge should be unique for a given frame of reference. Since more than one metric solution for the same frame can produce the same deflection angle, this means that an invalid space－time metric can produce the correct deflection angle for a light ray. To demonstrate this with an unambiguous example, we consider a new extreme case that there is no space contraction in the radius direction while the conditions of asymptotic flatness and the requirement for gravitational red shifts are satisfied. This solution has a distinct characteristic of "space expansion" in the other directions. Nevertheless, it turns out that, in spite of requiring far more subtle calculations, the resulting deflection angle of a light ray is the same. An interesting property of this new solution is that its event horizon corresponds to an arbitrary integral constant. Thus, this calculation demonstrates beyond doubt that an unphysical solution can produce the correct first-order approximation of light bending. This makes it clear that there is a main difference between local effects such as the gravitational red shifts and the local light speeds, which are not gauge invariant, and integrated effects such as the bending of light, which can be (restricted) gauge invariant.