Conservation of energy in gravitational red shift

[DeG]

When an em-wave red shifts due to leaving a gravitational field it loses energy, right? Where does that energy go, into gravitational potential energy of the photon? If so does this mean that the photon can "fall" back on the gravitating body to reclaim this energy? What happens to the energy if it escapes the gravitational field? I've also been told to think of red/ blue shift in terms of differing clock rates in the vicinity of a gravity field and that clock rates virtually stop at the event horizon of a black hole. Does this mean a photon that gets "sucked" into a black hole has an infinite frequency at the EH, since time has gone to zero?

 

[bcrowell]

The first thing to realize is that GR doesn't have conservation of energy in general. We have a FAQ about this: https://www.physicsforums.com/showthread.php?t=506985

However, GR does have conservation of energy in static spacetimes and in asymptotically flat spacetimes.

You have to be a little careful in relativity about specifying what you mean by "escapes the gravitational field." This only has a clear meaning in an asymptotically flat spacetime, where it means escaping to infinity.

So basically the answer to your question is that in an asymptotically flat spacetime, energy is conserved.

A photon that is emitted in the outward direction from a point outside the horizon will never return. But if that photon is reflected back down by a mirror, then yes, it will reclaim its energy.

Does this mean a photon that gets "sucked" into a black hole has an infinite frequency at the EH, since time has gone to zero?

This is completely dependent on what coordinates you're using, i.e., it's not really a well-defined question. There are different answers according to observers free-falling through the event horizon, an observer hovering above the event horizon, and an observer at infinity. But basically if a photon with energy E falls into a black hole, nothing all that surprising happens. The black hole eats the photon, and its mass increases by E/c².