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danR said:
How to account for the energy depends on what bookeeping system one is using.
In the Newtonian bookeeping system,a falling physical object is loosing potential energy and gaining kinetic energy. For the photon, it's still loosing potential energy (more or less,it could be argued that we are using a variant of Newtonian theory here, because standard Newtonian theory gets a bit strained in dealing with light. But still, that's the standard Newtonian way of dealing with falling objects, you've got a kinetic energy, and a potential energy, and the sum is the total energy.
In GR, we need another bookeeping system. There are at least three that might apply, but I'm only familiar enough with one of them to tell you how it "keeps the books".
The three that could apply are the ADM system, which keeps tract of the ADM energy, the Bondi system, which keeps tract of the Bondi energy, and the Komar system, which keeps tract of the Komar energy. These are all different definitions of "Energy", but unfortunately energy isn't the simple thing it is in GR as it is in Newtonian physics. This may be confusing, but it is what it is, I can only try to mention it to people, and perhaps to point them at references (which are usually over their heads, though there are a few good popularizations out there like the sci.physics.faq on energy in GR). But I digress.
It's common to use the names Bondi, ADM, and Komar energy, but it's not particularly common to call it a "bookeeping system", that's more or less an analogy I'm making to make the idea understandable.
OK - I've wandered a bit, let's get back on track. How do we handle energy in the Komar sense? Well, we don't really have a direct concept of "potential energy", but what we do instead is very similar. We CAN express the energy in this system as in integral of the energy density (though interestingly enough the integral isn't often unique), and what we do is to say that energy deep in a gravity well, counts for less towards the total energy of the system.
This is rather similar to what we'd say if we had a concept of "potential energy", but we don't. Mainly because there isn't any sort of tensor field we can think of which could store said energy, and people have for the most part realized that non-tensor approaches towards "fields" aren't really physical.
The factor by which it counts less can be thought of as the local time dilation factor, as long as you use coordinates that respect the underlying symmetry of the problem. A coordinate-independent description is possible, but it tends to confuse people, alas, and we've already had a few complaints on the thread that it was getting too technical. So I'll avoid mentioning it unless there's some specific interest, people who need to know can probably take a good guess at this point (or maybe not, in which case they'll have to ask and the people who get confused with it will have to live with it, I guess).
So, there you have it. The blue shifted photon , in some local sense, has more energy than the redshifted photon. But when you add it's energy to "the books", for accountng purposes you you you say that it contributes less to the book value than it's local value. Another way of saying it is that the "book value" of energy is the energy it would have at infinity.
And that's how you account for energy in the Komar system. More or less, I've been deliberately rather informal for the purposes of trying to explain it in terms that most people will be able to understand. Probably the biggest and most dangerous oversimplification that I've made is to assume that you can account for energy in terms of adding up piecies (i.e. via some integral) at all. It's usually possible to do this, and it's familiar, but it's not necessarily a unique process in GR.
[add]The other thing I've oversimplified, because it doesn't really contribute to the problem at hand, is the notion of how pressure affects the bookeeping.
I wish I had a better understanding of the ADM and Bondi systems to provide a similar explanation - or perhaps even to say that a similar explanation doesn't exist - but at t his point, I don't.
I will point out that arguing over the energy as it's defined in Newtonian physics isn't going to get the thread anywhere. Not that it usually stops people from doing it. From my viewpoint,though, if you want to understand how GR deals with energy, you actually have to try to lean how it deals with it. It's NOT necessarily the same way that Newtonian theory does it, and you might have to make a few mental adjustments. If you don't really have the background for it, it might be better to wait until you do if you want really detailed information and understanding without any errors. If you don't or can't, you'll have to make do with popularizations such as posts like this and the sci.physics.faq, which may get you pointed more or less in the right direction, but might be missing a few points that later turn out to be important to your understanding.
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