Although we've been over this before, Fredrick hasn't, so there's probably a few things I should say.
Let's say we have a small body orbiting the sun. If we use Newtonian mechanics, it's reasonably obvious that the momentum of the small body is not conserved. The momentum of the system as a whole is conserved, but not just the momentum of just the small body.
The energy of the small orbiting body is conserved, as long as one takes into account the gravitational potential energy. But the energy of the small orbiting body is not conserved if one does not add the gravitational potential energy to the small bodies kinetic energy.
So this is how it works in Newtonian mechanics - let's jump to relativity.
If we let the energy-momentum 4 vector of a small object orbiting a large object be P, then the quantity
P
a k
a
is a constant of motion. Here k
a is the "Killing Vector" that represents the timelike symmetry of the system. There are several ways of determining what the Killing vectors are, one of them is that they must satisfy "Killings equation"
\nabla_a k^b + \nabla_b k^a = 0
This is equivalent to saying that the covariant derivative \nabla_a k^b must be a completely antisymmetric tensor. Note that a unit vector in the time direction in the exterior vacuum region of the Schwarzschild metric is a Killing vector. This makes it a timelike Killing vector, which yields a conserved energy. There are space-like Killing vectors in the Schwarzschild metric as well - these yield conserved momenta (i.e. the angular momentum of the orbiting body). This is an example of Noether's theorem in action - time symmetries give conserved energies, space symmetries give conserved momenta. Noether's theorem applies to this situation because geodesics are an extremum of an action.
Because P
a k
a is a conserved quantity for an object following a geodesic, and because k
a is a unit vector in the time direction for the Schwarzschild metric in Schwarzschild coordinates, P
0 is a constant of motion for a particle orbiting a massive body in the Schwarzschild metric in Schwarzschild coordinates.
Opinions vary on what name to give this quantity, some people here like to call it the energy of the particle, but I'm not totally convinced that this is correct. This has been the topic of a reasonably endless debate. Note that the simplest form of the expression as written occurs only when Schwarzschild coordinates are used (it's coordinate dependent, not coordinate independent).
Names aside, it's very useful to know that this quantity is conserved.
However, this quantity is not the same as the total energy of the system. If the "small" orbiting body is the Earth, this does not give the total energy of the Earth-sun system, for instance.
The problem of finding the energy of a system in General Relativity is not the same as the problem of finding conserved quantites for orbits (particles following geodesics). It's a different problem entirely, though there are some similarites. It turns out that the energy of a system in GR can be defined if either one of two conditions is met. One condtion which permits a conserved energy to be defined is if the system has a time-like Killing vector, as the Schwarzschild metric does. When this condition is satisfied, one has both a definition of energy, and an observer-independent notion of "where" the energy is. In this special case one can separate the total energy of the system out in terms of kinetic energy and gravitational binding energy, as one does for the Newtonian problem.
The second, more general condition that conserves energy in GR is is that the system exists in an asymptotically flat space-time. If this condition is met, and there is no time-like Killing vector, the total energy of the system can be defined, but the energy can no longer be localized as it was in the previous case. To be really precise, there are at least two notions of energy in asymptotically flat space-time, the Bondi energy and the ADM energy. These two notions differ in how the "bookeeping" of the energy stored in gravitational waves is handled. These two different notions of energy can otherwise be shown to be equivalent, though it's a difficult enough job that Wald,s textbook "General Relativity" for instance, only gives references to where this is proved rather than proving it in the textbook itself.
The sci.physics.faq "Is energy conserved in General Relativity" is also a useful reference (and less technical than what I wrote), it is located
here
[add]
There's another notion of conservation of energy in GR that's worth mentioning. This is a differential conservation law which applies to the stress energy tensor. That makes a total of 4 different notions of energy and it's conservation that have been discussed so far!. The basic equation is
\nabla^a T_{ab} = 0
This is yields 4 equations (for b=0,1,2,3, the sum over the repeated indiex "a" is implied). These equations can be interpreted as a conservation of energy in an infinitesimal region of space-time, and as a conservation of momentum in an infinitesimal region of space-time. Unfortunately, these equations do NOT have anything useful to say about a finite region of space-time. A useful web page discussing the stress-energy tensor and how these equations correspond to the conservation of energy in an infenitesimal cube is located http://people.hofstra.edu/faculty/Stefan_Waner/diff_geom/Sec12.html
Garth is unhappy with the fact that energy (total system energy in a non-infinitesimal volume of space-time) is not conserved for arbitrary space-times in general relativity, and has created an alternate theory of gravity called SCC in which energy is conserved for arbitrary space-times. I'm sure he'll be happy to post the link to his papers.
