I'm going to take another shot at the more technical approach. But I've said it all before. I can still hope that it will "make sense" this time.
If a time-like killing vector exists it is possible to define a concept that behaves like energy, the covariant time-component of the 4momentum vector but in many cases it is the contravariant time component that is defined as energy
It's true that E
0 is a constant of motion for particles following a geodesic. But this is a red herring. It does not lead to good defintion of the energy of a system. This should be no surprise, it's not even coordinate independent.
I'm assuming that is what we are interested in, yes? The energy of a system, not a constant of motion of a particle following a geodesic?
Several good and equivalent defintions for the energy M
of a system with a timelike killing vector k
a are:
a surface intergal
<br />
M = -\frac{1}{8 \pi} \int_S \epsilon_{abcd} \nabla^c k^d<br />
here \epsilon_{abcd} is the Levi-civita tensor normalized to be a volume element of the space-time, this is the surface intergal of a two form.
a volume intergal
<br />
M = \frac{1}{4 \pi} \int_{\Sigma} R_{ab} n^a k^b dV<br />
here R
ab is the Ricci, and n
a is a unit future perpendicular to the volume element dV.
<br />
M = 2 \int_{\Sigma} (T_{ab} - \frac{1}{2} T g_{ab}) n^a k^b dV<br />
here T
ab is the stress energy tensor, g
ab is the metric tensor, and n
a remains the unit future perpendicular to the volume element dV.
Note that this last expression illustrates why you
can't just intergrate T
ab overe a volume of space and expect to come up with a
system energy . (Except as an approximation that's only valid in the weak field case).
These can all be found in Wald, pg 288-291
Now that we've talked about the easy case of a system with a timelike killing vector, let's talk about the harder case - what if you don't have a timelike killing vector.
Well, the idea is simple. If you don't have a killing vector that's timelike everywhere, maybe you have a killing vector that's asymptotically timlike, at infinity. We can then apply the first definition directly. If turns out that if you have an asymptotically flat space-time that's a vacuum at infinity, you do have killing vectors which are asymptotically timelike. To formulate this rigorously requires a defintion of conformal infinity. In this case, we also have to specify "which infinity" (it's null infinity, in the jargon of conformal infinity). It turns out
Fortunately ... the asymptotic symmetry group of null infinity has a preferred 4 parameter subgroup of translations, so the notion of "an asymptotic time translation" is well defined.
So, if I may conclude with a few words
asymptotic flatness, asymptotic flatness, and asymptotic flatness, are the three keys to energy conservation in standard GR.
.
