If space-time, i.e. x and t, has certain symmetries, for example if they are much like a static box, unchanging by what you put inside, then p and E are conserved, which can come in very handy. You can use this to solve some questions much faster than by calculating what a system actually does in detail, which may be difficult. All you need is to know that in the end, E must still be the same.
Energy – something mysterious?
Esoterics tell us that “everything is energy”. E is somehow mysterious, much more mysterious than p, and the conservation of E has become a dogma. E has become like god’s urine, a mystic ever lasting essence. However, we know for almost a century now that space is not static and that there is no reason to even expect p or E to be fundamentally conserved. But humans are religious by nature and E is holy now. There are many who come up with desperate attempts to save E from the antichrist.
They try to rescue holy E by coming up with some sort of reservoir, some potential where it came from, some latrine bucket where it went to. It is inside the gravitational potential maybe, although there is no gravity in general relativity, just curved space-time. Or it gets smeared around the cosmic horizon and made negative so that the overall E of the cosmos is zero and thus truly conserved - even if the universe itself should disappear, holy E remains the same.
I think the answer is psychology. Calm down, have a cup of warm chocolate, and look at it rationally. E is not even supposed to be conserved if space-time is not static.
E is often closely related to curvature. Think of particles being knots in space-time (not true, but still). More curvature of space time, more E. Also quantum mechanically: If a wave function has more curvature (higher frequency), then the associated E is higher. There may be more overall E in a space-time that expands - there is surely more opportunity for curvature.
“Laws” of nature are not some holy rules handed down. They are symmetries. Angular momentum is quantized. Why? It is momentum inside a space that has a certain symmetry, here being rotational symmetry leading to the angle of zero degrees being the same as 360 degrees. Angle space is periodical, thus the momentum that is paired with it, the angular momentum, is quantized. Is this a holy quantization? No! Take the symmetry away, and momentum is no longer quantized. It is that easy.
Classically, space-time is static, and so E is conserved, which means if some is missing, we better find out where it went. However, energy in general relativity is not even expected to be conserved. If it is missing and we know well why it is missing, why not?
If E is not conserved, a perpetual motion machine becomes possible. The universe is a perpetual motion machine of the worst kind. It not just creates E, it creates space out of nothing, violates the second law of thermodynamics, all of it. Why should it not? Certain “laws” are applicable when certain symmetries are present for the system under investigation. These symmetries that describe the physics in our backyard may not hold for the universe as a whole. They should not even be expected to hold.
I like to put in these "not even expected" terms to counteract the usual lore that certain circumstances are terribly counter intuitive and unexpected. For example, we should not even expect two twins having taken completely different paths through space-time to have aged equally! The twins meeting again and having aged the same way needs some symmetry that insures such an unlikely feat. Them being the same age is what is actually unexpected.
Previous entries on energy in general relativity and the boring universe: