I What is the Simpler Name for Conserved Property of Spacetime?

[KetilT]

If we have come to realize that energy conservation is not the most general conservation law in our spacetime, isn't it odd that we don't have a simple name for the "real deal"?

I bumped into this thought through Noether's theorem, which relates symmetries in fields to conservation of all kinds of charges for particles. It also applies to symmetries of spacetime, and the most general form seems quite non-trivial:

According to general relativity, the conservation laws of linear momentum, energy and angular momentum are only exactly true globally when expressed in terms of the sum of the stress–energy tensor (non-gravitational stress–energy) and the Landau–Lifshitz stress–energy–momentum pseudotensor (gravitational stress–energy)

(Source: Wikipedia)

This is a matter of semantics as much as physics, and I hope you don't mind me posting such a thread. The question is, if this quantity had a short name, what do you think it should be?

 

[fresh_42]

... if this quantity had a short name, what do you think it should be?

To me your suggestion sounds like the hidden desire to get something else "conserved" in return for something conserved being lost.
But that's only my layman impression.

 

[KetilT]

My hidden desire is to make the deep workings of this universe more understandable. Naming doesn't change the nature of anything, of course, but it makes it easier to manipulate the concept in one's mind. I find it very hard to picture the sum of the stress-energy tensor and the Landau–Lifshitz stress–energy–momentum pseudotensor.

Perhaps I am mistaken in wanting to assume that this quantity is anything, something physical that can divide and move around, but is always the same. This might be true for a litre of orange juice that gets added and removed from so many glasses, but perhaps not true for the quantity in question. The same goes for energy itself.

The pressing question then becomes, is the concept of energy/stress-energy as an "eternal substance" a misnomer? Is conservation among its many forms just a mathematical happenstance derived from the formulation of our equations? Or does it gain a common substance through its effect on the curvature of spacetime?

 

[PeterDonis]

if this quantity had a short name

In the general case, there isn't one. What that Wikipedia article leaves out is the fact that the sum of the stress-energy tensor and the Landau-Lifshitz pseudotensor doesn't have any meaningful physical interpretation in most spacetimes. It does in a certain restricted class of spacetimes (basically the ones that describe an isolated object surrounded by empty space), and in those cases it just allows you to define a globally conserved energy, momentum, and angular momentum, as the article says, that correspond reasonably well to our intuitive notions of what those quantities are. But this correspondence only holds in that restricted class of spacetimes.

is the concept of energy/stress-energy as an "eternal substance" a misnomer?

No. But it is a "substance" that, in the general case, is only conserved locally, not globally. That is, if you look at any small local volume of spacetime, there will be a local conservation law that is obeyed, that the amount of "substance" going into that volume of spacetime will be exactly equal to the amount coming out. (Remember that this is a spacetime volume, i.e., a 4-dimensional volume, so "going in" includes going in from the past and "coming out" includes coming out to the future, as well as going in or coming out through the spatial "sides" of the volume.) Mathematically, this is expressed as the covariant divergence of the stress-energy tensor (just that tensor, no pseudotensors) being zero. This is always true (it's guaranteed by the Einstein Field Equation), regardless of what global properties the spacetime does or does not have.