Energy conservation in general relativity

KleinMoretti · Feb 15, 2024

Is energy conserved in general relativity? I have read most of the posts here that address this. But it isn't clear to me, what most people say is that energy is conserved locally but it can't be defined globally, some people say this means that energy is not conserved in GE while others argue that it is. I have also come across these articles that talk about this,<https://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/>, <https://bigthink.com/starts-with-a-bang/expanding-universe-conserve-energy/> , <https://www.forbes.com/sites/starts...ue-to-the-expanding-universe/?sh=4715052c3efa> so my question is what does it mean when people say energy can't be defined and why do people say energy is conserved while others say its not, also are the answers given in those correct

PeroK · Feb 15, 2024

Let me ask a different question. What, precisely, does conservation of energy mean? And, why should energy be conserved in the first place? ("Because that's what I was told in high school" is not an acceptable answer!)

Ibix · Feb 15, 2024

Stress-energy is locally conserved. This is unarguable in GR. It means that if you build a small four dimensional box, the stress-energy that goes in to it (either from the same spatial volume in the past or just across the spatial sides of the box) also leaves (into the future or flowing out of the walls) and vice versa. In layman's terms this means that you will never see a paperclip pop into existence in front of you, nor it suddenly speed off into the night without something hitting it.

The problems start when you try to work out the total energy content of the universe. There are a number of technical issues with how you define such a thing that mean we only know how to do it in some spacetimes - and the ones that describe our universe at the large scale are not among those. It's not that we don't know how to define energy, it's the total energy we don't know how to define - a situation exacerbated by the fact that some models are infinite with matter everywhere, so it's not entirely clear that there is a sensible answer.

There are possible solutions from invoking Hamiltonian approaches, but the answer is then always zero total energy. In this approach, in my limited understanding of it, you end up defining an "energy of the gravitational field" that is always minus the energy in the universe, so the total is zero. There are also various pseudo-tensor aproaches about which I know even less. There's no general agreement that this kind of approach is even valid - I think (!) it implies a preferred frame, which isn't really vanilla GR.

KleinMoretti · Feb 15, 2024

PeroK said:

Let me ask a different question. What, precisely, does conservation of energy mean? And, why should energy be conserved in the first place? ("Because that's what I was told in high school" is not an acceptable answer!)

to me conservation of energy means that we can't create something out of nothing, as for why it should be conserved I don't really know,

KleinMoretti · Feb 15, 2024

Ibix said:

Stress-energy is locally conserved. This is unarguable in GR. It means that if you build a small four dimensional box, the stress-energy that goes in to it (either from the same spatial volume in the past or just across the spatial sides of the box) also leaves (into the future or flowing out of the walls) and vice versa. In layman's terms this means that you will never see a paperclip pop into existence in front of you, nor it suddenly speed off into the night without something hitting it.

The problems start when you try to work out the total energy content of the universe. There are a number of technical issues with how you define such a thing that mean we only know how to do it in some spacetimes - and the ones that describe our universe at the large scale are not among those. It's not that we don't know how to define energy, it's the total energy we don't know how to define - a situation exacerbated by the fact that some models are infinite with matter everywhere, so it's not entirely clear that there is a sensible answer.

There are possible solutions from invoking Hamiltonian approaches, but the answer is then always zero total energy. In this approach, in my limited understanding of it, you end up defining an "energy of the gravitational field" that is always minus the energy in the universe, so the total is zero. There are also various pseudo-tensor aproaches about which I know even less. There's no general agreement that this kind of approach is even valid - I think (!) it implies a preferred frame, which isn't really vanilla GR.

I see that Sean Carroll article frequently referenced in this issue, is his take a valid one?

Ibix · Feb 15, 2024

KleinMoretti said:

to me conservation of energy means that we can't create something out of nothing,

That's essentially what the local conservation of stress-energy does.

KleinMoretti · Feb 15, 2024

Ibix said:

That's essentially what the local conservation of stress-energy does.

but then when we talk about there not being global conservation doesn't that imply that is no longer true.

Hill · Feb 15, 2024

There are two different notions of energy conservation, local and global. The local one is well defined and true, the global one is not well defined.

Bosko · Feb 15, 2024

This question has been answered by Emmy Noether.
https://en.m.wikipedia.org/wiki/Noether's_theorem

PeroK · Feb 15, 2024

KleinMoretti said:

to me conservation of energy means that we can't create something out of nothing

That's not very precise. If you throw a ball up, then it stops (instantaneously) at its highest point. It's lost all it's kinetic energy. On the way back down it regains all or most of its kinetic energy. That kinetic energy was regained out of nothing, in a sense. So, to ensure conservation of energy, we invent GPE (Gravitational Potential Energy) and then the books balance. But, is GPE really "something". Or, is it just a mathematical invention?

KleinMoretti said:

as for why it should be conserved I don't really know,

The deep reason is time-translation invariance and Noether's Theorem. In an expanding universe, we no longer have time-translation invariance, so we no longer have the mathematical basis for conservation of energy.

PeterDonis · Feb 15, 2024

KleinMoretti said:

but then when we talk about there not being global conservation doesn't that imply that is no longer true.

No. It implies that there is no well-defined "global energy" to be conserved. Basically, that the idea of "just add up all the local pieces to get the global energy" does not work; there is no well-defined way to do the adding up.

There are two cases where there is a well-defined way to do the adding up: asymptotically flat spacetimes and stationary spacetimes. Asymptotically flat spacetimes describe an isolated object surrounded by vacuum, so it makes sense that there is a well-defined notion of "total energy" for the isolated object--it occupies a bounded region. (There is a wrinkle here, though, because there are actually two distinct ways of doing it, called the ADM energy and the Bondi energy. The difference is how energy radiated away to infinity is accounted for.) Stationary spacetimes have time translation invariance, so Noether's theorem applies and we can define a total energy (this is called the Komar energy).

However, the spacetime that describes our universe is neither asymptotically flat nor stationary, so neither of the above special cases apply.

Ibix · Feb 15, 2024

KleinMoretti said:

but then when we talk about there not being global conservation doesn't that imply that is no longer true.

...so you can get energy in odd ways in GR, except in the cases Peter described. But you typically need ropes long enough to stretch between galaxies, so on the everyday scale you don't.

pervect · Feb 21, 2024

KleinMoretti said:

to me conservation of energy means that we can't create something out of nothing, as for why it should be conserved I don't really know,

If you are familiar with the idea of the conservation of charge being expressed by the vanishing of a divergence in a vacuum, namely ##\nabla \cdot E = {\rho} / {\epsilon}## then we have a divergence in GR that vanishes (the divergence of the stress-energy tensor), and thus we can say that it's "conserved" in a similar sense.

There's an old book, "Div, Grad, Curl and all that", that talks about how the vanishing of a divergence can be thought of as a conservation law in an informal way.

If you are looking to associate a single number, "energy", to a physical system, however, the answer is more complicated. There are cases we can imagine, such as an isolated system with appropriate boundary conditions were we can answer "yes". However, the universe as a whole in standard cosmological models in not one of these cases - among other issues, it's infinite, so it's not as if it's some finite region surrounded by a vacuum. Other technical issues, involving the concept of "transport" (namely parallel transport) are probably more important but harder to describe informally.

So you can get answers varying from "the conservation of energy it's built into the field equations" to answers "energy is not conserved at all", depending on what you mean by "the conservation of energy". Unfortunately, there isn't any good alternative to understanding the technical details to disambiguate the problem :(.

Energy conservation in general relativity

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