Conservation of Energy and General Relativity

[physicsquantum]

I was reading through some main stream scientific literature, and I came across Sean Caroll's "Energy Is Not Conserved" post. Essentially, he contends that through general relativity energy is not conserved, at least not in conventional manner of thinking about energy.

Anyways, some portions of the post confused me. He says that "energy is conserved in general relativity, it’s just that you have to include the energy of the gravitational field along with the energy of matter and radiation and so on."

However later on in the post he also states that "energy isn’t conserved; it changes because spacetime does."

I'm hoping that someone could possibly read through this if they're interested and clear it up for me: http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/comment-page-2/#comments

Furthermore, if second claim is correct, would this not only disprove the conservation of energy but also the conservation of mass due to mass-energy equivalence?

Appreciate all responses and look forward to receiving clarification on the post

 

[samalkhaiat]

I was reading through some main stream scientific literature, and I came across Sean Caroll's "Energy Is Not Conserved" post. Essentially, he contends that through general relativity energy is not conserved, at least not in conventional manner of thinking about energy.

Anyways, some portions of the post confused me. He says that "energy is conserved in general relativity, it’s just that you have to include the energy of the gravitational field along with the energy of matter and radiation and so on."

However later on in the post he also states that "energy isn’t conserved; it changes because spacetime does."

I'm hoping that someone could possibly read through this if they're interested and clear it up for me: http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/comment-page-2/#comments

Furthermore, if second claim is correct, would this not only disprove the conservation of energy but also the conservation of mass due to mass-energy equivalence?

Appreciate all responses and look forward to receiving clarification on the post

Read my posts in
https://www.physicsforums.com/threads/stress-energy-tensors-in-gr.787233/

 

[PeterDonis]

He says that "energy is conserved in general relativity, it’s just that you have to include the energy of the gravitational field along with the energy of matter and radiation and so on."

However later on in the post he also states that "energy isn’t conserved; it changes because spacetime does."

These are not two different things he's saying; they are descriptions of two different ways of describing the same physics in ordinary language. (See further comments on that below.) He prefers the second way; others prefer the first way. Read that part of the article again, carefully; you will see that he says, among other things:

We all agree on the science; there are just divergent views on what words to attach to the science.

In other words, the question you're asking isn't about physics, but about terminology. The physics is the same either way.

would this not only disprove the conservation of energy but also the conservation of mass due to mass-energy equivalence?

No. "Conservation of energy" is not a precise term; it can mean at least two different things. This is a general problem with trying to describe scientific theories in ordinary language: ordinary language is not precise, while scientific theories are. In this case, the two possible meanings are:

(1) Local energy conservation: energy can't be created or destroyed in any small local piece of spacetime. This is true, and nothing in Carroll's article contradicts it. In fact, he talks about it explicitly, when he says:

There is still a single important equation, which is indeed often called “energy-momentum conservation.” It looks like this

$$
\nabla_\mu T^{\mu \nu} = 0
$$

The details aren’t important, but the meaning of this equation is straightforward enough: energy and momentum evolve in a precisely specified way in response to the behavior of spacetime around them.

Notice that here too we see the imprecision of ordinary language: this equation, which has a precise theoretical meaning, can be described as "energy-momentum conservation" or as "energy and momentum evolve in a precise way in response to the behavior of spacetime around them", which doesn't sound a lot like "conservation" since "evolve" implies "change". But at any rate, this equation is what prevents things like perpetual motion machines from working, so it captures the ordinary intuitive meaning of "energy conservation".

(2) Global energy conservation: if we add up all of the energy in spacetime at one instant of time, and then do the same thing again at a later instant of time, the two sums should be the same. This is not always true; in fact, in many cases the sum is not even well-defined. (Carroll is referring to the latter issue when he says that there is no such thing as a "density" of gravitational energy.) The fact that this is not always true is what Carroll means by the title of his blog post. (Note that there are cases in which it is true; those are the cases in which the property Carroll calls "time translation invariance" holds. But there are many cases where it doesn't; the most important is cosmology, since the universe as a whole is not time translation invariant.)