# Energy stored in space-time or space?

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## Main Question or Discussion Point

I was talking to a graduate physics student about the issue of energy conservation in an expanding universe. I paraphrased the argument against energy conservation as follows -
Suppose we have a photon in outer space that is very far from earth. The universe is expanding (by this I meant that in some regions very far from earth, space is expanding). Let's assume that the photon does not make make contact with anything made out of matter. Well, since space is expanding around it, the wavelength of the photon must increase, and if the wavelength increases, then the energy of the photon decreases. So it seems that energy is destroyed in this process. I asked him where the energy goes, and his response was that the energy is stored in space (or space-time). I am quite skeptical of this response, because as far as I'm concerned, energy only makes sense as a property of matter. The only exception I can think of is gravitational potential energy.

My questions are -
1. Is the idea of energy being stored in space (or space-time) in the way in which he proposed possibly true? Is it plausible? Why, or why not?

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This post by Sean Carroll is a good exposition of the viewpoint the student you talked to appears to be taking:

http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/
I personally prefer this:

"'there’s energy in the gravitational field, but it’s negative, so it exactly cancels the energy you think is being gained in the matter fields.'"

Because I keep hearing about how Lagrangian mechanics can be used to derive relativity, and I've seen the Euler-Lagrange equation, and I've seen the Hamiltonian, how you can get the latter from the former, and they seem to be fairly dependent upon energy.

So, I'd personally prefer Sean Caroll's explanation that "does not actually increase anyone’s understanding," because to me, it definitely does. Or at least it feels more natural. Why can't gravitational fields have negative energy? Minkowski wrote that time can be thought of as imaginary space, and space as imaginary time (well, specifically what he said was: "Thus the essence of this postulate may be clothed mathematically in a very pregnant manner in the mystic formula 3⋅105 km = √(-1) secs.").

If we can have that, why can't gravitational fields have negative energy?

Orodruin
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Because I keep hearing about how Lagrangian mechanics can be used to derive relativity, and I've seen the Euler-Lagrange equation
Then you also know that energy is not necessarily a conserved quantity in Lagrangian mechanics.

Then you also know that energy is not necessarily a conserved quantity in Lagrangian mechanics.
Yeah but ∂L/∂t = 0 looks better. ;)

I am a bit out of my league here. In fact the furthest I ever gotten in looking at Lagrangian mechanics WAS the conservation laws, but I do have one conceptual problem here: I’ve seen (and worked through) the proof using Lagrangian mechanics that shows that time symmetry of the laws of physics implies energy conservation.

How would this not apply to general relativity? Don’t we assume that the laws of physics do not change just because spacetime changes?

Dale
Mentor
How would this not apply to general relativity? Don’t we assume that the laws of physics do not change just because spacetime changes?
It does apply to general relativity. The thing is that the “law of physics” referred to is the Lagrangian, and certain space times have a Lagrangian which is not symmetric under time translations.

PeterDonis
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2019 Award
Don’t we assume that the laws of physics do not change just because spacetime changes?
The law of physics in this case is the Einstein Field Equation. It's the same everywhere in spacetime. But the stress-energy tensor and the Einstein tensor are not. The EFE doesn't say those tensors are constants; it just says that, however they vary in spacetime, they both vary the same way, so they are equal at every event.

So my last series of questions on this, if you don't mind: is it a completely arbitrary choice about whether you assume (1) negative energy in the gravitational field, which cancels energy being gained or (2) energy is just not conserved.

Is that like with synchronization conventions? You just make the choice? If so, is one more convenient than the other?