# Is energy conserved?

1. Feb 16, 2015

My understanding at present is that if a system of interacting particles is analysed using classical physics or special relativity energy is conserved, but if that same system is analysed using general relativity energy is not conserved. So is it conserved or not?

Looking at it another way, energy is conserved if spacetime is static but not conserved if spacetime is evolving. It's apparently believed that spacetime is evolving so energy is not conserved. Or is it conserved? Help!

Is the conservation of energy principle an approximation only which works well in certain conditions such as smallish localised areas or is there some other resolution to this apparent paradox?

Thank you

2. Feb 16, 2015

### Orodruin

Staff Emeritus
Energy conservation is something local and in GR it is not necessarily true that it can be extended to a global concept. You might simply not be able to ask the question "what is the total energy of the Universe?" However, the conservation of energy is replaced by the divergence of the energy-momentum tensor being zero, so you cannot go all out crazy with energy non-conservation even in GR.

3. Feb 16, 2015

### HallsofIvy

Even in special relativity, it is "total mass-energy" that is conserved, not mass or energy separately.

4. Feb 16, 2015

### Orodruin

Staff Emeritus
I do not agree with this. Energy, including the rest energy due to mass, is conserved in special relativity and this has a well defined meaning. Simply take a surface of simultaneity in a given frame and integrate the time-time component of the energy-momentum tensor over it and it will be the same regardless of the time defining the surface. That you have to include the masses of your constituents in order to obtain this is a different issue altogether.

5. Feb 16, 2015

### DrGreg

6. Feb 16, 2015

### Staff: Mentor

This doesn't make sense. SR is just a special case of GR, so if SR applies to a given system, it must give the same answers as GR gives, since GR applied to that same system just is SR.

But this isn't the same system analyzed two different ways (with SR and GR). It's two different systems. A static spacetime is a different system from a non-static spacetime. So there's nothing mysterious about the fact that energy conservation works differently in the two systems; they're different systems.

7. Feb 17, 2015

### pervect

Staff Emeritus
What do you mean by energy? There are certain technical definitions of energy that ARE conserved in GR, but they have prerequisites (such as static spacetimes, or asymptotically flat spacetimes) before they are able to be calculated. There isn't a single universal definition of "energy" in GR that always gives a conserved quantity.

It would seem to me from the tone of your question that you're not familiar with the technicalities. I can't blame you for that, really, but I'm at a loss to answer a question about "energy being conserved" if we don't have a mutual understanding of what "energy" is.

Probably the most readable introduction is what Dr. Greg already quoted, the sci.physics.faq reference.

It might also be helpful to say things like "ADM energy and Bondi energy are defined and conserved in asymptotically flat space-times, while Komar energy is defined and conserved in static space-times.

I suppose it might be helpful to note that if you add up non-gravitational sources of energy, you won't get a conserved quantity unless you include something that's equivalent to the Newtonian idea of "gravitational potential energy". But GR doesn't have a single clear idea to replace this Newtonian idea, though it does have some ideas of how to define conserved energies in special circumstances.

8. Feb 17, 2015

Thank you very much Orodruin. That's largely clarified things but I still have some problems one of them best exemplified by the following question:

Is it true that there exists certain problems that can be solved but which require the application of the conservation of energy principle for their solution?

If it is true then I assume that GR can't be applied to the problem because it doesn't necessarily recognise energy conservation. Does this mean that such problems do not lie withinin the domain of applicability of GR or could there be some other reasons why GR does not work

9. Feb 17, 2015

Thank you Orodruin, Halls of Ivy and DrGreg. I need to do some more research on this and you have provided some pointers about where to look. The FAQ referred to by yourself DrGreg looks particularly promising.

10. Feb 17, 2015

Thanks for your reply PeterDonis but I am just learning this stuff and I find your replies to be a bit contradictory. I Might be misunderstanding the points you have made. You say that SR and GR "must give the same answers" but then say that "energy conservation works differently in the two systems". If it works differently does it still give the same answers?

The following is a quote from Sean Carrolls blog referring to GR: (google "energy is not conserved")

"When the space through which particles move is changing the total energy of those particles is not conserved"

That's the thing that confuses me because in SR energy is conserved evidenced, for example, by nuclear energy

(Sean Carrol is a theoretical cosmologist from Caltech who amongst other things specialises in GR)

11. Feb 17, 2015

Thank you pervect.
By energy I mean how it is defined in classical physics in terms of work done and how it relates to particle events and interactions. I'm interested in things such as potential/kinetic energy changes, particle collisions, nuclear reactions and so on. As an example consider a high energy electron electron collision. Can GR be used to analyse the event and would it give the same answers as SR?

12. Feb 17, 2015

### Staff: Mentor

Suppose you have three different situations, A, B, and C. And suppose that A and B are different scenarios without tidal gravity while C involves tidal gravity.

An SR energy analysis gets a certain result for A, and a SR energy analysis gets a certain result for B. The two results differ because A and B are different scenarios.

A GR energy analysis gets a certain result for A, and a GR energy analysis gets a certain result for B. Again, the two results differ as above. The GR result for A agrees with the SR result for A and the GR result for B agrees with the SR result for B.

Scenario C cannot be analyzed with SR at all, so GR is required. Depending on the details there may not be any globally conserved energy available for the analysis.

13. Feb 17, 2015

### Staff: Mentor

Perhaps this thread is the place to clarify a point that I always found cloudy. Conservation of energy in GR as discussed in the above referenced FAQ, and conservation of energy related to zero-point energy and increasing volume of space. Are those properly two separate subjects, or are they the same with the "cosmological constant" representing the average of what happens at the micro level?

14. Feb 17, 2015

### Staff: Mentor

Any time that SR can be used GR can also be used and will give the same answer.

Additionally, there are scenarios in which SR can't be used. In those GR can be used, but in some of them there is no globally conserved energy.

15. Feb 17, 2015

Thank you DaleSpam

I can see that the results differ for different scenarios but not that they agree for the same scenario. Here's another quote from the Sean Carroll blog:

"If that spacetime is standing completely still, the total energy is constant; if its evolving, the energy changes in a completely unambiguous way".

To me that suggests that the SR energy analysis and GR energy analysis give different results.

16. Feb 17, 2015

### Staff: Mentor

That is two different scenarios. A static spacetime and a non static spacetime. A vs B, not SR vs GR.

17. Feb 17, 2015

### Staff: Mentor

When they are used to analyze the same system, yes.

When the systems are different, yes.

That's because in any system that can be analyzed using SR, "the space through which particles move" is not changing. In such a system, as Carroll says, energy is conserved.

If "the space through which particles move" is changing (for example, in the universe as a whole, which is expanding), you can't use SR to analyze the system. You have to use GR, and you will find, as Carroll says, that energy is not conserved.

18. Feb 17, 2015

### DrStupid

What does that mean for mass and energy to be conserved separately or not?

19. Feb 17, 2015

Thank you both. The last comment above and Dalespams comment summarise the difficulty I'm having here. Basically I want to analyse a single scenario and not two scenarios. If I think about a moving proton I see that as a single scenario. I can imagine it moving through spacetime and not two different spacetimes.

Spacetime is whatever it is and the proton moves through it. How can spacetime be changing and not changing? How can energy be conserved and not conserved? (Reminds me of quantum superpositions.)

I'm still puzzling over this but it will start to make some sort of sense if, for example, SR and GR each had its own domain of applicabilty, perhaps to do with the scale of the event. If something like this is the case I still see difficulties.

20. Feb 17, 2015

### Orodruin

Staff Emeritus
This depends on what your actual spacetime is. If it is sufficiently close to a Minkowski spacetime, then SR will suffice and the GR approach will just mimic it. If it is not sufficiently close, then only GR will be applicable.

I want to make a comparison to parallel lines on a sphere. If you are studying a sufficiently small portion of the sphere, the deviations from Euclidean space will be small and parallel lines will not cross. Euclidean space would be fine for approximating this behaviour. But if you look at distances comparable to the curvature, then all straight lines will cross and Euclidean space is not sufficient to make the description.