Energy Conservation Paradox: Is It True or Not?

[Dadface]

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

 

[Orodruin]

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.

 

[HallsofIvy]

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

 

[Orodruin]

not mass or energy separately.

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.

 

[DrGreg]

You may find the Usenet Physics FAQ "Is Energy Conserved in General Relativity?" relevant.

 

[PeterDonis]

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.

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.

Looking at it another way, energy is conserved if spacetime is static but not conserved if spacetime is evolving.

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.

 

[pervect]

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?

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.

 

[Dadface]

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

 

[Dadface]

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.

 

[Dadface]

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.

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)

 

[Dadface]

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.

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?

 

[Dale]

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?

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.

 

[anorlunda]

You may find the Usenet Physics FAQ "Is Energy Conserved in General Relativity?" relevant.

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?

 

[Dale]

Can GR be used to analyse the event and would it give the same answers as SR?

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.

 

[Dadface]

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.

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.

 

[Dale]

"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.

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

 

[PeterDonis]

You say that SR and GR "must give the same answers"

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

but then say that "energy conservation works differently in the two systems".

When the systems are different, yes.

"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

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.

 

[DrStupid]

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

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

 

[Dadface]

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

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.

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.

 

[Orodruin]

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.)

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.

 

[PeterDonis]

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.

But "spacetime" in itself is not a well-defined term. There are many different possible spacetimes. You have to specify which one you are thinking of before you have a well-defined scenario. Sure, within that scenario, "spacetime" refers to the particular spacetime you picked, not to all the possible ones in general. But you have to pick one; you can't just say "spacetime" without qualification, because that doesn't pick out one particular one from all the possibilities. And which one you pick will affect whether energy is conserved in your scenario; see below.

Spacetime is whatever it is

No, it isn't. "Spacetime" is a general term that refers to all the different possible 4-dimensional geometries that are consistent with the laws of GR. One particular 4-dimensional geometry, called "Minkowski spacetime", is the one for which SR is valid (and "GR" applied to this spacetime is identical to SR). But there are many others. In order to analyze a particular scenario, you have to pick, not just the objects in the scenario (such as a moving proton), but the spacetime the scenario takes place in as well.

 

[Dadface]

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.

But "spacetime" in itself is not a well-defined term. There are many different possible spacetimes. You have to specify which one you are thinking of before you have a well-defined scenario. Sure, within that scenario, "spacetime" refers to the particular spacetime you picked, not to all the possible ones in general. But you have to pick one; you can't just say "spacetime" without qualification, because that doesn't pick out one particular one from all the possibilities. And which one you pick will affect whether energy is conserved in your scenario; see below.



No, it isn't. "Spacetime" is a general term that refers to all the different possible 4-dimensional geometries that are consistent with the laws of GR. One particular 4-dimensional geometry, called "Minkowski spacetime", is the one for which SR is valid (and "GR" applied to this spacetime is identical to SR). But there are many others. In order to analyze a particular scenario, you have to pick, not just the objects in the scenario (such as a moving proton), but the spacetime the scenario takes place in as well.

Thank you again for your responses but I'm still not convinced. Let me illustrate why by means of an example:

Let a proton move from a place A to a different place B and let the electrical potential difference between the two places be equal to V.

The resulting change of electrical potential energy will be equal to to eV and that change is independant of the route taken. Or so I thought. Now it seems that the change of PE is not necessarily equal eV but depends on the geometry of the spacetime within which the event takes place. The key thing about the change of PE is the potentials at the two places A and B so I can't see how geometry comes into this. I think I need to look at the whole thing in greater detail.

 

[Dale]

Let a proton move from a place A to a different place B

"Place A" and "Place B" don't mean anything unless you specify the spacetime.

 

[PeterDonis]

The resulting change of electrical potential energy will be equal to to eV

That's because that's the definition of "change in electrical potential energy".

and that change is independant of the route taken.

Sure, because "electrical potential energy" is defined as a function of position. But, as DaleSpam pointed out, "position" itself does not have a well-defined meaning unless you specify the spacetime.

Now it seems that the change of PE is not necessarily equal eV but depends on the geometry of the spacetime within which the event takes place.

No. The change of PE is equal to eV because, as I said above, that's the definition of "change in electrical potential energy". What depends on the geometry of the spacetime is how you pick out Place A and Place B.

The key thing about the change of PE is the potentials at the two places A and B so I can't see how geometry comes into this.

It comes in when you try to define what the "places" are. See above.

I think I need to look at the whole thing in greater detail.

Yes, you do.

 

[jimgraber]

Hi Dadface,

In most cases, GR is a minor correction, and you can treat it as a perturbation, using the asymptotically flat rules prevect mentioned. You get a good definition of gravitational energy, and it is FAPP conserved. This is even true if you are working with a black hole or two (or a neutron star) and the curvatures are nontrivial. (In this case the perturbations become much larger, but can still be controlled with the asymptotically flat formalisms.)

Where things go south and gravitational energy can't even be consistently defined, much less conserved is when you work with cosmology. (Hint: Sean Carroll works primarily on cosmology.)

The messier your cosmological model and the weirder its topology, the more trouble you are in.

The expansion/contraction issue is also potentially troublesome.The above summary is a little bit oversimplified, but I think it is reasonably close to the mainstream view.
Jim Graber

 

[Dadface]

"Place A" and "Place B" don't mean anything unless you specify the spacetime.

With the point I'm making you don't need to specify the spacetime. If the proton moves through a potential difference V the change in potential energy will be eV regardless of the geometry or structure or any other properties of the regions through which the proton moves. This change of potential energy is central to the conservation of energy principle which is what I'm interested in here.

Thank you

 

[Orodruin]

With the point I'm making you don't need to specify the spacetime.

Yes you do. In order to even specify a potential, you need to specify the space-time, otherwise there are no points for the potential to take a value in.

Edit: That is, if you do not specify a space-time, you cannot specify points A and B.

 

[Dadface]

That's because that's the definition of "change in electrical potential energy".

True but I can't see the point of this comment. The definition is informed by observations and conforms to observations those observations being relevant to energy conservation. Thats why I used it.



Sure, because "electrical potential energy" is defined as a function of position. But, as DaleSpam pointed out, "position" itself does not have a well-defined meaning unless you specify the spacetime.

But as I pointed out in a post above I can quantify the event without specifying position or spacetime. For example:

"Change of potential energy as a result of moving through a potential difference V is equal to eV".

I can equate this change to work done.



No. The change of PE is equal to eV because, as I said above, that's the definition of "change in electrical potential energy". What depends on the geometry of the spacetime is how you pick out Place A and Place B.

And as I said above that definition is based on observations. It works pretty well.

I would need to refer to geometry to pick out the places A and B but not to make the statement above which is written in inverted commas.



It comes in when you try to define what the "places" are. See above.

Agreed but only when I would wan't quantify the event in greater detail. The point is that the change of potential energy can be equated to work done regardless of geometry, structure and anything else that is necessary to consider when there is a wish quantify the event in greater detail.

 

[Dale]

If the proton moves ...

Again, "moves" is not meaningful without specifying the spacetime.

In fact, not only is it necessary to specify the spacetime, but you also need to specify a coordinate chart. What is "move" in one chart is "at rest" in another, and what is a scalar potential V in one chart is a vector potential A in another.

You seem to think that we are trying to trick you into specifying the spacetime. We are not. The requirement to specify the spacetime is a fundamental requirement which has always been there. You have simply never paid attention to the requirement because you have always assumed a flat spacetime and not realized that you did so. However, not realizing it does not make it any less of a requirement.

 

[PeterDonis]

Dadface, as a side note, please use the quote feature as it is designed. If you just bold your own comments inside a quote of someone else's, it breaks the quote feature for everybody else.

I can quantify the event without specifying position or spacetime.

No, you can't, because in order to quantify the event you have to measure the particle at particular positions. You can't measure electrical potential without measuring position, because electrical potential is a function of position.

that definition is based on observations.

Observations that require measuring positions. See above.

 

[Dadface]

Again, "moves" is not meaningful without specifying the spacetime.

In fact, not only is it necessary to specify the spacetime, but you also need to specify a coordinate chart. What is "move" in one chart is "at rest" in another, and what is a scalar potential V in one chart is a vector potential A in another.

You seem to think that we are trying to trick you into specifying the spacetime. We are not. The requirement to specify the spacetime is a fundamental requirement which has always been there. You have simply never paid attention to the requirement because you have always assumed a flat spacetime and not realized that you did so. However, not realizing it does not make it any less of a requirement.

Thanks DaleSpam,

You have probably gathered that I have extremely limited knowledge of SR, different types of spacetime and so on. I haven't even read the opening chapter of any book on the subject and I doubt if I will get the time, or indeed have the inclination to do so. Because of that it seems I have been using the wrong terminology in different comments I have posted here. Despite my lack of knowledge I think the question I asked is a reasonable one. In a nutshell is energy conserved?

Let me try to rephrase the question about potential by referring to a thought experiment which can easily be done as a real experiment :

I have a charged variable capacitor with overlapping plates and I manually increase the separation of the plates. If I make no assumptions at all about the type of spacetime the experiment is carried out in I can calculate the increase in the energy stored in the field of the capacitor and I will know that this energy came from some of the energy that I used to increase the separation. I will feel confident that energy has been conserved.

Now I can imagine being told that the calculations I made might be in error because I didn't specify the nature of the spacetime in which I carried out the experiment. But I carried out my experiment "here", where I am. This is the spacetime I exist in and where I make my observations. Will my calculations be meaningless because I didn't specify the spacetime of where "here" is.

My understanding was that GR considers the universe on a grand scale and assumes an expanding space but as scales reduce static spacetime and Minkowski space is approached. If it is something like that i can understand it but I know I've probably used the wrong terminology.
Thank you

 

[PeterDonis]

If I make no assumptions at all about the type of spacetime the experiment is carried out in I can calculate the increase in the energy stored in the field of the capacitor and I will know that this energy came from some of the energy that I used to increase the separation. I will feel confident that energy has been conserved.

This calculation does make an assumption about the type of spacetime the experiment is carried out in. It assumes that spacetime is flat, at least to a good enough approximation in the region of spacetime in which the experiment took place. (At least, it does if you are doing the calculation the way I strongly suspect you are, since it's the way that is taught in electrodynamics class.) You just didn't realize that you were making that assumption. But if you show me the mathematical formula you use to calculate the increase of energy, I will show you where it makes that assumption, even if you didn't realize it.

(Hint: you said you "manually increase the separation of the plates". "Separation" means "distance between", and "distance" involves the geometry of spacetime. The formula I strongly suspect you would use to make the calculation implicitly assumes that spacetime, at least the region of it covered by the capacitor, is flat. It's easy to miss this because flat spacetime means that a lot of coefficients that would be present in a general curved spacetime are equal to 1, so they drop out of the formulas and it can seem like there's no spacetime at all in there, when actually there is.)

 

[PeterDonis]

My understanding was that GR considers the universe on a grand scale and assumes an expanding space but as scales reduce static spacetime and Minkowski space is approached.

You've got the right idea, yes. The usual way of stating this is that, in a small enough region of spacetime, we can always construct a "local inertial frame" in which that small region of spacetime looks just like a small region of Minkowski spacetime. As long as we restrict our attention to things happening within this local inertial frame (note that "local" means local in time as well as in space, so we mean things happening in a small enough region of space over a short enough interval of time), we can ignore the fact that spacetime globally might be curved. But that doesn't mean we don't make any assumptions about spacetime; it means that we assume spacetime is flat, or at least it is "flat enough" (whatever curvature might exist is too small to matter) in the small region we are dealing with.

 

[Dadface]

Dadface, as a side note, please use the quote feature as it is designed. If you just bold your own comments inside a quote of someone else's, it breaks the quote feature for everybody else.
No, you can't, because in order to quantify the event you have to measure the particle at particular positions. You can't measure electrical potential without measuring position, because electrical potential is a function of position.
Observations that require measuring positions. See above.

Thanks for your advice about the quote feature. I don't know how to use it to best effect yet but I will find out.

With regard to your second point, I had previously referred to the generally accepted fact that in (approximately Minkowskian) space, if the potential difference is V the work done is eV. I had taken V as a known piece of information just as I would have accepted the validity of Coulombs law if I wanted to calculate values of V. The point is that I can equate eV to the work converted in moving through the E field. The equation is one of numerous other equations that expresses energy conservation. It works very well here but I want to know does it work equally well in other geometries.
A more rigorous quantification of the event depends on a knowledge of loads of other variables including the permittivity of the space.

With regard to your third point the observations do indeed require measuring positions not only in this case but in numerous other cases. Does that mean that laws and equations involving positions, for example Coulomb's law, work only in this approximately flat geometry.

 

[Dale]

You have probably gathered that I have extremely limited knowledge of SR, different types of spacetime and so on. I haven't even read the opening chapter of any book on the subject and I doubt if I will get the time, or indeed have the inclination to do so. Because of that it seems I have been using the wrong terminology in different comments I have posted here. Despite my lack of knowledge I think the question I asked is a reasonable one. In a nutshell is energy conserved?

It is a reasonable question and you have received the correct answer and didn't like it.

I understand that you don't know this stuff and may not have the time to read up on the background material, but then don't try to argue with the people who have. Also, you cannot expect to ask a question about a theory and then exclude answers because they use that theory. The problem isn't your question, it is your refusal to listen to the answers.

If I make no assumptions at all about the type of spacetime the experiment is carried out in I can calculate the increase in the energy

You cannot do any such calculations at all without making assumptions about the type of spacetime. You may not realize that the assumptions are there, but those assumptions are there nonetheless. When you use a formula you are implicitly making all of the assumptions on which that formula is based.

Now I can imagine being told that the calculations I made might be in error because I didn't specify the nature of the spacetime in which I carried out the experiment. But I carried out my experiment "here", where I am. This is the spacetime I exist in and where I make my observations. Will my calculations be meaningless because I didn't specify the spacetime of where "here" is.

You specified the spacetime by choosing the formulas that you used in your calculations. The usual formulas were developed for the spacetime "here", so by using them you automatically have specified the spacetime "here".

My understanding was that GR considers the universe on a grand scale and assumes an expanding space but as scales reduce static spacetime and Minkowski space is approached. If it is something like that i can understand it but I know I've probably used the wrong terminology.

Yes, this is the "local" vs "global" that was explained in post 2 and also in the FAQ linked in post 5.