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.

 

[Dale]

It works very well here but I want to know does it work equally well in other geometries.

Asked and answered.

Does that mean that laws and equations involving positions, for example Coulomb's law, work only in this approximately flat geometry.

Yes

 

[pervect]

The following quote from MTW's "Gravitation" $20.4 "Why the energy of the gravitational field cannot be localized" may be of some help.

Omitted is an introduction in which the autors introduce a 'straw-man' approach to gravitational energy using pseudotensors that the authors proceed to shoot down.

To ask for the amount of electromagnetic energy and momentum in an element of 3-volume makes sense. First, there is one and only one formula for this quantity. Second, and more important, this energy-momentum in principle "has weight." It curves space. It serves as a source term on the righthand side of Einstein's field equations. It produces a relative geodesic deviation of two nearby world lines that pass through the region of space in question. It is observable. Not one of these properties does "local gravitational energy-momentum" possess. There is no unique formula for it, but a multitude of quite distinct formulas. The two cited are only two among an infinity. Moreover, "local gravitational energy-momentum" has no weight. It does not curve space. It does not serve as a source term on the righthand side of Einstein's field equations. It does not produce any relative geodesic deviation of two nearby world lines that pass through the region of space in question. It is not observable.

Anybody who looks for a magic formula for "local gravitational energy-momentum" is looking for the right answer to the wrong question. Unhappily, enormous time and effort were devoted in the past to trying to "answer this question" before investigators realized the futility of the enterprise. Toward the end, above all mathematical arguments, one came to appreciate the quiet but rock-like strength of Einstein's equivalence principle. One can always find in any given locality a frame of reference in which all local "gravitational fields" (all ChristofTel symbols disappear. No Christoffel symbols means no "gravitational field" and no local gravitational field means no "local gravitational energy-momentum."

Let me paraphrase the argument. If you consider some 1m^3 volume on the surface of the Earth, if you use usually coordinates of static objects, there is a gravitational field in the form of the Christoffel symbols present. I will try to explain the techincal language by saying the Christoffel symbols represent, among other things, the weight you read on a scale, or the weight you feel on your rear when you sit in a chair. In short, what most people think of as "gravity", the same notion that Newton had, of gravity as a force.

However, we can equally imagine a free-falling observer. This observer won't feel any "gravitational field" - they will feel second-order tidal forces, but, being in free fall, they won't feel any weight pulling them down in their chair.

In Newton's theory, the force of gravity is an actual force, and you can look at what work this force does. In GR, gravity is curved space-time, and you can always find an observer, moving along a geodesic, who doesn't experience any force. One of the important features of the theory is its observer independence, there point is there isn't any formulation of the "gravitational field" in the sense of forces you feel on your backside that is observer independent, so trying to leverage off the Newtonian ideas doesn't get anywhere. The basic issue is that GR is observer independent, while the concept of weight (technically, Christoffel symbols) is not.

Back to MTW:

Nobody can deny or wants to deny that gravitational forces make a contribution to the mass-energy of a gravitationally interacting system. The mass-energy of the Earth-moon system is less than the mass-energy that the system would have if the two objects were at infinite separation. The mass-energy of a neutron star is less than the mass-energy of the same number of baryons at infinite separation. Surrounding a region of empty space where there is a concentration of gravitational waves, there is a net attraction, betokening a positive net mass-energy in that region of space (see Chapter 35). At issue is not the existence of gravitational energy, but the localizability of gravitational energy. It is not localizable. The equivalence principle forbids.

This is the part that says even though we can't come up with an observer independent notion of "the gravitational field", much less any way to come up with "how much energy the gravitational field has", we can't ignore the whole idea of "gravitational energy" and get an overall conserved quantity.

What we are left with is that there are ways to get a conserved quantity, but they numbers we get are not observer independent when we consider some specific locatoin - we can't assign the gravitational energy (that we need to include to have a conserved quantity) any specific location, the best we can do is come up with an overall number. This is the case, at least, without specifying some particular "preferred class" of observers. We need to include it to have the books balance, but the detailed assignment of energy we get when we do this is different for different observers.

There are some interesting ideas which I would loosely describe as specifying a preferred class of observers, the so-called "De-donder gauge". See for instance http://ptp.oxfordjournals.org/content/75/6/1351.full.pdf. I'm not sure how popular this idea is, I suspect not very because it seems to be limited to advanced papers rather than something you read in your average GR textbook. If you read a textbook on GR, you'll probably see something about ADM, Bondi, and Komar masses, but not little or nothing on the DeDonder gauge. But on the plus side, my understanding is that you get a notion of energy that's defined without the special requiremcan coents (of asymptotic flatness or stationary space times. Furthermore, when you do have these special requirements met, you get comparable numbers to the ADM, Bondi, and Komar formulae. I'm afraid I'm not quite sure if the "comparable numbers" are exactly the same. There are a few details of the comparsion process that need to be specified, at a minimum one would need to compensate for the fact that the Bondi approach doesn't include the energy stored in gravitational waves.

 

[bhobba]

Hi Guys

Nice thread.

One thing I want to mention is the modern conception of energy is based on Noethers theorem and the symmetries of the system which is usually associated with the symmetries of an assumed inertial frame it resides in. Since GR is based on space-time curvature Noether's theorem lacks applicability and you can't define energy using it so of course conservation is problematical:
http://motls.blogspot.com.au/2010/08/why-and-how-energy-is-not-conserved-in.html
'The main lesson here is that general relativity is not a theory that requires physical objects or fields to propagate in a pre-existing translationally invariant spacetime. That's why the corresponding energy conservation law justified by Noether's argument either fails, or becomes approximate, or becomes vacuous, or survives exclusively in spacetimes that preserve their "special relativistic" structure at infinity. At any rate, the status of energy conservation changes when you switch from special relativity to general relativity.'

Thanks
Bill

 

[bhobba]

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?

See my post above.

Its due to the modern definition of energy which is based on symmetries and Noether's beautiful theorem. Those symmetries are lacking in GR.

In fact it was while investigating this very issue ie local energy conservation in GR, she discovered her very important theorem:
http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html
'Energy conservation in the general theory has been perplexing many people for decades. In the early days, Hilbert wrote about this problem as 'the failure of the energy theorem '. In a correspondence with Klein he asserted that this 'failure' is a characteristic feature of the general theory, and that instead of 'proper energy theorems' one had 'improper energy theorems' in such a theory. This conjecture was clarified, quantified and proved correct by Emmy Noether. In the note to Klein he reports that had requested that Emmy Noether help clarify the matter.'

Thanks
Bill

 

[Dadface]

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.)

Yes and thank you .Things are beginning to clarify now. I think I know where you're going with the capacitor calculation.

 

[Dadface]

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.

All of that is very unfair because it's untrue.

I like every answer in this thread which has been relevant. Each one has contributed to the level of understanding I have now. Even the answers that I have yet to fully understand have taught me things such as the necessity to get familiar with the terminology used.
I have asked additional questions for example to get greater clarification but I have not argued. I have not excluded a single answer and I have not refused to "listen" to the answers. In fact I have read through this thread several times and researched on the net trying to get a better grasp of some of the answers given.

 

[Dadface]

In connection with what has been described before I would like to ask another question which may be so naive or perhaps even meaningless that I'm reluctant to send it. But here goes anyway:

If I'm now sitting in a place where the geometry of spacetime can be described as being approximately Minkowskian, at what places can I go to (perhaps as thought experiments) where the geometry of spacetime is different?
Thank you.

 

[jimgraber]

Hi Dadface,
As I hinted before, your best choice is to go back to the beginning of the universe, if it is not flat.
The next best choice is to go really close to a black hole, especially a small one.

Jim Graber

 

[jimgraber]

A second thought:
There are really (at least) two issues here;
First, where is spacetime non trivially curved?
Second, where is energy not conserved, or at least hard to define?
These issues are related in a complicated way, but they are not the same.
Best again,
Jim Graber

 

[jimgraber]

A third thought:
If you go really close to a small black hole, you can also worry about loss of information, Hawking radiation and firewalls.
Still more fun, for those who care.

 

[Dale]

If I'm now sitting in a place where the geometry of spacetime can be described as being approximately Minkowskian, at what places can I go to (perhaps as thought experiments) where the geometry of spacetime is different?

This depends on the sensitivity of your experiment. "Local" means "a small enough region of spacetime that tidal effects can be neglected". The more sensitive your measurement, the smaller the allowed region.

To worry about energy non conservation you need a spacetime that is not static with large tidal forces. For example around a binary pair of black holes or neutron stars.

 

[Dadface]

Hi Dadface,
As I hinted before, your best choice is to go back to the beginning of the universe, if it is not flat.
The next best choice is to go really close to a black hole, especially a small one.

Jim Graber

A second thought:
There are really (at least) two issues here;
First, where is spacetime non trivially curved?
Second, where is energy not conserved, or at least hard to define?
These issues are related in a complicated way, but they are not the same.
Best again,
Jim Graber

A third thought:
If you go really close to a small black hole, you can also worry about loss of information, Hawking radiation and firewalls.
Still more fun, for those who care.

That's really helpful jimgraber so thank you very much. You have mentioned places whose structures differ enormously from the sructure of this place. As a result of thinking about your examples I'm assuming that it is the structure of the place that determines the geometry of the spacetime in the vincinity of that place. Is my assumption correct? If so I should have realized it earlier.

 

[PAllen]

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.
...

You allude to this a little later, but I thought it worth pointing out that Bondi energy (when defined) is not meant to be a conserved quantity. Instead, it specifically does not include radiation reaching asymptotic infinity. It thus decreases slowly for most non-stationary systems. Meanwhile, ADM energy (when defined) is conserved, and the delta between ADM and Bondi energy is taken to be radiation escaping to asymptotic infinity.

 

[Dadface]

This depends on the sensitivity of your experiment. "Local" means "a small enough region of spacetime that tidal effects can be neglected". The more sensitive your measurement, the smaller the allowed region.

To worry about energy non conservation you need a spacetime that is not static with large tidal forces. For example around a binary pair of black holes or neutron stars.

Thank you DaleSpam. jimgraber mentioned places which can be visited by the means of a thought experiment. I wonder if you or anyone else can tell me if there are places where you can actually go to even if just in principle. As an example suppose we wanted to measure the mass of the electron. At present this is measured to ten decimal places.
m = 9.10938291(40) The uncertainty is highlighted by the use of brackets

 

[Dale]

I wonder if you or anyone else can tell me if there are places where you can actually go to even if just in principle. As an example suppose we wanted to measure the mass of the electron. At present this is measured to ten decimal places.
m = 9.10938291(40)

I am confused now. I thought that you wanted to know about the conservation of energy. What is the mass of the electron thing about?