Local conservation of energy

[Danijel Gorupec]

I have questions about local conservation of energy. I am trying to better understand consequences that follow from it.
 

How I understand it, the 'local conservation' means that not only energy is conserved, but is conserved in every local region. It cannot just at once decrease in one region, and increase in a region far away without anything happening in between.
 

So, is the local conservation of energy a true law? It seems correct to me, but I cannot remember if I ever heard about it during my formal education (looong time ago)... [BTW, is it also valid regarding quantum mechanic? Specifically what I have in my mind is the wave function collapse - does the collapse has at all anything to do with energy distribution?]

 

Next... My understanding is that one of direct consequences of the law of local conservation of energy is that we are then capable to define a value of 'energy flow' (through space) and a value of 'energy density' (in space). Am I correct?
 

I have troubles understanding this 'energy density'. How much is this distribution of energy throughout space a physical thing. Should I expect that with a very sensitive equipment I should be able to measure a curvature of space that this energy density is producing... thus actually measuring energy density? (I understand that energy density should curve the space as well as mass do).
 

Is energy density tied to one frame of reference or is it the same in all frames of reference? It seems to me that if you change the frame of reference, the energy density distribution changes too. Take for example a kinetic energy.
 

Really, "where" is the kinetic energy? I mean, how is its density distributed in space? Where is gravitational potential energy? Where is electrostatic potential energy? I guess, If the energy density is for real, it should be possible to answer these questions. No?
 

I guess energy density could be in fields. For example, kinetic energy could be distributed in gravitational field of a massive object (its mass increases with speed). For electrostatic potential energy I even know some formulas that say how energy is distributed in electric field... But what do you say?

[mathematic]

How local do you mean?  For example, when you boil water, the water in the pot has more energy, but outside you were putting it in.

[Markus Hanke]

9 hours ago, Danijel Gorupec said:

So, is the local conservation of energy a true law?

Yes it is. But is very important to remember that this is a purely local conservation law - while it always holds at every point, it may not hold in an extended global region.

9 hours ago, Danijel Gorupec said:

BTW, is it also valid regarding quantum mechanic? Specifically what I have in my mind is the wave function collapse - does the collapse has at all anything to do with energy distribution?

Yes, it also holds in quantum mechanics, so long as we are dealing with a closed system of course. After collapse, the wave function describes eigenstates of the Hamiltonian operator (which encapsulates the energy dynamics of the system), and the total energy itself takes on allowable values that are eigenvalues of the Hamiltonian. Generally, due to boundary conditions, the spectrum of the Hamiltonian operator is discrete, so the eigenvalues are discrete as well.

9 hours ago, Danijel Gorupec said:

Next... My understanding is that one of direct consequences of the law of local conservation of energy is that we are then capable to define a value of 'energy flow' (through space) and a value of 'energy density' (in space). Am I correct?

Yes, you are. That’s the stress-energy-momentum tensor.

9 hours ago, Danijel Gorupec said:

I have troubles understanding this 'energy density'. How much is this distribution of energy throughout space a physical thing.

I am unsure what you mean by this, but it is quite physical in the sense that it has measurable consequences.

9 hours ago, Danijel Gorupec said:

Should I expect that with a very sensitive equipment I should be able to measure a curvature of space that this energy density is producing... thus actually measuring energy density?

Yes, exactly correct. Energy-momentum is equivalent to local spacetime curvature, and vice versa, via the Einstein equations. This can be measured, at least in principle.

9 hours ago, Danijel Gorupec said:

Is energy density tied to one frame of reference or is it the same in all frames of reference? It seems to me that if you change the frame of reference, the energy density distribution changes too. Take for example a kinetic energy.

Again you are correct, energy density is an observer-dependent quantity. However, the mathematical object that describes sources of gravity isn’t just energy density, it’s the full stress-energy-momentum tensor (energy density is one of its components). As being a tensorial quantity, all observers agree on it.

9 hours ago, Danijel Gorupec said:

Really, "where" is the kinetic energy? I mean, how is its density distributed in space? Where is gravitational potential energy? Where is electrostatic potential energy? I guess, If the energy density is for real, it should be possible to answer these questions. No?

The energy-momentum tensor, just like all tensors, is a purely local quantity. That means these energies are located exactly where you perform their respective measurements (I know this sounds trivial, but it really isn’t if you think about it in more detail). However, you need to remember that potentials cannot be physically measured, only their gradients can. Furthermore, the energy inherent in gravity itself is not localisable (which is why it isn’t part of the energy-momentum tensor, but encapsulated in the non-linear structure of the field equations themselves). 

9 hours ago, Danijel Gorupec said:

For example, kinetic energy could be distributed in gravitational field of a massive object (its mass increases with speed). For electrostatic potential energy I even know some formulas that say how energy is distributed in electric field

Electromagnetic fields and gravity are quite different in many respects, and this is one of them. You can tell exactly how energy is distributed in an EM field, because electromagnetism obeys a field equation that is linear. The same is not true for gravity, in that the Einstein equations are highly non-linear; hence gravitational self-energy is not localisable. However, you can still localise sources other than gravitational self-interaction, which is everything that is encapsulated in the energy-momentum tensor. This tensor is itself precisely defined via Noether’s theorem, so this is all very well defined mathematically.

[Strange]

1 hour ago, Markus Hanke said:

Yes, exactly correct. Energy-momentum is equivalent to local spacetime curvature, and vice versa, via the Einstein equations.

So, rather than the common description that energy-momentum causes spacetime curvature, it sounds like you are saying that energy-momentum and spacetime curvature are the same thing; just different ways of representing it. That is quite insightful.

[Danijel Gorupec]

Thanks Markus, I will try to read about the stress-energy-momentum tensors and then maybe I will ask for some additional clarifications. At first look the subject seems very difficult.

BTW, what do you mean by "the energy inherent in gravity itself"? I am worried if it needs to be treated separately.

[Markus Hanke]

7 hours ago, Strange said:

So, rather than the common description that energy-momentum causes spacetime curvature, it sounds like you are saying that energy-momentum and spacetime curvature are the same thing; just different ways of representing it. That is quite insightful.

Not all of curvature per se, but the particular aspect of curvature that is represented by the Einstein tensor. And yes, that is indeed equivalent to energy-momentum. That’s exactly what the equation is telling us:

[math]\displaystyle{G_{\mu \nu}=\kappa T_{\mu \nu}}[/math]

The two are physically equivalent, up to a proportionality constant. That is so because both the Einstein tensor and the energy-momentum tensor obey the same conservations laws; what’s more, it can be shown that the Einstein tensor is in fact the only tensorial function of the metric that does this. So as you said, they are just two ways to look at the same thing.

5 hours ago, Danijel Gorupec said:

BTW, what do you mean by "the energy inherent in gravity itself"? I am worried if it needs to be treated separately.

You don’t need to worry, because it does not need to be treated separately. The physical meaning of gravitational energy is a self-interaction of the gravitational field with itself; colloquially speaking one could say that gravity itself is also a source of gravity. This is encapsulated not as a separate source term (which would be mathematically impossible), but rather in the structure of the field equations themselves. Mathematically this system of equations is highly non-linear, and it is precisely this mathematical non-linearity that physically corresponds to a self-interaction. So you don’t need to worry about it, or include it separately - it’s already accounted for in the field equations themselves.

Edited June 30, 2018 by Markus Hanke

[MigL]

The typical way we determine energy conservation id by calculating the energy flux into/out of a region of space-time.
As such, it can only ever be local. Global energy conservations would involve flux in and out of all space-time. And since space-time is not embedded, this calculation fails.

I believe E Noether was working on the global conservation problem when she came up with her famous theorem linking conservation laws to symmetries.

[Markus Hanke]

4 hours ago, MigL said:

As such, it can only ever be local. Global energy conservations would involve flux in and out of all space-time.

By “global” I meant some region of spacetime that is large enough so that curvature cannot be neglected, not necessarily the entire universe. It is the presence of curvature that precludes the existence of a conservation law in such a region.

[MigL]

But isn't that simply a mathematical problem with the model ?
Because of the way we express energy-momentum in GR, we can only solve for the case where the metric is flat and static ?
IOW, not an actual physical problem, just one that resists mathematical handling.

Please elaborate.

[swansont]

13 minutes ago, MigL said:

But isn't that simply a mathematical problem with the model ?
Because of the way we express energy-momentum in GR, we can only solve for the case where the metric is flat and static ?
IOW, not an actual physical problem, just one that resists mathematical handling.

Please elaborate.

Energy is not an invariant. It will not be the same in different frames, which is a real physical issue, not a math one.

On 6/29/2018 at 4:55 PM, Danijel Gorupec said:

I have questions about local conservation of energy. I am trying to better understand consequences that follow from it.

 

As you might have discerned from the answers thus far, the details of the answers you get depend on the context.Newtonian physics will differ from relativity which will differ from QM. 

 

 

 

Quote

So, is the local conservation of energy a true law? It seems correct to me, but I cannot remember if I ever heard about it during my formal education (looong time ago)... [BTW, is it also valid regarding quantum mechanic? Specifically what I have in my mind is the wave function collapse - does the collapse has at all anything to do with energy distribution?]

 

 

 

In QM you need to look at it in light of the uncertainty principle. So it's conserved, as long as you are not looking on scales that violate ∆E∆t > hbar/2

 

Quote

 

 Is energy density tied to one frame of reference or is it the same in all frames of reference? It seems to me that if you change the frame of reference, the energy density distribution changes too. Take for example a kinetic energy.

Really, "where" is the kinetic energy? I mean, how is its density distributed in space? Where is gravitational potential energy? Where is electrostatic potential energy? I guess, If the energy density is for real, it should be possible to answer these questions. No?

 

 

Energy density is generally not looking at KE

 

 

 

 

 

[Markus Hanke]

2 hours ago, MigL said:

But isn't that simply a mathematical problem with the model ?
Because of the way we express energy-momentum in GR, we can only solve for the case where the metric is flat and static ?
IOW, not an actual physical problem, just one that resists mathematical handling.

Please elaborate.

The maths do reflect the physics, in this instance - and it would be a major issue if they didn’t, since that would render GR useless in its entirety.

At the heart of this issue is what the concept of energy-momentum physically means. To answer this, we must look at where the energy-momentum tensor actually comes from, and that is Noether’s theorem. What it tells us is that every continuous local symmetry is equivalent to a conserved local quantity; specifically, if a small enough local system is invariant under time translations, then there will be a conserved quantity associated with that system that reflects its total energy - that’s precisely the energy-momentum tensor. But the thing now is that - in general - only patches of Minkowski spacetime are time-translation invariant; if there is spacetime curvature, this symmetry does not exist, and hence neither does a consistent notion of energy-momentum associated with that region. What’s more, Noether’s theorem itself is only valid in Minkowski spacetime, too.

Physically this means that energy-momentum conservation holds only locally, in small enough patches of flat spacetime. In larger curved regions it does not hold - not in the sense of it being violated, but in the sense of the very concept of energy-momentum conservation being meaningless. Saying that energy-moment should be conserved in curved spacetime simply does not make any physical or mathematical sense, right from the get-go. 

Mathematically speaking, it is no problem to solve the integral that sums the divergence of the energy-momentum tensor over an extended region of curved spacetime. It is in fact trivially easy, since it becomes immediately apparent that you are left with terms that do not vanish (a very insightful exercise to do, suitable even for beginners). That’s just a reflection of the underlying physics, not any issue with the maths.

Edited July 3, 2018 by Markus Hanke

[MigL]

Thanks for the explanation Markus.

[Markus Hanke]

13 hours ago, MigL said:

Thanks for the explanation Markus.

No worries, and you’re welcome. I hope it makes (at least some kind of) sense.