How GR Resolves the Conservation of Momentum and Energy

[julian]

We know that the reason energy and momentum are conserved is b/c of Noether's theorem...time translational invariance implies energy conservation and space translational invariance implies momentum conservation.

Now in a curved spacetime you can still form conserved quantities - energy and momentum if the spacetime has Killing's vector fields.

However, GR is invariant under active diffeomorphisms (Einstein's Hole argument) and general active diffeomorphisms will destroy any Killing vector field. And hence energy and momentum are no longer physically meaningfull quantities? In Rovelli's book "Qunatum Gravity" he emphasizes this. For example he talks about the problem of defing the vacuum state when energy is not meanifull.

Anyway, so GR states that gravity is determined by mass or energy...but energy doesn't have physical meaning anymore?

 

[PeterDonis]

However, GR is invariant under active diffeomorphisms (Einstein's Hole argument) and general active diffeomorphisms will destroy any Killing vector field.

Not sure what you mean by this. If a geometry has a Killing vector field, it has it regardless of which coordinate chart you describe the geometry with. Diffeomorphisms are just transformations from one coordinate chart to another; they don't change the actual geometry. A given Killing vector field might look a lot simpler in one chart than in another, but that doesn't change whether it's a Killing vector field or not.

And hence energy and momentum are no longer physically meaningfull quantities?

If there are no corresponding Killing vector fields in the spacetime, yes (with some caveats, see below). But again, that's a feature of the geometry independent of coordinate charts.

In Rovelli's book "Qunatum Gravity" he emphasizes this. For example he talks about the problem of defing the vacuum state when energy is not meanifull.

I haven't read the book, but I've read a number of Rovelli's papers, as well as other literature on quantum field theory in curved spacetime. Everything I've read says what I said above: if a spacetime doesn't have a time translation Killing vector field, you can't define an associated energy, so you have a problem trying to define the quantum vacuum state. More generally, in spacetimes with more than one timelike Killing vector field, such as Minkowski spacetime, you can define different notions of energy and hence different "vacuum" states. This is how the Unruh effect arises.

Anyway, so GR states that gravity is determined by mass or energy...but energy doesn't have physical meaning anymore?

GR states that curvature is determined by the stress-energy tensor, via the Einstein Field Equation. That statement has physical meaning in any spacetime; you don't need any Killing vector fields to define the curvature tensor or the stress-energy tensor. But if there is no time translation Killing vector field in a particular spacetime--i.e., a particular solution of the EFE--then there won't be a conserved total energy either. Whether or not there will be a "physically meaningful" notion of energy depends on how you want to interpret the 0-0 component of the SET, or more generally the double contraction of the SET with a given observer's 4-velocity, which defines the "energy" that is measured locally by that observer.

 

[julian]

People try to argue that active diff invafriance has no physical meaning...BUT it implies that energy and momentum are no longer viable physical quantities. But usual physics has been based on these concepts! As Rovelli points out - we have to learn how to do phyics all over again...

 

[julian]

The gauge transformations of GR are not coordinate transformations - see thread on the Hole argument. Active diffeomorphims are when you simultaneously drag the gravitational and matter fields over the blank manifold...which is most definitely NOT a coordinate transformation.

 

[PeterDonis]

People try to argue that active diff invafriance has no physical meaning...BUT it implies that energy and momentum are no longer viable physical quantities.

Please re-read what I said above. Diffeomorphism invariance is a separate question from whether there is a meaningful notion of energy or momentum in a spacetime. The latter depends on the presence of appropriate Killing vector fields; if they are there, they are there regardless of which coordinate chart you use. So you can have diffeomorphism invariance with a spacetime, such as Schwarzschild spacetime, that has Killing vector fields and therefore has a viable definition of total energy (in the case of Schwarzschild spacetime it's the Komar mass).

But usual physics has been based on these concepts!

GR isn't "based on" the concepts of energy and momentum as defined by Noether's theorem; they aren't fundamental, they are just derived quantities that appear in spacetimes (particular solutions) that have the appropriate Killing vector fields.

 

[PeterDonis]

The gauge transformations of GR are not coordinate transformations - see thread on the Hole argument.

Can you link to the thread? The forum search tool isn't finding anything useful.

Active diffeomorphims are when you simultaneously drag the gravitational and matter fields over the blank manifold...which is most definitely NOT a coordinate transformation.

Ah, I see; you mean something different by "diffeomorphism", something that can actually change the geometry. But if you change the geometry, you change the solution of the EFE that you are talking about, so I don't understand how the statement that "GR is invariant" under these types of transformations means anything, except that the EFE itself is still valid--you've changed the particular solution but not the general equation. But obviously if I change the solution I can change from a geometry that has Killing vector fields to one that doesn't. So what?

 

[julian]

Energy of a black hole - how do you define that? By a quasi-local notion... Energy has no local interpretation in GR.

 

[PeterDonis]

Energy of a black hole - how do you define that? By a quasi-local notion... Energy has no local interpretation in GR.

If by "energy" you mean "energy stored in the gravitational field", then yes, there is no "local interpretation" because there's no tensor that describes it. But there is a tensor, the SET, that describes "locally" the energy (and momentum, and pressure, and stress) due to all non-gravitational matter and fields. In the case of a black hole, the SET is zero everywhere (except inside the matter that originally collapsed in the past to form the hole).

 

[julian]

Can you link to the thread? The forum search tool isn't finding anything useful.Ah, I see; you mean something different by "diffeomorphism", something that can actually change the geometry. But if you change the geometry, you change the solution of the EFE that you are talking about, so I don't understand how the statement that "GR is invariant" under these types of transformations ...?

Both you and Einstien had this problem! In fact he spent years trying to get out of the 'Hole argument' argument only to return to it and resolve it. The answer is that the coincidence between the values of the gravitational field and the matter field are preserved under active diffeomorphisms and SO this has physical meaning...from this you can formulate a notion of matter being located with respect to the grav field...this is what Rovelli means when he says in GR is about fields living on top of fields.

Give me second to find the Hole argument link. Cheers.

 

[julian]

If by "energy" you mean "energy stored in the gravitational field", then yes, there is no "local interpretation" because there's no tensor that describes it. But there is a tensor, the SET, that describes "locally" the energy (and momentum, and pressure, and stress) due to all non-gravitational matter and fields. In the case of a black hole, the SET is zero everywhere (except inside the matter that originally collapsed in the past to form the hole).

But for SET to satisfy enery-momentum conservation also depends on the existense of Killing vector fields...I think we agree that sometimes these notions don't correspond to conserved quantities...part of my question was if they are not invariant quantities how are we to interpret them?

 

[PeterDonis]

The answer is that the coincidence between the value of the gravitational field and the matter field is preserved under active diffeomorphisms and SO this has physical meaning...from this you can formulate a notion of matter being located with respect to the grav field...this is what Rovelli means when he says in GR fields live on top of fields.

Hmm. I found this paper by Rovelli on arxiv, entitled "Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance":

http://arxiv.org/pdf/gr-qc/9910079v2.pdf

Section 4 specifically talks about passive and active diffeomorphism invariance. Let me see if I understand what it's saying by giving two examples. Both examples start with the standard Schwarzschild exterior coordinate chart on the exterior vacuum region of Schwarzschild spacetime (i.e., the region outside the horizon). Call that spacetime S and that chart SC.

(1) A passive diffeomorphism is a transformation from SC to some other chart on the same manifold, for example the ingoing Painleve chart, PC. The statement that GR is invariant under passive diffeomorphisms is that the transformation SC -> PC does not change the underlying geometry of S; SC and PC may assign different coordinate 4-tuples to the same events, but all geometric invariants will be the same in both. And, of course, SC on S and PC on S will both be solutions of the EFE; on the surface they will look like different solutions, but computing the geometric invariants tells us that they both describe the same underlying geometry.

(2) An active diffeomorphism is a transformation that retains SC but changes the underlying spacetime from S to something else. For example, we could retain SC but change the underlying spacetime to Minkowski spacetime, M. After this transformation, the metric, and hence all geometric invariants, will look very different in terms of SC on M than they did in terms of SC on S. However, the statement that GR is invariant under active diffeomorphisms means that, if SC on S is a solution of the EFE, so is SC on M. They are *different* solutions, with different geometric invariants, but they're both solutions. So now we have two different geometries described using the same chart.

Am I understanding this correctly?

 

[PeterDonis]

But for SET to satisfy enery-momentum conservation also depends on the existense of Killing vector fields

If by "energy-momentum conservation" you mean *local* conservation, this is not correct. The covariant divergence of the SET is always zero, identically. That only depends on the Bianchi identities, which are automatically satisfied by the LHS of the EFE (the Einstein tensor), and therefore must be satisfied by the RHS as well (the SET).

I think we agree that sometimes these notions aren't conserved...part of my question was if they are not invariant quantities how are we to interpret them?

That depends on the specific problem you're trying to solve. There isn't a single interpretation that always works. Sometimes there isn't any; sometimes there is no useful notion of "energy" beyond the locally measured one (contraction of the SET with 4-velocity).

 

[julian]

If by "energy-momentum conservation" you mean *local* conservation, this is not correct. The covariant divergence of the SET is always zero, identically. That only depends on the Bianchi identities, which are automatically satisfied by the LHS of the EFE (the Einstein tensor), and therefore must be satisfied by the RHS as well (the SET).

From the book 'Anvanced general relativity':

"As a second example suppose $T^{ac}$ represents the (symmetric) energy-momentum tensor of a continuous distribution of matter, satisfying $\nabla_c T^{ac} = 0$. If $K$ is a Killing covector define the associated current as $J^a = T^{ac} K_c$. Then by an almost identical caculation one may verify that $\nabla_a J^a$, i.e., the current is conserved."

About active diff invariance, could you have a look at what I said in that thread.

 

[PeterDonis]

From the book 'Anvanced general relativity':

This says that the covariant divergence of the SET is zero, as I said. The rest holds if there is a Killing vector, which I've also agreed with.

About active diff invariance, could you have a lok at what I said in that thread.

Which thread? Can you post a link?

 

[julian]

This says that the covariant divergence of the SET is zero, as I said. The rest holds if there is a Killing vector, which I've also agreed with.



Which thread? Can you post a link?

"the current is conserved" is the important part...normal physics relies on the fact that we have energy-momentum current conservation.


Not sure how to post links, it was called "What do points on the manifold correspond to in reality?".

 

[PeterDonis]

"the current is conserved" is the important part...normal physics relies on the fact that we have energy-momentum current conservation.

GR only relies on the covariant divergence of the SET being zero. If there's a conserved current, it makes the calculations easier, but that doesn't mean a conserved current is required to do physics.

If by "normal physics" you mean "physics in most common scenarios, where there is a conserved current", then your statement is true, but only because it's basically a tautology: "physics where there is a conserved current relies on current conservation".

Not sure how to post links, it was called "What do points on the manifold correspond to in reality?".

Just copy the URL from your browser's address bar and paste it into the post; the forum software automatically surrounds it with a url tag. Like so:

https://www.physicsforums.com/showthread.php?t=587239&highlight=points+manifold+correspond+reality

 

[julian]

https://www.physicsforums.com/showthread.php?t=587239

 

[Dale]

GR is invariant under active diffeomorphisms and general active diffeomorphisms will destroy any Killing vector field. And hence energy and momentum are no longer physically meaningfull quantities?

Your second part doesn't follow from your first and Noether's theorem. What follows is that energy is not conserved in general, i.e. under general active diffeomorphisms. This should be obvious.

But many non-conserved quantities are nonetheless physically meaningful.

Anyway, so GR states that gravity is determined by mass or energy.

No it doesn't. It states that gravity is determined by the stress-energy tensor, which is a very different thing.

 

[Dale]

(2) An active diffeomorphism is a transformation that retains SC but changes the underlying spacetime from S to something else. For example, we could retain SC but change the underlying spacetime to Minkowski spacetime, M. After this transformation, the metric, and hence all geometric invariants, will look very different in terms of SC on M than they did in terms of SC on S. However, the statement that GR is invariant under active diffeomorphisms means that, if SC on S is a solution of the EFE, so is SC on M. They are *different* solutions, with different geometric invariants, but they're both solutions. So now we have two different geometries described using the same chart.

Am I understanding this correctly?

That is my understanding also. An active diffeomorphism involves things like mass magically appearing or disappearing, so it should be no surprise that things like gravitational PE and total energy are not conserved under general active diffeomorphisms.

 

[julian]

Your second part doesn't follow from your first and Noether's theorem. What follows is that energy is not conserved in general, i.e. under general active diffeomorphisms. This should be obvious.

But many non-conserved quantities are nonetheless physically meaningful.

That's why there was a question mark...I'm wondering if the notions of energy and momentum have become so ambiguous that they don't have anymore physical meaning than do points of spacetime...I mean with the introduction of SR spacetime points became ambiguous to a certain extent, but in GR they lost all objective physical meaning.

No it doesn't. It states that gravity is determined by the stress-energy tensor, which is a very different thing.

Was being a bit sloppy there. Need to refine my question.

 

[Dale]

I mean with the introduction of SR spacetime points became ambiguous to a certain extent, but in GR they lost all objective physical meaning.

I disagree with this too. Spacetime points (aka events) are well-defined geometric objects in the manifold. There is nothing ambiguous about them at all.

 

[julian]

I disagree with this too. Spacetime points (aka events) are well-defined geometric objects in the manifold. There is nothing ambiguous about them at all.

I am kinda quoting Einstein here. You agree that spacetime points loose some objective meaning cus of the principle of special relativity...what do you think is going to happen with the principle of general relativity - i.e. that the rules of physics take the same form in all reference systems, not just inertial frames??

 

[Dale]

I am kinda quoting Einstein here.

"Kinda quoting" is also known as "mis-quoting".

You agree that spacetime points loose some objective meaning cus of the principle of special relativity.

I don't know what "lose some objective meaning" means, and I have even less of an idea how you go from the principle of relativity to that. So, no, I do not agree.

I think you may be confusing events in the spacetime manifold with their coordinates.

 

[julian]

"The principle of relativity" by Einstein and others,"That this requirement of general covariance, which takes away from space and time the last remnant of physical objectivity, is a natural one, will be seen from the following reflexion".

See Rovelli "Quanyum Gravity" p.74 "The disapearence of spacetime".

 

[DrGreg]

"That this requirement of general covariance, which takes away from space and time the last remnant of physical objectivity, is a natural one, will be seen from the following reflexion".

(Note: you can see the quote in context at Google Books.)

Note this refers to "space and time", not "spacetime". My interpretation of this is therefore "the objectivity of space" and "the objectivity of time", certainly not "the objectivity of spacetime".

 

[Dale]

"The principle of relativity" by Einstein and others,"That this requirement of general covariance, which takes away from space and time the last remnant of physical objectivity, is a natural one, will be seen from the following reflexion".

See Rovelli "Quanyum Gravity" p.74 "The disapearence of spacetime".

If you take the quote in context it is clear that Einstein is talking about coordinates, not events in the manifold. As I said above, I think you are confusing the two.

 

[stevendaryl]

Anyway, so GR states that gravity is determined by mass or energy...but energy doesn't have physical meaning anymore?

There are two different, but related, concepts that are being mixed up in this statement.

One is a local statement, and one is a large-scale statement.

The local statement of conservation of energy can be expressed as a differential equation, the continuity equation:

∂/∂t ρ = - ∇.j

where ρ is the energy density, and j is the energy flux (transfer of energy per unit cross-sectional area per unit time). Now, in flat spacetime, we can define

E_total = ∫ρ dV

where the integral is over some closed surface S. Then energy conservation can be expressed as:

The rate of change of E_total is equal to the energy flux into surface S,
which can be written as the integral equation:

d/dt ∫ρ dV = ∫j.dS

The left-side is a volume integral over the volume enclosed by the surface, and the right-side is a surface integral over the surface.

In flat spacetime, the differential form of the law of conservation of energy is equivalent to the integral form. In curved spacetime, the local form continues to hold, but the integral form doesn't necessarily hold except in special circumstances.

I don't know what the best way to describe why not, but here's a hand-wavy explanation: One problem with the integral form, which you can see immediately, is that it involves on the left-hand side, a derivative with respect to t. So there must be a notion of time t common to the entire region enclosed by the surface S in order to make sense of the integral form. In contrast, the local form only requires a local notion of t,which always exists by considering a local inertial frame.

Anyway, energy density is always defined in GR, but total energy in a region may not be well-defined.

 

[stevendaryl]

I disagree with this too. Spacetime points (aka events) are well-defined geometric objects in the manifold. There is nothing ambiguous about them at all.

I would have said that before about 10 minutes ago, but now I'm not so sure. Take look at this paper http://arxiv.org/pdf/gr-qc/0610105.pdf

A quote:

In the language of manifolds, Einstein’s line of reasoning
on how to avoid the hole argument translates to the fact that, at least inside
the hole, the space-time points are not individuated independently of the metric
field.

I'm not sure that I understand exactly what is meant there.

 

[julian]

If you take the quote in context it is clear that Einstein is talking about coordinates, not events in the manifold. As I said above, I think you are confusing the two.

If you take 'events' to be spacetime coincidents determined by physical reference systems...yeah...what is Noether's theorem tied to? Abstract notions or real clocks and rulers?

I think you can give definite meaning to the matter SET if you use material reference systems. Gravity on the other hand is a bit more tricky...you can always go to a system in free fall and get rid of the grav field... that is not an active diff transformation...so yeah I agree what was said before about defining energy-momentum density for the grav field.

 

[julian]

There are two different, but related, concepts that are being mixed up in this statement.

One is a local statement, and one is a large-scale statement.

The local statement of conservation of energy can be expressed as a differential equation, the continuity equation:

∂/∂t ρ = - ∇.j

where ρ is the energy density, and j is the energy flux (transfer of energy per unit cross-sectional area per unit time). Now, in flat spacetime, we can define

E_total = ∫ρ dV

where the integral is over some closed surface S. Then energy conservation can be expressed as:

The rate of change of E_total is equal to the energy flux into surface S,
which can be written as the integral equation:

d/dt ∫ρ dV = ∫j.dS

The left-side is a volume integral over the volume enclosed by the surface, and the right-side is a surface integral over the surface.

In flat spacetime, the differential form of the law of conservation of energy is equivalent to the integral form. In curved spacetime, the local form continues to hold, but the integral form doesn't necessarily hold except in special circumstances.

I don't know what the best way to describe why not, but here's a hand-wavy explanation: One problem with the integral form, which you can see immediately, is that it involves on the left-hand side, a derivative with respect to t. So there must be a notion of time t common to the entire region enclosed by the surface S in order to make sense of the integral form. In contrast, the local form only requires a local notion of t,which always exists by considering a local inertial frame.

Anyway, energy density is always defined in GR, but total energy in a region may not be well-defined.

Local notion of time is always defined? If you formally go from the action principle to the Hamiltonian princple of GR you find that the Hamiltonian (which generates time evolution) vanishes, implying there is no time evolution. This is why you get comments like the whole universe happans at once and the so-called 'problem of time' in quantum gravity. (This is related to the hole argument by the way).

 

[samalkhaiat]

Hmm. I found this paper by Rovelli on arxiv, entitled "Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance":

http://arxiv.org/pdf/gr-qc/9910079v2.pdf

Section 4 specifically talks about passive and active diffeomorphism invariance. Let me see if I understand what it's saying by giving two examples. Both examples start with the standard Schwarzschild exterior coordinate chart on the exterior vacuum region of Schwarzschild spacetime (i.e., the region outside the horizon). Call that spacetime S and that chart SC.

(1) A passive diffeomorphism is a transformation from SC to some other chart on the same manifold, for example the ingoing Painleve chart, PC. The statement that GR is invariant under passive diffeomorphisms is that the transformation SC -> PC does not change the underlying geometry of S; SC and PC may assign different coordinate 4-tuples to the same events, but all geometric invariants will be the same in both. And, of course, SC on S and PC on S will both be solutions of the EFE; on the surface they will look like different solutions, but computing the geometric invariants tells us that they both describe the same underlying geometry.

(2) An active diffeomorphism is a transformation that retains SC but changes the underlying spacetime from S to something else. For example, we could retain SC but change the underlying spacetime to Minkowski spacetime, M. After this transformation, the metric, and hence all geometric invariants, will look very different in terms of SC on M than they did in terms of SC on S. However, the statement that GR is invariant under active diffeomorphisms means that, if SC on S is a solution of the EFE, so is SC on M. They are *different* solutions, with different geometric invariants, but they're both solutions. So now we have two different geometries described using the same chart.

Am I understanding this correctly?

Your understanding of diffeomorphism was correct until you read Rovelli. Rovelli (as always) talks about a "dead fish in the sea". How about asking him for a mathematical (without reference to any (fuzzy) physics) definitions of passive and active Diff.
What is called diffeomorphism in mathematics, physicists call the Group of general coordinate transformations.
If active Diff (what ever that may be) is not equivalent to passive Diff (what ever that may be) then your math is wrong.
So, you better off without Rovelli and his "dead fish".

Sam

 

[stevendaryl]

Local notion of time is always defined? If you formally go from the action principle to the Hamiltonian princple of GR you find that the Hamiltonian (which generates time evolution) vanishes, implying there is no time evolution. This is why you get comments like the whole universe happans at once and the so-called 'problem of time' in quantum gravity. (This is related to the hole argument by the way).

You're mixing up two different things. The hamiltonian in the hamiltonian form of GR is not the same as the stress-energy tensor, which is what people normally mean by energy (roughly speaking, it's the masses of particles, plus their kinetic energies, plus the energies of the various fields). The stress-energy tensor is the source of gravitational curvature.

The stress-energy tensor is well defined at every point in spacetime.

 

[samalkhaiat]

Local notion of time is always defined? If you formally go from the action principle to the Hamiltonian princple of GR you find that the Hamiltonian (which generates time evolution) vanishes, implying there is no time evolution...

You need to read about time-evolution in constraiant systems. No time evolution means no dynamics. I hope you know that GR and all other gauge theories DO describe time evolution.

Sam

 

[George Jones]

However, GR is invariant under active diffeomorphisms (Einstein's Hole argument) and general active diffeomorphisms will destroy any Killing vector field.

I don't see this.

Commutators of vector fields are preserved by push-forwards of diffeomporphisms, so push-forwards preserve Lie derivatives and thus Killing's equation.

 

[julian]

You need to read about time-evolution in constraiant systems. No time evolution means no dynamics. I hope you know that GR and all other gauge theories DO describe time evolution.

Sam

I understand that GR is a completely constrained system and you must identify a dynamical variable as the 'clock'...there is realtional evolution but not absolute...see Rovelli 'forget time'. It's great if you can deparamerterise a constrained system but this isn't always possible globally.