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## energy in GR

 Quote by julian 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.

 Quote by julian 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.

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 Quote by PeterDonis (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.

 Quote by DaleSpam 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 dont 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.

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 Quote by julian 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.

 Quote by DaleSpam 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??

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 Quote by julian I am kinda quoting Einstein here.
"Kinda quoting" is also known as "mis-quoting".

 Quote by julian 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.

 "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".

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 Quote by julian "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".

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 Quote by 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".
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.

 Quote by julian 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

Etotal = ∫ρ dV

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

The rate of change of Etotal 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.

 Quote by DaleSpam 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.

 Quote by DaleSpam 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.

 Quote by stevendaryl 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 Etotal = ∫ρ dV where the integral is over some closed surface S. Then energy conservation can be expressed as: The rate of change of Etotal 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).

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 Quote by PeterDonis 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

 Quote by julian 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.

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 Quote by julian 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

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 Quote by julian 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.