energy in GR

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?

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

(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 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.

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

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.

I am kinda quoting Einstein here.

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

"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".

"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".

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

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.

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.

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.

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.

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?

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).

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...

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