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EDIT: As some confusion has appeared, I want to make another clear question. If gravitational energy is meaningless in general relativity (since it is the geometry), how can one come up with the graviton, the quanta of the gravitational field?

I would like to understand the concept of gravitation's energy (and maybe momentum) in term of Eintein's General Relativity and its non-quantum extension. Since the action of matter and metric is:

$$S=\int d^4x(\sqrt{-g}R+\mathcal{L}_{matter})$$

it's understood that the energy momentum of matter is $$T_{\mu\nu}\sim\frac{\delta(\sqrt{-g}\mathcal{L}_\text{matter})}{\delta g^{\mu\nu}}$$

while $$G_{\mu\nu}\sim\frac{\delta(\sqrt{-g}R)}{\delta g^{\mu\nu}}$$

where $G_{\mu\nu}$ denotes Einstein's tensor. How can we come up with a energy term of the metric?

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  • $\begingroup$ What do you mean by "energy term of the metric"? $\endgroup$ – Jim Aug 12 '14 at 17:04
  • $\begingroup$ I mean the energy of gravitational field, or graviton, or gravitational wave,... Any concept means the energy of gravitation. $\endgroup$ – quangtu123 Aug 12 '14 at 17:18
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    $\begingroup$ It's impossible to define a local, generally covariant gravitational field energy. You might be interested in Is Energy Conserved in General Relativity? from the Usenet Physics FAQ. $\endgroup$ – benrg Aug 12 '14 at 18:12
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    $\begingroup$ The Einstein field equations is essentially a way of equating energy with gravity. Gravity is usually on the left side and energy is on the right side. But under standard general relativity, gravity is not itself energy. Which is why it is unclear to me exactly what you are asking $\endgroup$ – Jim Aug 12 '14 at 18:29
  • $\begingroup$ Related: physics.stackexchange.com/q/41662/2451 and links therein. $\endgroup$ – Qmechanic Aug 12 '14 at 18:32
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(add my comments as an answer)

Per @Jim's comment, i would say Einstein's equations are a way of relating energy and geometry (which in turn is interpreted as gravitation). But is important to understand a difference between energy and (underlying) geometry, because when geometry turns into gravitation things can get weird.

There seems to be a confusion around energy (of matter/fields/stresses etc..), geometry (as in metric/curvature) and finally the interpretation of geometric curvature and metric as gravitation

As such, there is no gravitational energy-momentum tensor (it is meaningless and also not covariant). Think of it like it all happens on a manifold which may be curved, like doing "physics" on a sphere, the energy of the "physics" is different from the sphere upon which they manifest.

On other hand, for conceptual reasons, one can say that any enegy which could be attributed to the gravitational metric is (should be) already present in the standard energy-momentum tensor $T_{\mu\nu}$. This is another way to describe Einstein;s original gravitational field equations. Note that there is an ambiguity involved, as in what order does the geometry affect energy (movement) and movement (energy) affects geometry. A standard approach is that: matter/energy tells space how to curve and then space tells matter how to move.

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  • $\begingroup$ Can you then also explain how can graviton appear as the quanta of geometry? How can it have energy> $\endgroup$ – quangtu123 Aug 13 '14 at 4:35
  • $\begingroup$ @Mr.T, well gravitons are not verified experimentally afaik. Second there are many flavors of gravitons out there, i'm afraid i cannot answer that intelligently at this moment (im reluctant of gravitons). The question is about the classical GR (not a quantum version of it) $\endgroup$ – Nikos M. Aug 13 '14 at 4:39
  • $\begingroup$ @Mr.T, what you ask in essense is a quantum gravity theory (which has GR as a limit) and is not inconsistent, to my knowledge the various (existent) theories do not meet these requirements. $\endgroup$ – Nikos M. Aug 13 '14 at 6:08
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You may define a conserved total stress-energy tensor (matter + gravitation).

The main problem is that a conserved total stress-energy tensor is not covariant, and that a covariant stress-energy tensor is not conserved. Said differently, $\nabla^\mu T_{\mu\nu}=0$, which is a covariant equation, does not represent a conservation law, while $\partial^\mu( \sqrt{-g}\mathcal T_{\mu\nu})$ represents a conservation law, but is not covariant (the quantity $\mathcal T_{\mu\nu}$ is not a tensor, but a pseudo-tensor).

While some pseudo-tensor may be defined (for more details, see energy-momentum pseudotensor), they are not always simple to manage, for instance, sometimes, they need cartesian coordinates to be meaningful.

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"If gravitational energy is meaningless in general relativity ..."

This is a false premise. Gravitational energy can be understood perfectly well in general relativity. The problem is just that many people who answer here have failed to understand it.

"(since it is the geometry)"

There is no meaning whatsoever to the statement "gravitational energy is the geometry"

For better answers search this site for the many questions on this topic, but beware that the top answer is usually rubbish on the topic of energy in general relativity on this site. Look at the other answers to see who makes most sense.

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  • $\begingroup$ When I said it is the geometry, I mean Gravity is the geometry. Then I think geometry has no normal energy $\endgroup$ – quangtu123 Sep 17 '14 at 12:02
  • $\begingroup$ Why do you think that geometry has no energy? The geometry is dynamic so it has energy. $\endgroup$ – Philip Gibbs - inactive Sep 17 '14 at 12:39
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Trimok is correct. Insofar as gravitation is a field theory with physical things represented by conserved quantities, there is no conserved energy tensor in GR. Another way to look at it is to simply assume energy must also exist in GR, and then deduce that because there is no tensor representation of it, then it must be non-local. Energy can disappear here and appear there without having traversed the intervening space. This was Dirac's point of view (see his little book on GR).

My own point of view is that GR as it stands is not a complete theory. Maxwell-Lorentz electrodynamics is a complete theory - one has field equations driven by conserved currents and from these you can construct a conserved energy tensor. One side says how the fields act on currents, the other how currents generate fields. GR does not have this two-sided aspect - one knows how energy, the current aspect of GR, generates the gravitational field but not how the field acts on energy, because the latter does not have a conservation law, even globally.

The solution to this impossible situation sought by Einstein his entire life is to remove the dichotomy between field and current. He hoped to find a representation of matter in which matter itself appears as an aspect of geometry and is not put in by hand on the right side. Such a theory would appear like

f(R);m = 0

where f(R) is some combination of curvature covariants in some broader manifold and ;m is covariant differentiation in that manifold. It turns out that Weyl's extension of Riemannian geometry provides just such a representation, but in 6 and not 4 dimensions. So one can have a 6d world with no posited matter, a vacuum - this world then splits in the large into a 4d world with the other 2 dimensions representing matter that is "in" that 4d world. But conceptually, both apparently empty space and apparently dense matter are just different states of a more encompassing 6d vacuum. You can read about it here:

http://www.researchgate.net/publication/227346279_Gravitation_and_Electrodynamics_Over_SO(33)

-drl

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    $\begingroup$ Or, you can declare a fixed background, or an outer boundary with a timelike or null killing vector, and then you have a perfectly fine notion of energy. $\endgroup$ – Jerry Schirmer Oct 31 '14 at 17:04
  • $\begingroup$ Sorry, this has two problems.. 1) The fundamental new idea introduced by GR is background independence, or said another way, that geometry is dynamic and determined by the matter distribution and 2) as stated above, there is no definition of the local energy in gravitational field. You just cannot escape this fact with hand waving. -drl $\endgroup$ – D R Lunsford Mar 31 '15 at 16:01

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