Is spacetime a massless spin 2 field?

[kodama]

since a massless spin 2 field when perturbatively quantized gives rise to gravitons, which couple to everything and is identical to gravitation, is spacetime itself massless spin 2 field?

do virtual graviton exchange also modify time and space?

 

[ShayanJ]

massless spin 2 field when perturbatively quantized gives rise to gravitons, which couple to everything and is identical to gravitation

This is not completely true, see this paper!

 

[ohwilleke]

I agree with the paper cited by Shyan. The abstract of that paper states that:

From Gravitons to Gravity: Myths and Reality
T.Padmanabhan
(Submitted on 23 Sep 2004)
There is a general belief, reinforced by statements in standard textbooks, that: (i) one can obtain the full non-linear Einstein's theory of gravity by coupling a massless, spin-2 field hab self-consistently to the total energy momentum tensor, including its own; (ii) this procedure is unique and leads to Einstein-Hilbert action and (iii) it only uses standard concepts in Lorentz invariant field theory and does not involve any geometrical assumptions. After providing several reasons why such beliefs are suspect -- and critically re-examining several previous attempts -- we provide a detailed analysis aimed at clarifying the situation. First, we prove that it is \textit{impossible} to obtain the Einstein-Hilbert (EH) action, starting from the standard action for gravitons in linear theory and iterating repeatedly. Second, we use the Taylor series expansion of the action for Einstein's theory, to identify the tensor Sab, to which the graviton field hab couples to the lowest order. We show that the second rank tensor Sab is {\it not} the conventional energy momentum tensor Tab of the graviton and provide an explanation for this feature. Third, we construct the full nonlinear Einstein's theory with the source being spin-0 field, spin-1 field or relativistic particles by explicitly coupling the spin-2 field to this second rank tensor Sab order by order and summing up the infinite series. Finally, we construct the theory obtained by self consistently coupling hab to the conventional energy momentum tensor Tab order by order and show that this does {\it not} lead to Einstein's theory. (condensed).

Any theory in which a graviton couples to everything including itself almost by definition contradicts the Einstein-Hilbert equations, because in such a field (1) the energy of the gravitational field is localized, and (2) gravitational fields gravitate. Neither is true of the Einstein-Hilbert equations.

The EH gravitational curvature is like a photon, it doesn't self-interact. The conventional graviton that couples to everything including itself, is like a gluon.

FWIW, I suspect that EH is wrong for this reason, and I also suspect, in part based on some back of napkin calculations and heuristic reasons in papers by Deur such as http://arxiv.org/abs/1407.7496 and http://arxiv.org/abs/0901.4005, that the theory that one would generate from a massless spin-2 graviton that interacts with everything including itself is right, and moreover that this one modification to EH would single handedly resolve essentially all dark matter phenomena and all or a significant share of dark energy phenomena. But, this hasn't been proven at this point.

The abstract of the 2009 paper is as follows:

Implications of Graviton-Graviton Interaction to Dark Matter
A. Deur
(Submitted on 26 Jan 2009 (v1), last revised 6 May 2009 (this version, v2))
Our present understanding of the universe requires the existence of dark matter and dark energy. We describe here a natural mechanism that could make exotic dark matter and possibly dark energy unnecessary. Graviton-graviton interactions increase the gravitational binding of matter. This increase, for large massive systems such as galaxies, may be large enough to make exotic dark matter superfluous. Within a weak field approximation we compute the effect on the rotation curves of galaxies and find the correct magnitude and distribution without need for arbitrary parameters or additional exotic particles. The Tully-Fisher relation also emerges naturally from this framework. The computations are further applied to galaxy clusters.

The abstract of the 2014 paper is:

A correlation between the amount of dark matter in elliptical galaxies and their shape
Alexandre Deur
(Submitted on 28 Jul 2014)

We discuss the correlation between the dark matter content of elliptical galaxies and their ellipticities. We then explore a mechanism for which the correlation would emerge naturally. Such mechanism leads to identifying the dark matter particles to gravitons. A similar mechanism is known in Quantum Chromodynamics (QCD) and is essential to our understanding of the mass and structure of baryonic matter.

 

[PeterDonis]

This is not completely true, see this paper!

I'm not sure this paper is reliable. I think we've had previous PF threads on this, I'll try to dig some up.

The EH gravitational curvature is like a photon, it doesn't self-interact.

This is not correct. The Einstein Field Equations, which are the field equations derived from the EH Lagrangian, are nonlinear, so there are graviton-graviton vertices in the quantum field theory derived from this Lagrangian. Maxwell's Equations, which are the field equations derived from the massless spin-1 Lagrangian, are linear, so there are no photon-photon vertices in quantum electrodynamics.

The conventional graviton that couples to everything including itself, is like a gluon.

What field theory are you referring to with this term, "conventional graviton"?

based on some back of napkin calculations and heuristic reasons in papers by Deur such as http://arxiv.org/abs/1407.7496 and http://arxiv.org/abs/0901.4005,

It was shown in the 1960's and early 1970's that the field theory of a massless spin-2 field leads to the EH Lagrangian. It took some time because the theory is perturbatively non-renormalizable, so there are an infinite number of counterterms; but Deser figured out a neat way to sum all of them and get a closed form solution, which was the EH Lagrangian. The introduction to the Feynman Lectures on Gravitation gives a good summation of this work. I believe it is also covered in Weinberg's 1972 text on GR. These were not back of napkin calculations or heuristics; they were proven theorems.

 

[ohwilleke]

This is not correct. The Einstein Field Equations, which are the field equations derived from the EH Lagrangian, are nonlinear, so there are graviton-graviton vertices in the quantum field theory derived from this Lagrangian. Maxwell's Equations, which are the field equations derived from the massless spin-1 Lagrangian, are linear, so there are no photon-photon vertices in quantum electrodynamics.

Section 20.4 of leading textbook "Gravitation" by Misner, Thorne and Wheeler at page 467 is emphatic about this question:

To ask for the amount of electromagnetic energy and momentum in an element of 3-volume make sense. First, there is one and only one formula for this quantity. Second, and more important, this energy-momentum in principle "has weight." It curves space. It serves as a source term on the right hand side of Einstein's field equations. It produces a relative geodesic deviation of two nearby world lines that pass through the region of space in question. It is observable.

Not one of these properties does "local gravitational energy-momentum" possess. There is no unique formula for it, but a multitude of quite distinct formulas. The two cited are only two among an infinity. Moreover, "local gravitational energy-momentum" has no weight. It does not curve space. It does not serve as a source term on the right hand side of Einstein's equations. It does not produce any relative geodesic deviation of two nearby world lines that pass through the region of space in question. It is not observable.

Anybody who looks for a magic formula for "local gravitational energy-momentum" is looking for the right answer to the wrong question. Unhappily, enormous time and effort were devoted in the past to trying to "answer this question" before investigators realized the futility of the enterprise.

Toward the end, above all mathematical arguments, one came to appreciate the quiet but rock-like strength of Einstein's equivalence principle. One can always find in any given locality a frame of reference in which all local gravitational fields" (all Christoffel symbols . . .) disappear. No [Christoffel symbols] means no "gravitational field" and no local gravitational field means no "local gravitational energy-momentum."

Nobody can deny or wants to deny that gravitational forces make a contribution to the mass-energy of a gravitationally interacting system. The mass-energy of the Earth-moon system is less than the mass-energy that system would have if the two objects were at infinite separation. The mass-energy of a neutron star is less than the mass-energy of the same number of baryons at infinite separation. Surround a region of empty space where there is a concentration of gravitational waves, there is a net attraction, betokening a positive net mass-energy in that region of space (see Chapter 35). At issue is not the existence of gravitational energy, but the localizability of gravitational energy. It is not localizable. The equivalence principle forbids it.

The very definition of a graviton is that it is a localized bundle of gravitational energy. MTW, and the weight of consensus in a community of relativistic physicists taught this (other textbooks are in accord and the EH equations themselves support their conclusion), deny that this is possible in GR.

But, the fact that you can't localize gravitational energy in EH, does not mean that this feature of EH accurately describes the universe.

What field theory are you referring to with this term, "conventional graviton"?

When I say "conventional graviton", I an referring to a graviton that meets the following definition:

1. It has a zero rest mass.
2. It couples to every particle including itself that has mass-energy in proportion to its mass-energy.
3. It has spin-2.
4. It is always attractive.
5. It has a coupling constant equivalent in strength to Newton's constant, G, in GR.

I am not referring to a specific field theory, merely to any field theory that have a graviton consistent with that definition. There is a widespread assumption in the physics community that a graviton that is consistent with that definition uniquely defines a field theory that is equivalent in the classical limit to EH.

 

[PeterDonis]

The Deser paper showing the nonlinearity of the EH Lagrangian, and how it is derived from the field theory of a massless spin-2 field, is here:

http://arxiv.org/abs/gr-qc/0411023

 

[PeterDonis]

Section 20.4 of leading textbook "Gravitation" by Misner, Thorne and Wheeler at page 467 is emphatic about this question:

They are emphatic that there is no way to localize "energy in the gravitational field". But that is not the same as saying that "gravitational curvature is not self-interacting". For that to be true, the Einstein Field Equations would have to be linear, and they aren't.

When I say "conventional graviton", I an referring to a graviton that meets the following definition:

1. It has a zero rest mass.
2. It couples to every particle including itself that has mass-energy in proportion to its mass-energy.
3. It has spin-2.
4. It is always attractive.
5. It is a coupling constant equivalent in strength to Newton's constant, G, in GR.

Point 4 is redundant, it is implied by points 2 and 3. (Many field theory texts go into this; see, for example, Zee's Quantum Field Theory In A Nutshell. IIRC MTW also has a discussion of it in one of the exercises in a fairly early chapter.) But otherwise, you have just described the field theory that is shown, in Deser's paper, to lead to the EH Lagrangian.

There is a widespread assumption in the physics community that a graviton that is consistent with that definition uniquely defines a field theory that is equivalent in the classical limit to EH.

It's not an assumption, it's a conclusion, based on the work that culminated in the Deser paper I linked to.

 

[ohwilleke]

The Deser paper showing the nonlinearity of the EH Lagrangian, and how it is derived from the field theory of a massless spin-2 field, is here:

http://arxiv.org/abs/gr-qc/0411023

I am aware of Deser's paper (written in 1969 and uploaded to arxiv to make it more generally accessible in 2014), but not convinced that it is correct, for reasons including those of T.Padmanabhan (2004) above.

 

[PeterDonis]

I am aware of Deser's paper (written in 1969 and uploaded to arxiv to make it more generally accessible in 2014), but not convinced that it is correct, for reasons including those of T.Padmanabhan (2004) above.

Fair enough. It seems like this is an ongoing dispute in the physics community, so I don't think we're going to resolve it here.

As far as the question asked in the OP, I think the answer would have to be "reply hazy, ask again later".

 

[atyy]

Another paper about the uniqueness of Einstein gravity from spin 2 under certain assumptions is:

http://arxiv.org/abs/hep-th/0007220
Inconsistency of interacting, multi-graviton theories
Nicolas Boulanger, Thibault Damour, Leonardo Gualtieri, Marc Henneaux

 

[fzero]

Fair enough. It seems like this is an ongoing dispute in the physics community, so I don't think we're going to resolve it here.

As far as the question asked in the OP, I think the answer would have to be "reply hazy, ask again later".

Deser wrote another paper in 2009, http://arxiv.org/abs/0910.2975, which revisits the 1970 calculation and responds to what he calls the "misunderstandings" of Padmanabhan and others. As I find no citations to this Deser paper by Padmanabhan, I would assume that he has conceded the issue.

 

[PeterDonis]

The very definition of a graviton is that it is a localized bundle of gravitational energy.

It depends on whether you're talking about the massless spin-2 field itself, or about particular states of the field that we would ordinarily describe as "particle" states. But I think there's a way of phrasing this objection that avoids that ambiguity; see below.

MTW, and the weight of consensus in a community of relativistic physicists taught this (other textbooks are in accord and the EH equations themselves support their conclusion), deny that this is possible in GR.

Here's how I would phrase this: MTW and other textbooks and the general consensus in the GR community say that, if we write the Einstein Field Equation in its usual form...

$$
R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R = 8 \pi T_{\mu \nu}
$$

...then the tensor on the RHS, ##T_{\mu \nu}##, cannot contain any stress-energy due to the gravitational field itself. The reason for this is that the LHS has a covariant divergence that vanishes identically, by the Bianchi identities, and so the covariant divergence of the RHS vanishes identically as well. That is a nice property because it means that the "source" ##T_{\mu \nu}## is automatically conserved.

The objection you make about the graviton can then be phrased as follows: if the graviton is a massless spin-2 field, then there should be a stress-energy tensor associated with this field (derived in the usual way from the field's Lagrangian). So where is that stress-energy in the above equation?

The answer that the GR community gives is that it is there; it's just on the LHS, not the RHS. In other words, the "energy due to the gravitational field" is part of the Einstein tensor, as the EFE is usually written. If we really want the RHS of our equation to represent all of energy present, including the energy due to the graviton field, then we have to rearrange terms in the EFE to put "gravitational energy" on the RHS. That raises two issues:

(1) There is no unique way to do it; there are multiple possible "pseudo-tensors" that can be constructed to represent "energy in the gravitational field", i.e., multiple possible ways that we can take a piece of the Einstein tensor and move it to the RHS of the field equation.

(2) However we do it, we lose the property of both sides of the equation having zero covariant divergence; i.e., we lose automatic conservation of the source.

Neither of these issues show that the procedure just described is not valid; you can do it, and for some purposes it can be useful to do it. But they do show that you can't have everything; you can have "energy in the gravitational field" included in the "source", but only at the expense of having the source no longer be automatically conserved.

 

[ShayanJ]

I'm not sure this paper is reliable. I think we've had previous PF threads on this, I'll try to dig some up.

Could you find anything?

P.S.
I feel responsible to let @vanhees71 know about the responds to Padmanabhan's paper.

 

[PeterDonis]

Could you find anything?

Nothing that would add anything useful to this thread; previous threads just covered the same ground we've covered here.

 

[julian]

I think general relativists generally have a problem with this idea. Like how you can start with linear gravitons on Minkowski spacetime and get out the Schwarzschild solution? Here's a quote by Penrose that Ashtekar said had a deep impression on him:

"... if we remove life from Einstein's beautiful theory by steam-rollering it first to flatness and linearity, then we shall learn nothing from waving the magic wand of quantum theory over the resulting corpse."

 

[PeterDonis]

how you can start with linear gravitons

You don't. The Deser paper discusses this.

 

[julian]

Oh I thought you did and that Deser was the first person to complete some infinite iteration starting from interacting linear gravitons on Minkowski spacetime...O.K. I'll have a look at it.

 

[PeterDonis]

starting from interacting linear gravitons

There's no such thing. A linear field is not interacting. There have to be nonlinear terms to have a self-interaction of the field at all. Compare, for example (Deser discusses this too), a Maxwell field--linear, non-self-interacting (no photon-photon vertices in quantum electrodynamics) with a Yang-Mills field--nonlinear, self-interacting (there are gluon-gluon vertices in quantum chromodynamics). This is for the "pure" field, i.e., no other fields present (no electrons, no quarks, etc.). The "starting" Lagrangian for the graviton is like the Yang-Mills field in this respect, not the Maxwell field.

 

[ohwilleke]

A critical assumption of Deser's 2009 paper is that the strong equivalence principle holds perfectly. This isn't an unreasonable assumption but one of the main empirical points that would contradict this assumption would be the existence of a gravitational "fifth force" in the form of a Yukawa force that supplements the inverse-square like relationship in very weak fields and arises from graviton (or in the non-quantum regime, gravitational field) self-interactions.

Deur, drawing on QCD analogies, constructs just such a Yukawa force (with a potential function proportional to e^-mr/r) that would arise if self-interactions of gravitons took place and had a physical effect, which in turn produces dark matter and dark energy like effects with the right order of magnitude at pretty much any scale (in Deur's analysis some of dark energy effects are due to gravitons being diverted to interactions where dark matter phenomena are observed, and away from everywhere else, making gravitational pull in the "everywhere else" direction weaker by a comparable amount).

Deur himself doesn't articulate it that way, but the Yukawa force that he derives from graviton self-interactions implies a violation of the strong equivalence principle, and hence isn't necessarily in conflict with Deser (2009) so much as it rejects one of its axioms for the empirically motivated reason that dark matter phenomena may be manifestations of just the sort of Yukawa force which would contradict the strong equivalence principle empirically.

In general, most of more phenomenologically successful gravity modification theories by people other than Deur (e.g. TeVeS by Bekenstein and MOG by Moffat), share with the Brans-Dicke theory of gravitation, which is a scalar-tensor theory (unlike GR which is a pure tensor theory), an added scalar component, and often an additional vector component as well, to the standard GR tensor only theory.

It is also worth noting that neither particle based dark matter theories, nor modified gravity theories predict any discernible effects at the solar system scale where almost all strong equivalence principle experiments have been conducted. The mass of the solar system and its scale are just way too small for measurable effects to arise in them. One needs to look for deviations at the scale systems the size of galaxies or larger, not solar systems, to see measurable effects. Thus far, we haven't yet had sufficiently rigorous tests on the solar system scale to detect the predicted deviations from the strong equivalence principle (it also doesn't help that the mass distribution in the solar system is quite close to spherically symmetrical because the vast majority of mass is concentrated in the sun and the rest is dispersed circularly around the sun; while the effects can only be observed to the extent that you have a non-spherically symmetrical mass distribution in the system).