Spin-2 exchange particle and GR

[bombadil]

One of string theory's triumphs is the post-diction of a spin-2 field, which is supposed to represent gravity (GR). Yet, only in the weak field limit ("linearized") can it be shown that GR is a spin-2 field. So here's my question: why are string theorists confident that they have post-dicted GR if it isn't clear that GR is a spin-2 field?

Disclaimer: I'm in physics, but have had little exposure to GR and HEP, and I don't have any strong biases in this area, just curious.

-bombadil

 

[genneth]

The spin-2 is actually very robust. It simply implies a lack of moments with less symmetry than a quadrapole --- which is a robust prediction of GR (and even Newtonian physics --- mass always attracts).

 

[humanino]

Yet, only in the weak field limit ("linearized") can it be shown that GR is a spin-2 field.

No, you do not need that. In principle, the graviton field will couple to the energy momentum tensor, so even before you made your expansion, you know that it should be spin-2. In fact, there is no agreement on what the specific form of this expansion should be, that is what is the specific expression of the graviton field in terms of gravitational degrees of freedom. Even so, people can study general properties of the graviton as a spin-2 massless object.

 

[hamster143]

No, you do not need that. In principle, the graviton field will couple to the energy momentum tensor, so even before you made your expansion, you know that it should be spin-2. In fact, there is no agreement on what the specific form of this expansion should be, that is what is the specific expression of the graviton field in terms of gravitational degrees of freedom. Even so, people can study general properties of the graviton as a spin-2 massless object.

The issue here is that you can construct a perturbative QFT around Minkowski spacetime that looks like linearized gravity, and you can even claim that such a QFT arises naturally from string theory, but it's not at all obvious that such a field theory would be equal to GR in the classical continuum limit. In fact, many experts in general relativity would say that any attempt to "recreate" GR perturbatively from Minkowski spacetime is an exercise in futility. Because, for example, Minkowski spacetime has a definite and unique topology, and solutions of general GR equations are allowed to have any topology that locally satisfies Einstein equations and equation of state. That is what motivated alternative approaches such as LQG. You discard all conventional QFT mechanisms that depend on global Minkowskianness of spacetime, most notably Fourier transforms and momentum space, and you try to make a theory that is completely local.

This discussion is probably more suited for the "beyond SM" subforum.

 

[humanino]

I do not see why you introduced string theory in the discussion.

General relativity as an effective field theory: The leading quantum corrections

I describe the treatment of gravity as a quantum effective field theory. This allows a natural separation of the (known) low energy quantum effects from the (unknown) high energy contributions. Within this framework, gravity is a well behaved quantum field theory at ordinary energies. In studying the class of quantum corrections at low energy, the dominant effects at large distance can be isolated, as these are due to the propagation of the massless particles (including gravitons) of the theory and are manifested in the nonlocal/nonanalytic contributions to vertex functions and propagators. These leading quantum corrections are parameter-free and represent necessary consequences of quantum gravity. The methodology is illustrated by a calculation of the leading quantum corrections to the gravitational interaction of two heavy masses.

 

[bombadil]

I'm trying to understand (1) how significant it is that string theory has post-dicted a spin-2 field and (2) if this spin-2 field can be unambiguously associated with GR.

Here's my best guess about what's going on after reading your comments: spin-2 fields share certain generic properties that gravity does (see genneth). If (and this is a big if, right?) you can construct a QFT to describe GR, then it must be a spin-2 field (see humanino). However, it is unclear if such a QFT can be constructed, it has only been successfully constructed to "look like" linearized gravity so far (see hamster 143).

Am I getting warmer?

 

[humanino]

However, it is unclear if such a QFT can be constructed, it has only been successfully constructed to "look like" linearized gravity so far (see hamster 143).

It can certainly be constructed, it IS gravity there is no doubt, only issue is that it non-renormalizable, so it can still be used as an effective field theory at low energy (the unclear issue is whether it needs to be renormalized at all, as it may be non-trivially UV-finite).

 

[hamster143]

It can certainly be constructed, it IS gravity there is no doubt, only issue is that it non-renormalizable, so it can still be used as an effective field theory at low energy (the unclear issue is whether it needs to be renormalized at all, as it may be non-trivially UV-finite).

Not every massless spin-2 field deserves to be called gravity.

Do you get coupling with matter, in both directions (matter -> gravity and gravity -> geodesics) from string theory, naturally or otherwise? How do you get deflection of light by gravitating objects?

 

[blechman]

There's no doubt that it is gravity?

Do you get coupling with matter and back-effects on spacetime metric naturally from string theory? How do you get deflection of light by gravitating objects?

String theory automatically gives you the Einstein-Hilbert action (NON-linear, includes backreaction, etc!), that is certainly true; the couplings of the spin-2 massless mode of the string to other fields DO reproduce GR's predictions in the IR limit, that is true too. In that sense, string theory does "include" gravity.

 

[hamster143]

String theory automatically gives you the Einstein-Hilbert action (NON-linear, includes backreaction, etc!), that is certainly true; the couplings of the spin-2 massless mode of the string to other fields DO reproduce GR's predictions in the IR limit, that is true too. In that sense, string theory does "include" gravity.

Do you have a reference that shows how this coupling reproduces predictions of GR in both directions?

 

[blechman]

I'm not sure what you mean by "both directions" but look at your favorite string theory textbook. Polchinski derives the Einstein-Hilbert action. So does Green-Schwarz-Witten.

Also (and more importantly!) we have Feynman and Deser (and Coleman/Mandula, etc) from decades ago that tells us that a massless spin-2 particle MUST be the graviton (that is, its UNIQUE action is the Einstein-Hilbert action) and it must couple to the energy-momentum tensor. This is assuming things like unitarity and Lorentz invariance. It is the use of this theorem that allows you to see that as long as string theory has a massless spin-2 mode you KNOW that it contains gravity. And all string theories necessarily contain this mode.

The issue has never been whether or not string theory contains gravity; the issue is whether or not string theory's version of QUANTUM gravity is truly UV-complete (renormalizable/finite), and whether this theory is unique. To my understanding, this is where the fight between loops and strings lies. This is what exactly what humanino said earlier.

 

[hamster143]

By both directions I mean that not only must the field couple to the energy-momentum tensor, but it also must affect geodesics. In other words, that it is the metric and not just an arbitrary spin-2 field that happens to have Einstein-Hilbert lagrangian. The only thing Einstein-Hilbert action gives us is the dynamics of the field. For it to be GR, we also want the field to affect distances between points. I guess it would be sufficient if it replaced the (1,-1,-1,-1) metric in lagrangians of all other fields, e.g. [tex]\int F^{\mu\nu} F^{\rho\sigma} g_{\rho\mu} g_{\sigma\nu} \sqrt{-g} d^4 x[/tex] for electromagnetism.

 

[blechman]

This is exactly the statement that the massless spin-2 mode (graviton) couples to the energy momentum tensor! And this has been shown.

 

[Haelfix]

A Zee does the loose calculation for the graviton propagator in his qft textbook. There is no ambiguity (modulo some subtle gauge technicalities), it really is GR in the IR.

However it assuredly is not GR in the UV. Whatever 'it' is, is called quantum gravity (this is the DEFINITION of quantum gravity). We don't know how to solve the path integral in that case (b/c perturbation theory breaks down) so we need to figure out another way to solve the problem.

 

[hamster143]

This is exactly the statement that the massless spin-2 mode (graviton) couples to the energy momentum tensor! And this has been shown.

Yeah, on the second thought, I guess it is. Lagrangian density [tex]\frac{1}{2\kappa}R\sqrt{-g} - 1/4 F^{\mu\nu} F^{\rho\sigma} \eta_{\rho\mu} \eta_{\sigma\nu}[/tex] (which would seem to be the natural result of adding a spin-2 field to SM) would result in a free gravitational field ([tex]G_{\mu\nu}=0[/tex]) that's not coupled to anything. Lagrangian density [tex]\frac{1}{2\kappa}R\sqrt{-g} - 1/4 F^{\mu\nu} F^{\rho\sigma} \eta_{\rho\mu} \eta_{\sigma\nu} \sqrt{-g}[/tex] does not seem to reproduce the correct coupling.

I'd like to find out more about how exactly the spin-2 mode manages to sneak into other Lagrangians. I'm looking at Feynman's Lectures on Gravitation now ...

 

[bombadil]

I'm really enjoying this discussion even if 90% of it is over my head (which is good).

Here's my provisional understanding of what's being said:

(1) quantum gravity, or QG, (be it stringy or loopy) is necessarily a spin-2 field.
(2) The only successful perturbative description of it has been in the weak field limit (this is equivalent to saying QG has reproduced GR in the IR limit?)
(3) QG does not in general lend itself to perturbative methods.
(4) blah UV blah IR blah (I'm guessing UV and IR are fancy names for high and low energy?).
(5) No version of QG has been shown to reproduce all (or even most) of GR's strong field* affects (e.g. relativistic precession as explained and measured http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.2644v1.pdf" --this is a really cool article by the way)

So, my provisional conclusion is that string theories post-diction of a spin-2 field is pretty impressive, but QG has yet to reproduce GR in the strong field*.

*by strong field I'm talking about an intermediate energy range where linearized gravity does not apply, but GR is still valid (so the high energy behavior of *true* gravity hasn't caused it to deviate from GR yet). In other words, the strong field is where [itex]\phi/c^2 \sim 1[/itex] but not where [itex]\phi/c^2 \gg 1[/itex]

Please feel free to correct and comment on this. (and you really should check out the linked Science article, it's a unique strong field test of GR)

 

[Haelfix]

There's obviously a confusion about using the weak field method. It's essentially a trick, that's used in the middle of a calculation, and drops out later. Eg. just b/c you linearize the action at some point, doesn't mean that the resulting solutions are linear at the end!

When we say that we get GR in the IR, we really mean GR and not something else. Eg:
Guv = 8pi Tuv, where Guv is the Einstein tensor and Tuv is the stress energy tensor. This classical solution is valid for any value of the metric field, weak or strong and has the full geometrical significance that you would expect.

 

[hamster143]

Okay, I looked through Feynman's lectures on gravitation, and I read Deser's 1970 article, and I'm far from convinced. At best, what we have there is a statement that a massless spin-2 field that couples in a certain way to matter stress-energy tensor will look sort of like GR in continuum limit. (You can't even say that it MUST have Einstein-Hilbert action, because there can be terms like R^2 in the lagrangian, which aren't prohibited theoretically and are too weak to be measurable experimentally.) It's certainly not proven in either of these articles that the field must couple to the energy-momentum tensor. As far as I can tell, Deser actually assumes that the spin 2 field will have effect on geodesics.

 

[hellfire]

a massless spin-2 particle MUST be the graviton (that is, its UNIQUE action is the Einstein-Hilbert action)

The metric tensor has 10 independent components that reduce to 6 if one imposes local conservation of energy and momentum. The graviton, on the other hand, only has two degrees of freedom. Given this, how does it then that a theory of the graviton can be equivalent to general relativity at least in the linear approximation?

 

[blechman]

gauge invariance.

10 independent compoents
-4 components that can be gauged away (as you say)
-4 constraints from the EoM from these modes that you gauged away
=2 DoF.

The same thing happens for photons: 4-1-1=2 DoF.