QFT of Gravitons in Minkowski Space vs GR

[ohwilleke]

TL;DR Summary

How would a quantum field theory of gravity massless spin-2 gravitons in Minkowski space differ generically or qualitatively from GR, if at all, and if there is a difference, how would it be experimentally measurable?

A central feature of classical GR that it is background independent and operates via a curvature in space-time. As I understand it, this is not true of the other Standard Model forces which are consistent with special relativity and operate in Minkowski space, in which forces are transmitted via carrier bosons, but is not consistent with GR.

As I understand it, in GR, locally space-time is approximately Minkowski space, but not exactly.

If gravity were formulated as a theory of a massless spin-2 boson that couple's to a particle's mass-energy in the same Minkowski space that Standard Model forces are modeled in with a coupling constants of the appropriate strength, how would that differ from classical GR? (I'm not saying that this is the right way or not the right way to do quantum gravity, just trying to understand what is at stake.)

I'm looking for differences that would be generically true or qualitatively different from all such theories, and not a particular non-existent working quantum gravity theory or calculations, which don't exist.

I'm asking about how any difference, if there is a difference, would be experimentally measurable as a way of stating and understanding more concretely the difference between the space-time topology in which the SM and special relativity alone operate, and the space-time topology of classical GR.

If that is too broad or speculative, in what circumstances would the differences (if any) probably be most obvious?

 

[PeterDonis]

As I understand it, this is not true of the other Standard Model forces which are consistent with special relativity and operate in Minkowski space

QFT can be done in curved spacetime, so it is not strictly true that the Standard Model forces (no need to say "other" since gravity is not one of them anyway) can only be modeled in globally flat spacetime.

As I understand it, in GR, locally space-time is approximately Minkowski space, but not exactly.

I'm not sure what you mean by this.

If gravity were formulated as a theory of a massless spin-2 boson that couple's to a particle's mass-energy in the same Minkowski space that Standard Model forces are modeled in with a coupling constants of the appropriate strength, how would that differ from classical GR?

This question was investigated in the 1960s and early 1970s. The basic result is that the field equation for a massless spin-2 boson is in fact the Einstein Field Equation, so it is perfectly possible to interpret GR as the classical limit of the QFT of a massless spin-2 field on flat spacetime. However, there are at least two caveats:

(1) The resulting QFT is not renormalizable and is not considered a viable candidate for a fundamental theory; at best it can be treated as an effective field theory, valid below some cutoff energy scale (perhaps the Planck scale).

(2) If the QFT is built on an underlying globally flat spacetime, then any resulting EFE solution can only have the same topology as globally flat spacetime, i.e., $R^4$. So a solution like a black hole, for example, which has a different topology, would not be allowed. Whether this limitation is actually an issue depends on whether topologies other than $R^4$ are actually needed to model any observations, or whether models (at least effective ones, valid below some energy scale) with topology $R^4$ can always approximate the data closely enough.

 

[Elias1960]

Summary: How would a quantum field theory of gravity massless spin-2 gravitons in Minkowski space differ generically or qualitatively from GR, if at all, and if there is a difference, how would it be experimentally measurable?

While I have nothing to object to the answer given by PeterDonis, I would like to add a difference which is IMHO important. Namely, that the symmetry group of GR has to be broken in the field-theoretic approach. In defense of PeterDonis, I would like to add that I have a similar objection even to the original papers, say, Feynman, that they do not consider this difference adequately.

But even Feynman may be excused because Feynman's approach starts from the spin 2 field on the Minkowski background, thus, the starting point is a theory which, unlike GR, depends on the background, and is not background-independent. All the intermediate terms in his approach will also be background-dependent. One cannot start the hole discussion at any point during the whole process of approximation, except in the limit itself.

But if in every step of the approximation there exists a preferred background, what is the limit of this background, and how is it connected with the resulting Einstein equations? I have not found an answer in the papers. But I would guess that the answer is simply the straightforward one: The preferred background is the harmonic one.

If correct, the restriction is a little bit more serious than (2) of PeterDonis, namely to $\mathbb{R}^4$ we would have to add "with global harmonic coordinates on that $\mathbb{R}^4$".

On the other hand, I would like to weaken this point:

any resulting EFE solution can only have the same topology as globally flat spacetime, i.e., $\mathbb{R}^4$. So a solution like a black hole, for example, which has a different topology, would not be allowed.

If a full GR solution has non-trivial topology, it does not follow that the field theory has a problem with this. A solution of the field theory could cover only some part of the full GR solution covered by some particular harmonic coordinates. If this part is large enough to cover all that we can observe (say, by excluding parts behind horizons), there would be no reason to reject this from a field-theoretic point of view. Completeness of the GR solution could be considered as irrelevant. Say, during the collapse, the star becomes a frozen star, nothing changes there, everything stops, and all clocks will stop in the limit too. The part behind the horizon is simply thrown away. Time is the time of the background, the harmonic time coordinate, not what clocks show.

A central feature of classical GR that it is background independent and operates via a curvature in space-time. As I understand it, this is not true of the other Standard Model forces which are consistent with special relativity and operate in Minkowski space, in which forces are transmitted via carrier bosons, but is not consistent with GR.

Correct. And I see no way to obtain a background-independent GR from field theory. What is, instead, straightforward is to obtain a background-dependent variant of GR in harmonic coordinates.

If gravity were formulated as a theory of a massless spin-2 boson that couple's to a particle's mass-energy in the same Minkowski space that Standard Model forces are modeled in with a coupling constants of the appropriate strength, how would that differ from classical GR?

The difference would be background-dependence.

There are different ways to do this too. First, one can simply introduce the harmonic condition explicitly to the EFE. Then, we can introduce a term into the Lagrangian which breaks the diff symmetry and enforces harmonic coordinates. This second way would include modifications of the EFE too, caused by the additional term in the Lagrangian.

I'm asking about how any difference, if there is a difference, would be experimentally measurable as a way of stating and understanding more concretely the difference between the space-time topology in which the SM and special relativity alone operate, and the space-time topology of classical GR.

This is already more difficult. If one introduces terms into the Lagrangian which enforce harmonic coordinates $\square X^\mu = 0$, they could look like the term for a scalar field $X^\mu(x)$. This field would be massless dark matter. So, in principle observable, but in reality not. If it would be introduced as an explicit condition, outside the Lagrange formalism, one would not have even this difference. But the theory would be less beautiful without a Lagrange formalism.

A way to introduce here a Minkowski background would be to give the time component the wrong sign. $X^0(x)$ would become a ghost field. This would be the Lagrangian of the "relativistic theory of gravity" proposed by Logunov. The ghost would be (as massless dark matter) quite harmless, moreover, there would be global restrictions on its values ($X^0(x)$ has to remain a time-like coordinate), so that this may be not a decisive reason for rejection. This would give already quite different predictions: Stable gravastars with sizes slightly greater than their horizon, and a big bounce instead of a big bang. (Here, the wrong sign allows the $X^0(x)$ field to circumvent what for normal fields give nothing different from GR, in particular the big bang singularity theorem, and to create something which can stop the collapse.)

Beyond this, there remains evidence for non-trivial topology. Which is hard, given that, as considered above, one is free to cut away unobservable parts of solutions with non-trivial topology. So, one would need not simply wormholes, but traversable ones, so that cutting the nontrivial part does no longer work.

If the global universe would be curved, one could argue that this would have to violate homogeneity of the theory in the preferred coordinates. But what we see is flat on a large scale, as it would have to be for a homogeneous case.

 

[PeterDonis]

A solution of the field theory could cover only some part of the full GR solution covered by some particular harmonic coordinates.

Yes, but then only that portion of the "full" GR solution would be what is claimed to be "real". The full GR solution would be a mathematical structure part of which does not correspond to anything in reality.

I'm not sure how workable this view actually is, because the region of spacetime covered by the harmonic coordinates and with topology $R^4$ would be geodesically incomplete, whereas the original background Minkowski spacetime used to formulate the QFT is geodesically complete. Given that we already know that such a QFT can only be an effective theory in any case, this might not be much of an additional issue, though.

 

[Elias1960]

I'm not sure how workable this view actually is, because the region of spacetime covered by the harmonic coordinates and with topology $R^4$ would be geodesically incomplete, whereas the original background Minkowski spacetime used to formulate the QFT is geodesically complete. Given that we already know that such a QFT can only be an effective theory in any case, this might not be much of an additional issue, though.

I think so too.

If one uses a Lorentz-ether-like interpretation which identifies true absolute space and time with the background and decreases the role of the metric to a definition what clocks measure, the incompleteness reduces to some clocks with speed (in unmeasurable background time) decreasing exponentially so that they remain frozen forever. Conceptually this would be unproblematic. Quantum theory would use the background t in the Schroedinger equation, there is anyway no operator to measure it in QT.

In principle, it defines a possibility for empirical falsification by infalling into the BH candidate and reaching the part which has been cut. Unfortunately, the experimenter cannot inform us outside about the successful falsification.

Whatever, the restriction to $\mathbb{R}^4$ makes GR field theory a different theory, and the viability of GR and of field theory GR is in principle different.

 

[PeterDonis]

the incompleteness reduces to some clocks with speed (in unmeasurable background time) decreasing exponentially so that they remain frozen forever.

Actually, if we restrict to $R^4$ topology, I think it's impossible to have an event horizon, so there would not be any clocks that would be "frozen". Even models like gravastars, which use exotic matter to allow objects to be static while having apparent horizons, don't have actual event horizons: there are no regions of the spacetime which cannot eventually send light signals to infinity.

 

[Elias1960]

If we simply use harmonic coordinates as preferred, and cut the parts not covered by those which are reasonable before the collapse, then what will be cut is the part behind the horizon. So, there will be indeed no regions which cannot send light to infinity. But the infalling clock will approach what is the horizon in the geodesically complete solution, and even if it would have, at every finite moment of time, a finite speed, that speed would slow so fast that the measured time would remain finite even for infinite background time, $\tau = \int_{t_0}^{\infty} \frac{d \tau(t)}{dt} dt < \infty$.

In a theory of massive gravity like RTG there will be no reach of the horizon even in infinity, so that there will be simply an extremely slow clock on or inside a stable gravastar-like object with a fixed distance from horizon size.

So this depends on how field theory version is defined.

 

[PeterDonis]

the infalling clock will approach what is the horizon in the geodesically complete solution

You mean geodesically incomplete, I take it? If we just take Schwarzschild spacetime and cut out the part at and below the horizon, what remains is obviously geodesically incomplete.

there will be simply an extremely slow clock on or inside a stable gravastar-like object

Extremely slow, yes, but there will be no frozen clocks anywhere in such a solution.

 

[Elias1960]

You mean geodesically incomplete, I take it? If we just take Schwarzschild spacetime and cut out the part at and below the horizon, what remains is obviously geodesically incomplete.

Yes, this first possibility - adding the harmonic condition simply as an independent equation to the Einstein equations - would give geodesically incomplete solutions. But they would be complete on the background $\mathbb{R}^4$. Geodesical completeness would lead its power and importance if space and time is defined by the background, and the metric describes only clock showings.

Extremely slow, yes, but there will be no frozen clocks anywhere in such a solution.

Yes, in the RTG variant of the field theory BHs would be avoided without destroying geodesical completeness.

It is essentially a different theory, even if all one does is to enforce the same harmonic gauge via a symmetry breaking term in the Lagrangian.

 

[PeterDonis]

they would be complete on the background

No, they wouldn't. There would be geodesics that can't be extended to arbitrary values of their affine parameter; that's the definition of geodesically incomplete.

Geodesical completeness would lead its power and importance if space and time is defined by the background

No, because when you cut out part of the standard GR solution, you are cutting out part of the background as well. The geodesics will be incomplete in the cut off background just as they are in the standard GR metric. It just so happens that cutting out an open set of $R^4$ still gives you $R^4$, so the topology of the background doesn't change. But topology isn't everything.

It is essentially a different theory

Actually, I'm not sure it is. It's just a standard GR solution with exotic matter in the central region, which allows the solution to be stationary without a horizon for values of the surface radial coordinate smaller than the Buchdahl Theorem limit but larger than $2M$.

 

[Elias1960]

No, they wouldn't. There would be geodesics that can't be extended to arbitrary values of their affine parameter; that's the definition of geodesically incomplete.

There would be geodesic trajectories $X^{\mu}(T)$ so that
$$\int_{T=0}^\infty \sqrt{g_{\mu\nu}\frac{\partial X^\mu}{\partial T}
\frac{\partial X^\nu}{\partial T}} dT < \infty.$$
This would, indeed, be geodesic incomplete.

No, because when you cut out part of the standard GR solution, you are cutting out part of the background as well.

We can but there is no reason to do such things. But if we start, say, with almost Minkowski coordinates before a collapse, then a collapse begins, we continue to use the same preferred coordinates. And if they exhaust all the $\mathbb{R}^4$ before exhausting the geodesically complete GR solution, some part is cut out of the field theory, and one can essentially do nothing against it.

This is the case relevant for the BH collapse. The harmonic extension of Minkowski initial time during the collapse of a black hole would define an example of a harmonic time coordinate which becomes infinite. It would be a time coordinate which looks qualitatively like the original Schwarzschild time coordinate. The horizon forms where that harmonic time becomes infinite. So, no, I don't cut out any part of the background, I solve the equations on the background in harmonic gauge for the complete $\mathbb{R}^4$, for all $-\infty < X^\mu < \infty$, and if I succeed, this solution is complete from the field-theoretic point of view (even if not geodesically complete).
The field theory has simply a different notion of completeness of a solution.

Actually, I'm not sure it is. It's just a standard GR solution with exotic matter in the central region, which allows the solution to be stationary without a horizon for values of the surface radial coordinate smaller than the Buchdahl Theorem limit but larger than 2M.

Once GR allows nontrivial topologies and field theory insists on $\mathbb{R}^4$, they have different solutions, thus, are at least formally different theories. The question is how far this is observable.

I agree, many non-GR theories will be, in some covariant field-theoretic version, equivalent to GR with strange additional matter. But this will be a local equivalence at best. Say, preferred harmonic coordinates look formally like scalar fields following a scalar wave equation $\square X^\mu(x)=0$. But only a small subset of such fields define valid solutions - those where the four "scalar fields" define a global system of coordinates, with all allowed four-combinations of such field values $-\infty < X^\mu < \infty$ exists exactly a single event having these four values. Such global restrictions for field values cannot be reasonably defined in a GR-with-matter-fields context.

But these conditions missed in a GR-with-matter-fields interpretation of the same solution does not destroy the solution. So, it will appear also as a solution of the GR-with-matter-fields theory,

 

[PeterDonis]

Once GR allows nontrivial topologies and field theory insists on R4

But the point I'm making is that if you allow exotic matter there are static GR solutions (like gravastars) on $R^4$ topology that have surface radial coordinates in the range $2M < r < 2.25M$, i.e., between the horizon radius and the Buchdahl Theorem limit. "GR" is not the same as "GR restricted to solutions that obey the energy conditions".

 

[Elias1960]

Full agreement about this last point.