# How are gravitons compatible with general relativity?

1. General Relativity.
2. Explained by the graviton.

How are these two things compatible?

How can it be that gravity is explained perfectly through curving spacetime, and at the same time we want to understand it by thinking of a particle that mediates its interaction? Isn't this confusing?

• A third description — Newtonian gravity — works well enough for many cases, and a fourth description — involving post-Newtonian corrections to Newton’s theory — makes it work even better. Plus there is string theory, and loop quantum gravity, etc. You’ve got lots of choices. – G. Smith Apr 21 at 19:28
• – Jonas Apr 21 at 20:51
• Simple layman's (non-)answer: If we knew this, we'd have the solution to quantum gravity. – Barmar Apr 22 at 14:47
• The second explanation does not exist. – my2cts Apr 22 at 23:25
• It is possible to reverse-engineer a free theory of massless spin-2 particles ("gravitons") by consistently summing the 'induced' interaction terms into full, background-independent GR - arxiv.org/abs/gr-qc/0411023v3. – Avantgarde Apr 27 at 4:01

There is no working theory that has been, with complete consistency, started from gravitons, and ended up at 4D General relativity in its low energy limit.

But the idea of the whole endeavor is the same thing as Electricity and magnetism. You have the classical picture of E&M that is based around electric and magnetic fields, and then you have the photon-based version of that theory, given by QED, where perturbations of those fields get treated as "photons", and in a limit, a superposition of many of these states average out to the fields.

It's the same idea with some theory of gravity -- perturbations of the metric tensor are gravitational waves, and you'd conceptualize "small" perturbations as "gravitons", and in some low-energy macroscopic limit, a superposition of many graviton states would look like the macroscopic metric tensor.

• But, to our knowledge, E&M doesn't affect spacetime, and gravity by definition does. Do we have any scientific examples showing how an EM field can make a clock tick slower? It seems a bit of a stretch to say that just because we were successful in casting EM in a particle view, that we can automatically cast gravity in the same way. In fact, casting EM in the quantum framework in some ways makes it less compatible with GR, which only aggravates the problem raised by OP. – stix Apr 22 at 20:28
• @stix: in the "graviton" picture of the world, "spacetime" is a field that is expressible as a "background" and a "dynamical" part. The metric tensor if the analogue of the E and B field (really, it's the analogue of $A_{a}$) And I'm not saying that we can do this, if it was definitively possible, there would be an accepted quantum gravitational theory right now. I'm saying that this is what people are trying to do when they write down theories with gravitons in them. – Jerry Schirmer Apr 22 at 21:44
• @stix: In GR, also massless particles affect spacetime due to the mass-energy equivalence. Meaning there's in principle no difference, as far as spacetime being affected or not goes, between a collection of photons (which is the EM mediator) and a collection of protons and electrons. So it's as you say not the case that the EM field can make a clock tick slower directly, but excitations in the EM field can bend spacetime just like normal particles can - and the result will indeed be that the clock ticks slower, because bent spacetime is bent spacetime regardless of how & why it was bent. – Vegard Apr 23 at 2:54
• @stix : "But, to our knowledge, E&M doesn't affect spacetime, and gravity by definition does." I see that you are unfamiliar with Kaluza-Klein theory which expresses electromagnetism as curvature of a 5-manifold. And others have already expressed that energy in E&M fields bends spacetime in the Standard Model. – Eric Towers Apr 23 at 20:44
• @EricTowers But we also know Kaluza-Klein theory is wrong, or at the very least, incomplete. Shrug – stix Apr 23 at 21:22

The second description arises from attempting to quantise the gravitational field. A full quantisation, so far, has proven elusive.

However, we can look at the case of a weak gravitational field, linearise this and quantise. This suggests that the quanta of the gravitational field is a spin-2 particle which is named the gravition.

Whilst everyone knows that gravity is expressed as the curvature of spacetime, it's less well known that all the other forces - electromagnetism, the weak and strong forces - are also expressed by curvature. Here, we equip the spacetime points with additional symmetries and then take the curvature.

Since these forces are well known to be mediated by particles, the photons for the electromagnetic force, the W & Z bosons for the weak force and the gluons for the strong force, we already see that there is a consistent picture where curvature and particles are implicated. This, in a sense, is an aspect of wave-particle duality: waves are unlocalised, curve, and and are the substance of the world whilst particles are localised, go in straight lines, and are the quanta of interaction.

• This is the best answer of why physicists invented the concept of a graviton. A weak, linearized, quantized gravitational QFT would be mediated by a spin-2 particle against a flat Minkowski spacetime. If that was actual reality, and not an approximation, we'd be done. – lamont Apr 23 at 3:11

They are not! But we're trying to get there.

### Preamble

Gravity is a series of phenomena, and we're trying to find a law to explain it. Newton tried it, but as we know he missed some points. Then Einstein came, and we got General Relativity. As you probably know, GR is a wonderful theory that explains almost every aspect of gravity in great detail - but not everything. Since we also have another theory that explains almost every aspect of the other forces, QFT, someone said "why can't there be particles that mediate gravity?".
Now here's the tricky part. We would like this new, gravity-mediated-by-particles to solve the problems in GR without losing the progress we've already made, so we want this theory of gravitons to be compatible with GR. But what does this mean?

### Compatibility

To be compatible means that it must give the same description of the same stuff (at least), so in modern theoretical physics language they must have the same action (plus some little boundary conditions that aren't important for us now). Now, there are two ways for this to be the case: you can start from the GR action and try to describe its ingredients as particles, or you can try to invent particles and then twist the knobs so that you action will be the desired one: I will give you an idea of how examples of the two approaches can be carried out.
Please note that we have no idea whether gravitons exist, and if they do what is the correct description of them: these are all just (nice) ideas.

We pretty much know that approach #1 doesn't work. One can give a description of gravitons $$h_{\mu\nu}$$ as the oscillations of the metric $$g_{\mu\nu}$$ on the flat metric $$\eta_{\mu\nu}$$: $$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}, \quad |h_{\mu\nu}|\ll1$$ and one finds that these gravitons are massless, spin 2 particles (which is good). But performing calculations one finds that the coupling of these particles to matter particles is nonrenormalizable (which isn't necessarily bad, but it's not what we would like). So, these gravitons and GR would be compatible, but their description would be no use to us.

There are many theories that try to pursuit approach #2, one of them is String Theory. In string theory one starts very far from GR: let's say there are strings all over the places and the particles are just different vibrations of these strings. In all of the possible ways the string can vibrate (in addition to the already known particles) there is a massless spin 2 particle, that we can identify with the graviton. The created theory is renormalizable, but is this theory of gravitons compatible with GR? One can check by taking the low-energy limit of the string action and, imposing some symmetries, one finds the GR action! So yes, the ST gravitons and GR are compatible, meaning (as we already said) that having the same action would lead to the same description of the same stuff.

I think that poorly-thought-out popular descriptions of quantum field theory have created a distorted idea of what it really says about the nature of the world.

In quantum field theory, you start with a field, which essentially is spacetime. The field exhibits particle-like behavior, and this particle-like behavior of fields is the only kind of "particle" in the theory.

This behavior shows up for the same reason that simple harmonic oscillators, hydrogen atoms, and vibrating solids have discrete energy levels in a quantum world.

The gravitational field oscillates (gravitational waves), and regardless of the details of quantum gravity, it's hard to avoid the conclusion that those oscillations should be discrete and particle-like in the same way as other oscillations. That's the only sense in which gravitons are the particle of gravity in quantum gravity. The gravitational field is still a field.

Particles also show up as lines in Feynman diagrams. Feynman diagrams don't show an electron emitting or absorbing a photon at a particular place and time. Each Feynman diagram represents something more like an integral over all spacetime diagrams that have that general shape. And the result of integrating a particle-like path over all possible shapes, with wavelike interference between them, is a wave. Even classically, Huygens' principle implies that you can think of electromagnetic waves as taking all possible particle-like paths between emitter and receiver.

All of this is also true in string theory. String theory is essentially a research program to find the theory that is approximated by string diagrams in the same way that Feynman diagrams approximate quantum field theory. The string diagrams aren't supposed to be the ultimate reality; the ultimate reality is more field-like.

• What do you mean by stating that a field is essentially spacetime? – Deschele Schilder Apr 25 at 3:28

The 2 ideas are compatible because one is built out of the other.

If you think of quantum gravity as individual building blocks (quanta) that describe gravity in the micro world, then it should, in theory, be able to be scaled up to describe the macro world. Hence, you should be able to derive General Relativity through its quantum description.

I see no reason why they should not be compatible. You can say "is steel compatible with being put together to create skyscrapers?", and of course you would say yes, they are! So why not with gravity? If Gravitons are the basic building blocks (like steel), then why couldn't you build up the much larger theory (the skyscraper)?

Just like @Jerry Schirmer said:

"a superposition of many graviton states would look like the macroscopic metric tensor."

and this makes a lot of sense, at least to me.

Incidentally, I agree with @Jerry Schirmer and @Lorenzo Castagno where Electromagnetism was a classical theory that was later transformed into a quantum theory (QED). The same could be true of gravity, (where the analog is General Relativity) and it could be in the same position as Electromagnetism where it might be later quantized into a theory of quantum gravity.

The main difficulty (at least to me) lies in the fact that the stuff one has to quantize in general relativity is spacetime itself, while in the other three cases (the other three forces), it are fields in spacetime that are quantized.

You can say that the condensate of virtual gravitons looks the same as a metric tensor (like the condensate of virtual photons looks like the classical electromagnetic field) but still the difficulty remains how to couple these gravitons to the curvature of spacetime. They travel through flat spacetime but how the curvature of spacetime is influenced by the gravitons?

When two particles interact by exchanging gravitons, then the particles themselves absorb (or emit) the gravitons. Or when two massive bodies like the Sun and the Earth move through spacetime the condensate of gravitons is exchanged between the constituents of the Earth and the Sun. To say that the gravitons affect spacetime itself is another thing. So a condensate of gravitons indeed looks like a metric tensor, due to the tensor Nature of the gravitons.

When the particles have a very high kinetic wrt eachother they will form black holes upon collision. Which is another difference between normal quantized fields and gravity quantized "fields". You can incorporate the emergence of black holes into the field theories of colliding particles as one of the possible outcomes of the collision but how the gravitons turn (flat) spacetime into black hole is another thing.

You can of course say that the gravitons are quantized spacetime variations. But then they are non-pointlike structures.

In string theory gravitons are considered to be tiny vibrating closed strings (so non-point-like structures) travelling through a flat background spacetime which is why the theory is said to be background dependent. Are the closed strings quantized spacetime variations? Well, they are extended and they do carry tensor information. But how is this information given to the flat spacetime to make it curved? Is a flat spacetime with strings ("extended tensor structures") imposed on it a curved spacetime?
In contrast, Loop Quantum Gravity comes closer to quantizing spacetime itself, there is no background spacetime through which the gravitons travel. Spacetime is already quantized from the start. The theory is background independent. Gravitons are small spacetime distortions (so not closed strings in flat spacetime). But how are elementary particles represented, if not point-like or string-like? Is there some common interface with string theory?

So, is general relativity compatible with quantum gravity? Personally I think it is. That is if you quantize spacetime itself, to which LQG comes closest. String theory is not compatible with general relativity, even if some low energy approximation reproduces the curved spacetime of general relativity. This is simply because the theory describes the gravitational interaction as taking place in a flat spacetime.

Questions, questions, questions... Meant as an answer to the difficulty in quantizing gravity. Non-renormalizability, said to be a major obstacle to quantizing gravity, seems to be of less interest, though finding a renormalizable theory can help in understanding the quantizing of spacetime itself.