# Standard Model: gravity and metrics

I am struggling to understand the possible extension of the Standard Model with gravity (or for what it matters, bosonic string theory). Please forgive this silly question, as I am an engineer venturing into physics following Susskind's lectures.

Knowing there is no theory of gravity in SM, what are the possible reconciliations paths currently under research?

Let me explain my doubts.

As I understand GR explains how gravity is "not a force", but masses distort the space-time, and any particle will simply follow the most straight line under the metric due to the mass presence (for instance Schwarzschild or Kerr).

The SM, on the other hand, shows how forces are mediated through gauge bosons.

One extension of SM includes gravitons which are supposed to "mediate" gravity. As far as Susskind's lessons go, string theory is one possible way of including gravitons into being.

How can then be a non flat metric in the first place if gravity could be brought in a coherent extension of the SM?

I mean, particles "gravitate" around a mass because the metric is non flat, without any need for a gravity field. If we had a graviton, I don't understand how masses could distort space-time and make use of a gauge boson: in this case there should be a field, while GR does not imply a gravity field. I started from many answers here, and of course Wikipedia, but this particular point is obscure, it seems to me that the theories are mutually exclusive, either a metric description or a field one as in SM.

Thanks for any pointers!

• There is probably not such extension. Not only is gravity not a force, as you say correctly, the SM is not even a theory of the right quantum fields. It's just an ad-hoc effective model for the sub-TeV range. At this point we simply don't know what the vacuum really looks like. Commented Aug 6, 2016 at 17:45

I guess the answer depends on what approximations you are willing to make but we do not have a theory of quantum gravity readily available, so we can't really consider arbitrary quantum matter coupled with quantum gravity. (String theory is a sort of special case.)

If you are fine working with the classical limit of the standard model (which Andrzej Derdziński discusses in his terse book Geometry of the standard model of elementary particles), then you can couple it with gravity without a problem. It's just a few Yang-Mills fields. (Although, arguably with the classical limit, you may end up just investigating the EM field.)

But I am guessing the OP is interested in quantum matter content and quantum gravity. We don't really have a quantum theory of gravity.

Semiclassical considerations may be done, i.e., quantum matter content and classical gravity. This is investigated in Gordon McCabe's book The structure and interpretation of the standard model.

One extension of SM includes gravitons which are supposed to "mediate" gravity. As far as Susskind's lessons go, string theory is one possible way of including gravitons into being.

Well, the "graviton" can be understood as a perturbation of the metric tensor.

(This is how we generically think of particles in quantum field theory, as perturbations of a field.)

Deser proved (in the '60s?) it's okay, everything works out, and we using perturbations around even Minkowski spacetime works fine. (The intuition is basically the "Taylor series expansion about the Minkowski solution" converges for any perturbation.)

There's actually an intuition to understanding the terms from such a "Taylor expansion of the metric" as "Spacetime tells matter how to move, and matter tells spacetime how to curl up."

From the other end, even if we started with "an arbitrary spin-2 graviton" model, Feynman proved we end up recovering GR. (See Feynman's lectures on gravity for the sordid details.)

The moral is: there are many different ways to describe gravity, and at times some descriptions are more applicable to a problem than other descriptions.

How can then be a non flat metric in the first place if gravity could be brought in a coherent extension of the SM?

You can do quantum field theory in curved spacetime. Though the spacetime is "fixed".

For example, Stefan Hollands wrote a review paper computing the quantum Yang-Mills field in an arbitrary-but-fixed curved spacetime.

We can make it "slightly dynamic" by coupling gravity to matter "at the tree level". What this means is, when we draw Feynman diagrams of interactions, we do not permit graviton loops. This is "semiclassical gravity".

Another intuition for this is, we truncate the "Taylor series expansion" to the first term about some background (for terms coupling the metric to gravity). IIRC, the metric is still coupled to itself ("gravity still self-gravitates").

Thomas-Paul Hack's thesis reviews sordid details about the scalar and spinor fields in the semiclassical gravity setting.

But to turn on "full gravity" coupled to quantum matter, that is quantum gravity, which we don't have a theory of.

Any attempt to write down a quantum theory of gravity is going to have to decide whether it will start from general relativity or from quantum mechanics.

The latter way involves writing down some quantum theory, showing that this quantum theory contains a massless, spin-2 particle $$g_{ab}$$, and then, in some limit, this particle has a lagrangian density that begins to look arbitrarily close to $$\sqrt{|g|}\left(R + g^{ab}M_{ab}\right)$$, where $$M_{ab}$$ is the lagrangian density of the relevant matter fields. String theory reached this by starting out as a theory of strong interactions that just couldn't have those spin-2 modes removed from it. In a theory like this, the general relativity concepts you quote above like "motion along geodesics", and "the equivalence principle", arise as consequences of the theory in some low-energy limit, not as first principles of the theory.

If you do the former, you need to go and embark on a much more ambitious program of:

1. Starting from general relativity
2. quantizing the modes of general relativity
3. coupling matter to this

or

1. Start from general relativity
2. defining the rules of quantum mechanics on this curved background
3. showing how the quantum modes you're describing change the curved background

People have tried all three of these approaches to varying success, and with various problems. Generally, the "start from GR" approaches run into big problems when you try to pull the matter fields back in, while the "start from QM" approaches run into problems when you try and find the right low-energy limit.

it seems to me that the theories are mutually exclusive, either a metric description or a field one as in SM.

No. The Schwarzschild metric shows that there must not be exclusivity between the description of gravity as curved spacetime and as a field, because curved spacetime may be entirely described as gravitational time dilation, see the answer of John Rennie to this question. These are two different models for the same thing.