B Lagrangian Point in General Relativity

[PeterDonis]

There are of course the interaction terms between the $g_{\mu \nu}$ and the fields describing matter and radiation.

Calling these "interaction" terms is, IMO, a misnomer, classically speaking. Classically speaking, the only reason the metric appears in these terms is to lower or raise indexes on the fields in order to form scalars.

 

[vanhees71]

Sure, but there are also equations of motion for the other fields (or for the point-particle coordinates, if you use the point-particle model), and of course they contain the interaction with the gravitational field (the pseudometric).

Of course you are right in saying that these interactions are usually written on the left-hand side of the equation of motion. There the coupling to the gravitational field occurs in the terms with the Christoffel symbols. For point particles you get (using natural units with $c=1$)
$$m \mathrm{D}_{\tau} u^{\mu}=K^{\mu},$$
where
$$\mathrm{D}_{\tau} u^{\mu}=\mathrm{d}_{\tau} u^{\mu} + {\Gamma^{\mu}}_{\rho \sigma} u^{\rho} u^{\sigma}, \quad u^{\mu} = \mathrm{d}_{\tau} x^{\mu}.$$
The $K^{\mu}$ is the Minkowski force of all other interactions.

Now, if the interaction with the gravitational field where just as inertial forces in Newtonian mechanics, you could find coordinates, where the Christoffel symbols vanish identically, but that's not the case when "true gravitational fields" are present. Then you can make the Christoffel symbols vanish only at one arbitrary point in spacetime (weak equivalence principle).

That the Christoffel symbols occur in the description of the interaction of matter with the gravitational field is indeed natural from the point of view of gauge theories since it's the connection. That's analogous to the em. field, where the gauge connection is given by the Faraday tensor and the corresponding Minkowski force is
$$K_{\text{em}}^{\mu}=q F^{\mu \nu} u_{\nu}.$$

In Newtonian physics it's the same. What is usually called "inertial forces" belongs to the "covariant time derivative" for vectors wrt. a non-inertial reference frame. E.g., for a rotating frame
$$m \mathrm{D}_t \vec{v} = m (\mathrm{d}_t \vec{v}+\vec{\omega} \times \vec{v})=\vec{F}$$
with
$$\vec{v}=\mathrm{D}_t \vec{x} = \mathrm{d}_t \vec{x} + \omega \times \vec{x}$$
and $\vec{F}$ the "true forces".

It's only convention to bring the terms different from $m \mathrm{d}_t^2 x$ from the left to the right-hand side and call them "inertial forces". Of course, in Newtonian physics you can make all these inertial forces vanish, because there's by assumption always global inertial frames.

 

[vanhees71]

Calling these "interaction" terms is, IMO, a misnomer, classically speaking. Classically speaking, the only reason the metric appears in these terms is to lower or raise indexes on the fields in order to form scalars.

The important point going from special to general relativity is that the pseudometric becomes a dynamical field. The variation wrt. the pseudometric gives the Einstein field equations,
$$G_{\mu \nu}=\kappa T_{\mu \nu},$$
where $T_{\mu \nu}$ is the energy-momentum-stress tensor of the matter fields (including the em. field). The variation with respect to the matter fields gives the equations of motion for these fields including the interaction with the gravitational field.

This is a complete set of closed equations, describing the dynamics of the gravitational field and the dynamical degrees of freedom you choose to describe the matter (fields for continuum-mechanical descriptions like fluids or "dust" or spacetime coordinates for point particles) and the Maxwell em. field.

All this has nothing to do with anything related to quantum mechanics or quantum field theory. It's all classical field theory!