On the nature of forces in general relativity

Question

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EDIT:

I understand that it's not wise to fixate on Schroedinger words, however their meaning still remains obscure to me. Besides this my question on the possibility to abandon the concept of force in general relativity remains since I heard this statement in other occasions but never understood it.

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I am reading "Space-Time Structure" by Erwin Schroedinger. In the introduction he says that the idea of general relativity actually embraces every kind of dynamical interaction and that gravity is just the simplest case. In this context he states the following:

At any rate the very foundation of the theory, viz. the basic principle of equivalence of acceleration and a gravitational field, clearly means that there is no room for any kind of 'force' to produce acceleration save gravitation, which however is not to be regarded as a force but resides on the geometry of space-time. Thus in fact, though not always in the wording, the mystic concept of force is wholly abandoned. Any 'agent' whatsoever, producing ostensible accelerations, does so qua amounting to an energy-momentum tensor and via the gravitational field connected with the latter. The case of 'pure gravitational interaction' is distinguished only by being the simplest of its kind, inasmuch as the energy-momentum- (or matter-) tensor can here be regarded as located in minute specks of matter (the particles or mass-points) and as having a particularly simple form, while, for example, an electrically charged particle is connected with a matter-tensor spread throughout the space around it and of a rather complicated form even when the particle is at rest. This has, of course, the consequence that in such a case we are in patent need of field-laws for the matter-tensor (e.g. for the electromagnetic field), laws that one would also like to conceive as purely geometrical restrictions on the structure of space-time. These laws the theory of 1915 does not yield, except in the simple case of purely gravitational interaction. Here the defect can at least be camouflaged or provisionally supplemented by simple additional assumptions such as: the particle shall keep together, there shall be no negative mass, etc. But in other cases, such as electromagnetism, a further development of the geometrical conceptions about space-time is called for, to yield the field-laws of the matter-tensor in a natural fashion.

I cannot understand these statements at all.

First of all what I get from the first phrase in bold above is that any "source of acceleration" to which a test particle is subjected is just a contribution to the energy-momentum tensor. Hence if the energy-momentum tensor is divided in matter contribution $T_{\mu\nu}^{(matt)}$ and interaction contribution $T_{\mu\nu}^{(int)}$, the sum of these two will enter the Einstein equations and hence determine the metric and the geodesics that are followed by test particles. Hence what I get from that statement is that any "interaction" has the only effect of modifying the metric and hence test particles follow the geodesics for some opportune metric.

But, it seems to me, this is not true, for example, for electromagnetic interaction. In fact suppose that we have a charged test particle, that is a particle whose "spread out" energy momentum tensor (due both to the matter and the electromagnetic content) is negligible with respect to the one of the background, then the presence of an electromagnetic field $F_{\mu\nu}$, besides contributing to the gravity (metric), would also produce an additional interaction that modifies the geodesic equation of motion to a non-geodesic one \begin{equation} u^\alpha\nabla_\alpha u^\mu=\frac{q}{m}F^\mu_{\hspace{.2cm}\beta} u^\beta \end{equation} where $q$ and $m$ are the charge and the mass of the test particle. It is true that the covariant derivative contains some information about the presence of an electromagnetic field, but it does not contain all of it. Moreover it's not possible to have the motion of the test particle as a geodesic of some metric since geodesics should be unique once we specify the initial point and tangent vector, and particles with the same initial conditions but different charges cannot follow the same path.

So I am failing to see how "there is no room for any kind of 'force' to produce acceleration save gravitation".

As a consequence I cannot understand also what follows. I get that the energy-momentum tensor for a charged particle is more complicated than the one of a neutral particle, but I don't understand which "field-laws for the matter tensor" would allow us to reduce the acceleration due to electromagnetic interaction to the gravitational one. I am aware, though not in detail, of the existence of Kaluza-Klein theory, but Schroedinger explicitly refers to a "four-dimensional continuum" and I'm not sure that the content of his statement is of the same type of the Kaluza-Klein one.

Any help in the understanding of Schroedinger's statements would be greatly appreciated. More in general I would like to understand what it means that the concept of force can be abandoned in general relativity.

Answer 1

Any unified geometrized theory will have to use a geometry that's more complicated than Riemannian. The 1915 version of GR is relatively simple because it depends on the mass-mass ratio (gravitational mass to inertial mass) which we think is just 1. So particles of any (small) mass have the same geodesics.

In the EM version, this can't be true. Ignoring magnetism for the moment, the electric acceleration depends on the charge-mass ratio. Thus different kinds of particles must have different geodesics even at the same point in spacetime and with the same velocity. This probably requires at least a Finsler space.

Further, since we already know that acceleration + EP implies time dilation, it seems likely that such a theory must predict an EM time dilation. Certainly David Apsel (1978,1979,1981) does. By analogy with gravitational TD being approximately 1 + mgh/mc^2 (i.e. a function of the ratio of potential energy to rest mass energy), the (electrostatic part of) EM time dilation has to have magnitude 1 + qV/mc^2 (to first order for low voltages). This is quite easily testable, and Apsel 1979 proposed an experiment, but in 40 years nobody has done the test. DISCLAIMER: This is VERY non-mainstream physics. I am an advocate for it, but few people are. There is a small chance that Fermilab will grant a group including myself some beam time to test the theory in 2020 or 2021, but TRIUMF found the whole idea so laughable that they refused to even issue an application number.

The magnetic time dilation is too small to measure unless you have access to gigatesla fields or greater. That's why I ignore it above.