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My question is very simple. We all know that, for a test particle(classical) in a gravitational field, the motion is only determined by the geodesic lines(let's forget about the initial conditions for now), and has no dependence on the "structure" of the particle, such as spin(in the classical sense), charge, etc..

But let's now consider the motion of a test field, scalar, spinor, vector or tensor, under gravity. I know how to describe this kind of motion using the corresponding field equations, Klein-Gordon equation, covariant Dirac equation, or master equations accounting for any masslesss fields. Of course, the results do depend on the spin(quantum). But is there a way to think about this in a "equivalence principle" sense, which can satisfactorily account for the difference due to spin?

Thank you in advance!

night cat
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  • "equivalence principle" makes me think of equivalence of mass and energy. A photon has only energy. If it is deviated by gravitation, is energy treated as mass, and the Newton laws apply? If yes, there would be known a "gravitational blue shift" of direction that would compare to Raleigh effect, as it occurs to me at the moment. I have not heard of any in spite of apparently paramount significance. The conclusion is that there is no gravitational force need to deviate a photon from a star behind the sun by the sun (Einstein's verification showed: all wavelenghts just the same angle, wrong?). – Peter Bernhard Nov 21 '22 at 16:13

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We all know that, for a test particle (classical) in a gravitational field, the motion is only determined by the geodesic lines

Actually, that isn't quite right. The geodesic lines don't actually exist in any objective sense. They're abstract things that are used to model particle motion, but's it's wrong to think that light curves because it follows a geodesic. Einstein never said that, he said "the curvature of light rays occurs only in spaces where the speed of light is spatially variable". It's rather like sonar, actually.

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and has no dependence on the "structure" of the particle, such as spin

I'm afraid that isn't quite right either. Matter is deflected half as much as light. If you throw an electron past a star at close to the speed of light, its path curves, but the curvature is only half the curvature of the photon's path. That's because of the structure of the particle. Because of the spin. See Ned Wright's deflection and delay article which talks about Einstein changing his prediction to 1.75 arcseconds, which is twice the Newtonian deflection.

Of course, the results do depend on the spin(quantum). But is there a way to think about this in a "equivalence principle" sense, which can satisfactorily account for the difference due to spin?

Sure. You know about pair production and the wave nature of matter and the Einstein-de Haas effect which "demonstrates that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics". Just think of an electron as a photon going round and round in a closed spin ½ "spinor" path. Then simplify it further to a square path, like this:

enter image description here

Only the horizontals bend down. Not the verticals. So the electron is deflected half as much as the photon. The GR deflection of light is twice the Newtonian deflection of matter.

John Duffield
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