I know that we can't envision the spin of an electron envision as something spinning (unless the electron has a non-stringlike spherical structure), but the spin does interact with an electric field because of its charge. Can't we say that it interacts with the gravitational field due to its mass?
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3$\begingroup$ Not according to the General Relativity. However the spin interacts with gravity in an extension of GR known as Einstein-Cartan theory. The basic difference between the two is the fact that the later has a non zero torsion. $\endgroup$– DiracologyCommented Jan 25, 2017 at 14:45
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2$\begingroup$ Anyone aware of any experiments on this ? $\endgroup$– StephenG - Help UkraineCommented Jan 25, 2017 at 15:15
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1$\begingroup$ The spin interacts because of the charge and because it's angular-momentum-like (cf. Einstein-de Haas) and angular momentum of charged particles interacts with the magnetic field classically. An interesting and more precise question would be if the spin of massive particles contributes to the metric analogously to classical angular momentum in the Kerr metric. $\endgroup$– ACuriousMind ♦Commented Jan 25, 2017 at 17:00
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$\begingroup$ Particles with non-zero spin are not necessarily massive. I don't know much about general relativity, but something tells me, that spin coupling to gravitational field should be independent of mass, at least, in leading order. $\endgroup$– MsTaisCommented Jan 25, 2017 at 17:03
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$\begingroup$ @StephenG-I had the same question in my mind! $\endgroup$– Deschele SchilderCommented Jan 25, 2017 at 18:34
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1 Answer
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Here is a simple minded answer: The electron is an elementary particle in the table of the standard model which is a quantum mechanical theory. A gravitational interaction where the spin could play a role would need the quantization of gravity, which is a matter under research and not finalized.
In effective quantization of gravity, the corresponding quantum mechanical particle, waiting a niche in a future standard model , is the graviton, hypothsized to have spin 2 and be zero mass.
An electron graviton interaction then has the following Feynman diagram:
If the upper line one the left is an electron and the lower line descriptive of a charged mass, the electron interacts with a virtual photon with the electric field. The above is only the contribution from one type of diagram, a an exchange . Depending on the lower line's particulate structure more diagrams can contribute and also higher order ones. For a gravitational mass for the lower line on the right, the graviton would be the exchange particle.
The effect of spin comes in the matrix elements that describe the wavefunctions to be integrated , so as to get the crossection.
In a similar way that one says that, given the appropriate boundary conditions, the electron's spin interacts with the electric field, one cans say that :given the appropriate constants and a quantization of gravity, the electron's spin interacts with the gravitational field.
Spin is part of the total mathematical description in calculating the wavefunction of the system, and enters through the conservation of angular momentum in allowing or forbidding some diagrams.
The basic numbers that should be kept in mind are the coupling constants at the vertices, where for photon the vertex is depressed by 1/137, but for a graviton by 6*10^-39, which makes any individual particle state interaction not accessible to experiment.
