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In the study of electromagnetic fields and quantum electrodynamics we observe and theorize on the behavior of superconductivity and the Meissner effect. Has an analog of these behaviors been proposed to gravitational field theory and matter?

If so what possibilities could these propositions lead to?

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The gravitational analogue of magnetism is called "gravitomagnetism" (and the general mathematical analogy between Maxwell's equations and gravitation in general relativity is called gravito-electromagnetism), which deals with the gravitational interactions of currents of mass/energy, just as magnetism deals with interactions of currents of charge. According to the book The Measurement of Gravitomagnetism: A Challenging Enterprise by physicist Lorenzo Lorio, there is no gravitomagnetic analogue of the Meissner effect: see section 4.3 on pages 45-47 which can be read on google books here. Quoting just the introduction and conclusion of this section:

If however one wants to consider a real analog of the pure Meissner effect, one should treat the pure gravito-magnetic case. To be more definite, we should envisage a situation where matter can flow without friction in response to a gravito-electromagnetic field, i.e. matter in a pure superfluid state.

...in principle, in superfluids there is no analog of the Meissner effect in superconductors. In other words, recalling a remark by Pascual-Sánchez [142], we can say that superfluids in gravito-magnetic fields display a paramagnetic-like behavior rather than a diamagnetic-like one.

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