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There are notable similarities in the classical laws that govern the gravitational force and the electromagnetic force. Considering stationary point masses/charges, both the gravitational force and the Coulomb force follow inverse square laws: $$ F_g=G\frac{m_1 m_2}{r^2} $$ $$ F_C=\frac{1}{4 \pi \epsilon_0 }\frac{q_1 q_2}{r^2} $$ For the Coulomb force, each electrostatic charge has an electric field and it is the interaction of the two electric fields which create a force. This is analogous to the gravitational fields of masses, which also interact to bring about a gravitational force.

Magnetism has of course been shown to arise from the laws of electrostatics in a relativistic setting. See the Feynman Lectures on Physics, Volume II, Chapter 13-6: https://www.feynmanlectures.caltech.edu/II_13.html

If magnetism can arise just from the relativistic effects of moving electric charges, then why does gravitational magnetism not arise as a relativistic effect of moving masses? Physicists such as P. M. S. Blackett and Arthur Schuster did indeed propose notions of gravitational magnetism, although these were were not supported by experimental observations. Does the explanation, for why gravity and electromagnetism seemingly lack this commonality, require transcending the classical description of gravity?

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  • $\begingroup$ The OP appears to be referring to Blackett Effect. This is possibly off-topic as it's not mainstream physics (IMO). $\endgroup$ – StephenG Dec 5 '20 at 23:22
  • $\begingroup$ I refer to the Blackett Effect as just one of multiple propositions of gravitational magnetism that have been made. My question is about the more general behaviour of the gravitational force and why it differs from the electromagnetic force when it comes to its relativistic nature, despite following analogous classical laws in a static setting. $\endgroup$ – Alexander Kalian Dec 5 '20 at 23:25
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    $\begingroup$ Gravity and electromagnetism are very different. In the case of gravity, the inverse-square law is only an approximation, whereas it is exact in (classical) electrodynamics. Gravity is more precisely described by general relativity. A gravitational analog of magnetism does exist, sort of, as described in knzhou's excellent answer to the question What is the analog of the Aharonov-Bohm effect for general gauge fields and for gravity?. $\endgroup$ – Chiral Anomaly Dec 5 '20 at 23:36
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    $\begingroup$ Wikipedia has some things of interest. $\endgroup$ – jacob1729 Dec 6 '20 at 0:27
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    $\begingroup$ The gravitational Einstein-Infeld-Hoffman equations have velocity-dependent “magnetic” terms for moving masses, and I believe these are used to compute Solar System ephemerides. So I think the premise of your question is false. $\endgroup$ – G. Smith Dec 6 '20 at 1:31
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There actually is a "magnetic" gravitational effect, which is described by the "next better thing" between Newtonian gravity and General Relativity: the so-called "gravitoelectromagnetism" (GEM) theory, which I've also referred to as the Newton-Maxwell theory of gravity.

This pretty much does exactly as you suggest: it introduces, in addition to the Newtonian gravitational field $\mathbf{g}$, a further field $\mathbf{d}$ (for lack of a better letter, because it's two letters after B just as G is two letters after E), or gravimagnetic field, satisfying equations analogous to the full suite of Maxwell's equations:

$$\nabla \cdot \mathbf{g} = -4\pi G\rho$$ $$\nabla \cdot \mathbf{d} = 0$$ $$\nabla \times \mathbf{g} = -\frac{\partial \mathbf{d}}{\partial t}$$ $$\nabla \times \mathbf{d} = -\frac{4\pi G}{c^2} (\rho\mathbf{v}) + \frac{1}{c^2} \frac{\partial \mathbf{g}}{\partial t}$$

where $\rho$ is the mass density, and $\mathbf{v}$ is the velocity field (c.f. continuum mechanics). This field is the gravitational analogue of the magnetic field in electromagnetism, with the same relation to the gravitational field as the magnetic field bears to the electric field.

GEM/NMG is fully compatible with special relativity in that it is Lorentz symmetric, but it only approximates general relativity, in particular, it fails to meet the full demands of the equivalence principle and thus breaks down when you get to very strong fields like black holes. But I believe it is good enough to account for effects such as the anomalous precession of planetary orbits and other weak-field GR phenomena. Thus gravitation, in fact, does not behave differently after all, at least in the weak-field regime. The Lorentz symmetry mandates a gravimagnetic field, and one exists.

The so-called "Blackett effect" appears to be a hypothesis that you can generate an electromagnetic magnetic field, i.e. a $\mathbf{B}$-field, using only neutral matter. This has been experimentally falsified, apparently - it does not exist. The kind of field generated here is not a magnetic field that interacts with electric charge, but a different kind of field that interacts with mass just as gravitational fields do.

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