Gravitational analog of electromagnetic force

Is there a gravitational analog of electromagnetic force?
[Apart from the obvious "static" forces, ie electrostatic between fixed charges and gravitational between fixed masses.]
I am thinking of the classic situation of a moving charge (or current) creating a magnetic field which then affects other moving charges. So a question by analogy, does a moving mass have an additional field which is only detected by other moving mass ? It would be very very small I guess.

Are you trying to find things that the two types of force have in common? Google is your friend here. I tried "compare and contrast gravitational and EM forces" and there were loads of hits.
I don't think you can expect to get much out of PF responses to your question because any answers are going to be divergent (most likely correct but divergent). I think you will need to read around by yourself and come to your own conclusions. I'm not trying to dodge the question but from the way it's presented , I don't think you are sure what you actually want to know. Reading around can help you resolve this.

There's 'teleparallel gravity', which is still hypothetical. IIRC, it reckons space/time twisted up one way gives our familiar EM, in another gives gravity. As yet, the different 'flavours' are mutually exclusive...
Sadly, beyond a fascinating article in New Scientist a couple of years ago, I've seen nothing about it that was comprehensible without serious math...

Gravitational analog of electromagnetic force was proposed by Oliver Heaviside in 1893 as gravitoelectromagnetism: https://en.wikipedia.org/wiki/Gravitoelectromagnetism
Now it is seen as the first correction to Newton toward general relativity, as magnetism in EM necessary for Lorentz invariance, directly confirmed by Gravity Probe B.

Thanks everyone ! Well appreciated.
I (now) wonder if anything with a non-zero spin can be regarded as "fixed" apart from approximation. But that is a whole different question and topic. Thanks again

I think a traditional view is that gravity does not have "positive" and "negative" masses. These two are required for an electric current to create a magnetic field via Relativity (see Electricity and Magnetism by Purcell).

But the switch of sign between Coulomb and Newton seems a very difficult problem (I am recently thinking about) - having two masses as sources of curvature, taking them closer total energy should grow, what means repulsion.

E.g. in liquid crystals they get Coulomb-like interaction between topological charges:
https://pubs.rsc.org/en/content/articlelanding/2019/sm/c9sm01710k#!divAbstract
https://journals.aps.org/pre/abstract/10.1103/PhysRevE.76.011707
https://www.nature.com/articles/s41598-017-16200-z
Below is example calculation of effective Coulomb potential for such two topological charges in various distances ... but how to switch sign for Newton?
Is there such calculation of approximately Newton effective potential from general relativity Einstein-Hilbert action (->Hamiltonian)?

Can one express some GR scenarios as having a "gravitoelectric" and "gravitomagnetic" part? Yes, Is it helpful in making calculations? Almost never.

"Gravitoelectric" is Newton force approximation, "gravitomagnetic" are the first order corrections (necessary for Lorentz invariance): frame dragging, Lense-Thirring, e.g. tested by Gravity Probe B: https://en.wikipedia.org/wiki/Gravitoelectromagnetism
The big question are higher order corrections, their experimental verification.

Simple simulator for Kepler problem e.g. around rotating black hole with GEM approximation: https://demonstrations.wolfram.com/KeplerProblemWithClassicalSpinOrbitInteraction/


I have just found calculated Hamiltonian for Einstein-Hilbert (far nontrivial): page 24-27 of https://arxiv.org/pdf/2204.03537
But I cannot find further derivation of approximately Newton effective potential like above: place two masses in various distances, calculate energy-distance dependence from integration of Hamiltonian (?)