This wiki says: https://en.wikipedia.org/wiki/Frame-dragging
Qualitatively, frame-dragging can be viewed as the gravitational analog of electromagnetic induction.
I was wondering what exactly this means, and the wiki for electromagnetic induction doesn't seem to go into it. I wasn't able to Google anything that shed light on this analogy either.
What does this sentence mean exactly? How is gravitational frame dragging, analogous to electromagnetic induction?
The common factor is that the complete interaction includes velocity-dependent effects.
In the case of electromagnetism: the Lorentz force is velocity-dependent.
Newtonian gravity is purely position dependent, and that has implications for what has to be assumed about the speed of gravity.
A theory of gravity in which gravity is purely position dependent, and with a finite speed of gravity, is a theory that does not comply with the principle of conservation of momentum. (The problem is often referred to as 'force aberration'; with a finite speed of propagation (and pure position dependence) the force does not point instantaneously to the point where the attracting body is.)
Maxwell's equations for the electromagnetic field imply a particular speed for the propagation of the electromagnetic interaction: the speed of light.
Given that it was recognized that electromagnetic interaction propagates at a finite speed physicists began to explore possibilities of formulating a theory of gravity with a finite speed for the propagation of the gravitatitional interaction.
The physicists of the time recognized that a theory of gravity with a finite speed of propagation would have the potential of accounting for the anomalous precession of Mercury.
In order for such a theory to comply with conservation of momentum the interaction must include velocity-dependent effects, in just the right way.
The constraint of compliance with the principle of conservation of momentum narrows down the possibilities. In that sense it is very much a guiding constraint.
Einstein's 1915 General Relativity rendered all previous attempts at formulating finite-speed theory of gravity obsolete.
For more information:
1999 article by Steve Carlip:
'Aberration and the speed of Gravity'
Carlip discusses the interconnections between speed of propagation of an interaction, velocity-dependent effects of the interaction, and the emission of waves.
The easiest way to access the idea behind this analogy is to think in 4-vector, we know that the electric and magnetic fields can be represented in one being $A^{\mu}=(\phi/c, \vec{A})$ with $ \vec{B}=\vec{\nabla}\times\vec{A}\;\;,\vec{E}=-\vec{\nabla}\phi-\frac{\partial\vec{A}}{ \partial t} $
If we replace the electric field by the gravitational field ($\vec{g}=-\vec{\nabla}\Phi)$, we need to complete the 4D "representation" for this scalar field by a vector field which is the analog of the vector $\vec{A} $ that is related to the magnetic field.
The analogy and explanations here : https://en.wikipedia.org/wiki/Gravitoelectromagnetism
