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RC;
I think the best way to approach the idea of force is to look at it classically, IOW, classical Fields.
The term "frame dragging" is actually the same a "gravitomagnetism" and is used interchangeably.
So before getting into the almost semantical question of whether a Gravitomagnetic field IS a force; let me ask you an analogous question:
Do we refer to a MAGNETIC field as a force? Do we say a
magnetic field IS A FORCE?Not typically, because many times it does not produce a force. What we do say however, is that
A TEST particle, IF it has CHARGE, and
if it is MOVING through the magnetic field, ...
will EXPERIENCE A FORCE.IOW, for force to develop,
MOTION of a CHARGED particle is necessary; and that motion must have a component transverse to the direction of the field.
....according to the standard Lorentz formula ....(ignoring the electric field)...
F = qV X B .... (where V is the velocity of test charge q; and "X" is the cross product)
LIKEWISE; by analogy; (using the linearized Gravitational equations), the "Gravito-magnetic" field itself is not usually referred to as "a force", since similarly, a test mass must be
MOVING through the GM field in order to
"experience" a force.
....which can be seen by inspection of the analogous Gravitational Lorentz equation ....which (ignoring the newtonian gravity field) is simply...
F = mV X B(g) ... (where V is velocity of a test mass m; and 'X' is the cross product with the GM field, B(g).
Just like in the electromagnetic case where a
stationary charge "feels" no force from a magnetic field, so likewise a
stationary mass 'feels' no force from a Gravito-magnetic field...Motion of the test particle is necessary.
Likewise,... motion
parallel to the field lines will result in no force; a transverse component of motion is necessary.
Now if someone wants to "call" gravitomagnetism a force or not really doesn't matter to me; (you can call it an ice cream cone if you want),
as long as you recognize the above physical relationships.
Lunar