The fact that the linearized Ricci tensor can give rise to a wave equation makes the interpretation of two interacting fields, like the electric and magnetic fields of electromagnetism, attractive. This would mean that a neutral mass, just like electric charge, can give rise to a magnetic field when in motion (like electric current), or when it spins (like a magnetic dipole). But, there is no Lorentzian force that would act on the angular momentum of a neutral mass that would be responsible for its precession.
The similarity between a the geodesic equation of a neutral particle moving with a non relativistic velocity and Lorentzian force is striking with the exception that "mass replaces the charge...and the gravitomagnetic field exerts four times the force that we would expect from the electromagnetic analogy." (Moore, A General Relativity Workbook, p. 409.) The analogy with a Coriolis force leads to even more confusion where the non inertial frame is not specified.
A charged particle gives rise to a magnetic moment; a circularly moving mass gives rise to angular momentum. It was initially observed that a neutron has a non-zero magnetic moment indicating that neutral matter can possess magnetic moments. Later, though, it was found that the neutron possesses charges in the form of quarks each with a charge of 1/3. A better example would be a neutrino has angular momenta, but no magnetic moment. It's coupling to the photon is identically zero.
So a neutral mass cannot create a (gravito-) magnetic field. If it could, like that of a neutron, then the magnetic field would be the good old-fashioned one from electrodynamics. The Lorentz force acts only on charged particles so that it is inaccurate to say that its action on the angular momentum vector is responsible for precession. And why would a generalized geodesic equation contain both the Lorentz force and an analogous term with charge replaced by mass?
The working principle of a gyroscope is based on gravity. The tendency of a rotating object to maintain is orientation of rotation is a manifestation of the conservation of angular momentum. For any change in the axis of rotation will cause a change in the orientation in such a way that the angular momentum is conserved. Nature tries to eradicate torques!
Attempting to utilize this analogy it is claimed that "a spinning object (a gyroscope) has an analogous gravitomagnetic moment...By analogy, in a gravitomagnetic field, such a gyroscope should experience a torque...such a torque on a gyroscope causes it to precess around the field in the direction of an angular velocity" which is proportional to the gravitomagnetic field. (Moore, p 410)
However, Moore cinches the argument when he says "we want to find for a spinning object the gravitomagnetic moment that corresponds to the magnet moment iA for a simple loop of area A conducting a current i...Current is defined to be the charge per unit time passing a certain point. By analogy, the quantity that should replace i in the gravitomagnetic moment should be the mass per unit time that passes a certain point. For our hypothetical mass, this is m/T, where T= 2pi r/v is the time to complete the turn, and v is the particle speed around the axis.. The quantity that should substitute A is the area enclosed by the mass's circular orbit."
Charges come in two "flavors", mass in only one. You can't simply replace charge by mass and expect to get things like a Lorentz force, a magnetic field, etc. The geodesic equation, which Moore confuses with the Lorentz force contains a term proportional to the particle's speed, just like in the Lorentz force. But the coefficient is the antisymmetric component of the perturbed metric, h_(ij), where the indices i,j range over the spatial coordinate alone. Recall that in Harris's paper [AJP 59 (1991) 421] we had to throw out the time derivative in the expression for the Christoffel symbol of the first kind in order to get a similar expression in terms of peturbed metric which contain the time component, 0, i.e. h_(0i). Since the time component of the metric accounts for classical potentials, it seems hardly fitting to identify the antisymmetric component containing only spatial indices, h_(ij) with the gravitomagnetic field.
"The so-called Lense-Thirring precession of a gyroscope at a point in empty space provides a practical way to measure both the magnitude and direction of any gravitomagnetic field present at that location." The argument is then confused when Moore uses the earth as an example by considering a gyroscope in an equatorial orbit around the earth. Are we under the impression that the earth is a neutral mass? What about the molten spinning iron core of the earth's interior? The orbiting gyroscop's precession will be easiest to observe when the earth's "spin" lies in the equatorial plane for then it will be perpendicular to the magnetic field. To call the angular speed of precession twice the gravitomagnetic field is ludicrous since how do you distinguish it from the true magnetic field? Implicit is the assumption that the earth is neutral,
There is a second effect called "geodetic precession." Like the precession of Mercury it is proportional to the Schwarzschild radius. This is almost two orders of magnitude greater than the Lense-Thirring effect, and its numerical value had to be assumed by general relativity's value for geodetic precession in the LAGEOS satellites in order to obtain an accuracy of 10% in the Lense-Thirring measurements. And "because of stray charge problems the Lense-Thirring measurement is only good to about +/- 19%. (The same accuracy pertains to the measurement of Avogadro's number using Einstein's theory of Brownian motion of 1905!)
The magnetic moment that makes an angle with the direction of the magnetic field produces a potential energy. This potential energy arises from the fact that the Lorentz force acts on the moving charge. There is no Lorentz force acting on neutral matter. A torque will be exerted on the angular momentum vector, and this is responsible for the precession of angular momentum. This recalls Einstein's claim that you can always add a "smidgen" of charge to neutral matter without changing its properties.
Classically, anything with a a magnet moment should have a spinning charge in a circular orbit. This property of spin was attributed to charged particles like the electron and proton. Yet, you can't explain the existence of a magnetic field by a classically spinning proton. Spin is a quantum effect, and certainly lies outside the domain of general relativity.