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# The Pseudo-Forces of General Relativity

Updated: Jul 21

It has been known since Forward's work that centrifugal forces are candidates for anti-gravity since they act in the same line of the radius but in opposite directions. It is, however, not know that the Coriolis force is what is behind the Lense-Thirring "inertial dragging", which is actually an oxymoron since the frame in which it act is non-inertial.

General relativity contends that masses follow geodesics, which, because, the acceleration is the covariant derivative of the velocity, contains "extra" terms that depend on curvature and motion of the mass. If we consider the winds above a rotating earth, then the centrifugal force will affect the curvatative of the wind, while the Coriolis force will modify the direction of the wind motion. In terms of non-Euclidean geometries the latter is analogous to the Poincare disc model of hyperbolic geometry, where the inhabitants of the disc are ignorant to where there are on the disc because their measuring rods and clocks change with the dimensions of the bloke who is using to measure his time and position. What changes is their ratio when the Coriolis force is at work.

If, on the other hand, the rotation is non-uniform then the Euler force enters which is the moment of the time-rate-of-change of the angular velocity vector. This is a completely tangential force. If gravity is a radial force then this force will not have an effect or be affected by it.

In a previous blog, we showed that the electric vector of Ampere's law is the sum of a radial contraction in the direction of motion and the Coriolis force projected in the radial direction. This is equivalent to the centrifugal force which acts radially. The centrifugal force affects tcurvature and is natually position dependent. Since general reativity confuses geometry with gravity, this would be all that we may hope to describe the tractories with. Yet, the inertial frame dragging involves the Coriolis force. For it considers the vector potential to be the moment of angular momentum

A = 2 r^2 w X r'/r^2,

where w is the angular velocity vector, r' is the unit vector in the radial direction, and the 2 is thrown in for the spin of the putative graviton. The magnetic field is the curl of A

B = curl A = 2 (w -3(w.r')r').

In the weak field limit, therefore, the expression has the form of a magnetic dipole. But there is a catch here insofar as w.r' =0 so B=2 w, which is the usual result. (The proportionality constant would be the charge to mass ratio which we neglect.) In fact this realization led Mawell to his vortex theory of electromagnetism. The ratio w/r is constant in time, and the velocity of the vortex decreases as 1/r.

In the weak field limit, the general relativistic geodesic equation D u/ds =0, where is u a four-vector reduces to the Lorentz force

E = m(-1/r^2 + (dr/dt) X 2 w)

noting that the particle velocity (dr/dt) is used in the definition and not the velocity vector v. But this makes no difference for the transverse component of the force, -2m (dr/dt)v/r^2, pointing the v direction. The velocity v enters into the definition of the angular velocity as the curl of v.

Taking the curl of E gives

curl E = -2(dr/dt) w/r,

which is proportional to the vorticity vector. The rate of change of the vorticity vector is

dw/dt = v X curl w= -(dr/dt) w/r

since the dot product v.w=0. Combing the two equations we get

dw/dt = curl E

which is none-other than Faraday's law of induction.

This presupposes that the field is (1) non-uniform like that of a carousel that is starting up-or slowing-down, and (2) the gravitational field is no longer radial, but has a component pointing in the direction of the velocity v. Faraday's equation is transverse; the curl extracts out the non-radial component of the electric field, which, in our case is the gravitational field.

The effect of frame-dragging of an inertial frame is none other that the application of the Coriolis force to a mass orbiting a spinning body. This alters the trajectory of the mass, but it has nothing to do with curvature which is what the centrifugal for does. The projection of the Coriolis force in the radial direction results in the centrifugal force, and we have no reason to believe that gravity has anything but a radial component.

In relation to Newton's bucket experiment in which the bucket filled with water is suspended on a rope which is then twisted. When it is released the bucket rotates and the water will eventually rise on the walls of the bucket. The depression depends on where you look in the bucket and how fast the bucket is rotating. It does not depend on the velocity of the rotating water relative to the bucket. Hence, Newton concluded that rotation can be determined absolutely, and not relative to other bodies. The depth of the depression can be likened to the action of gravity which is really the action of the centrifugal force, hence, the name anti-gravity.

If a mass is suspended on a string, and enclosed in a mass shell, then if the shell is set spinning, the rod will be "dragged" along with it. But, the velocity at which the rod rotates will be different than the velocity at which the mass shell rotates. In all models that have been constructed from general relativity, beginning with Godal's model of a spinning universe in 1949, show that there is a detectable corotation of the rod inside the spinning mass shell. This violates Mach's principle which claims that no corotation should be detected. The effect measured is due to the Coriolis force which depends on the velocity at which the rod rotates.

However, if gravity is truly geometry then the curvature can be modified by the centrifugal force which acts in the radial direction and not along a transverse direction. The Coriolis force imparts an acceleration perpendicular to the radius and thereby results in a tangential component to the force. The non-uniform rotation results in the time rate of change of the vorticity vector which is a tangential force.

To associate these tangential components with the action of gravity is a leap in faith, and so far, there are no experiments that have shown that gravity acts in such a manner. No Faraday equation no gravitational induction, and consequently no gravitational waves