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The Pseudo-Forces Governing the Advance of the Perihelion

The Einstein modification of the Binet equation reads


u" + u = m/h^2 + 3m/c^2 u^2

where the gravitational parameter is m=GM, h the angular momentum per unit mass, and u=1/r. Converting this into a force law, the acceleration is

a = m/r^2 +3 mu w*2 r, (1)

where w is the angular velocity and mu= m/c^2 r, the analog of magnetic permeability. Now the question boils down to why the "3", for without it, it would the sum of Newton's law and the centrifugal force.


If we add on the Coriolis force, viz.,

a = m/r^2 - mu(-wXwXr+ 2 vXw),

where v is the velocity in the rotating reference frame, we get

a = m r'/r^2 + mu(w^2 r'+2v^2 r'/r - 2(dr/dt) v). (2)

If we project the acceleration in the radial direction we come out with Einstein's modification, (1). So the acceleration in (2) is not entirely radial. Taking the curl of (2) gives

curl a = -2 mu w/r (dr/dt).

Since a is the analog of the electric field, Faraday's law reads

dw/dt = curl a =-2 mu w/r (dr/dt). (3)


It has claimed in the Wikipedia article on the Lense-Thirring effect, that unlike the Coriolis force, the last term in (2) "is not fictional, but due to frame dragging induced by the rotating body. So, for example, an (instanteously) radially infalling geodeisc at the equation will satisfy [3]." The velocity dr/dt is identified as the observer's velocity, and if he is stationary, "there is no effect on the observer." That velocity is the velocity in the inertial frame, and the square of the difference betwee the velocities in the rotating and inertial frames is proportional to the centrifugal force.


So the "3" in Einstein's equation was a hint that the modification required the Coriolis force in addition to the centrirugal force. When projected along the radial direction the Coriolis force adds the "2" to the centrifugal force, Moreover, it also leads to the conclusion that (3) is the rate of change of the line of nodes. It is the couple due to the gravitational field acting on the planet at the center of gravity of the gravitational field. Here, there is nothing being "dragged", and it is normal Coriolis force that vanishes in an inertial frame. Finally, equation (3) can be looked upon as the conservation of angular momentum.


[Note that I have not been able to find any confirmation that there is a secular change in the nodes of Mercury.]


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