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On the Interactions of Point Masses

According to gravito-magnetic theory, the formula for the gravitational field (g) of a point mass (m1), whose motion up until time t < r/c is known, is(1,2)

g(r,t) = Gm1r/(r dot u)3 [u(c2-vret2) + r x (u x aret)].            (1)

In this equation r is the retarded distance from m1 to the point (r):

r = c(t-tr)    (1a)

and u is a utility velocity vector

u = c<r> - v,        (1b)

where <r> is the unit vector of r.

vret and aret are m1’s retarded velocity and acceleration respectively.

Note that r points from the retarded position of m1 to point r.

The formula for m1’s gravitomagnetic field is

O(r,t) = <r>/c x g(r,t).             (2)

The force experienced by a second mass, m2, at (r,t) and moving with velocity v2, is

F = m2(g + v2 x O).        (3)

m1, m2, g and O are mathematically imaginary. Thus the force, experienced by m2 in the field of m1, has a radial attractive component toward m1 and a transverse component found by application of the left-hand rule to v2 x O.

In order to calculate the motion of m2, Newton’s relativistically correct 2nd law must be applied.

F = d(m2v2)/dt.             (4)

In many practical applications where v2<<c,

F = m2a2     (5)

will give satisfactory results.

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1. See INTRODUCTION TO ELECTRDYNAMICS, Griffiths, Second Edition, Sect. 9.2

2. See “Roots of Gravitomagnetic Theory” on this website.