Isaac Newton thinking about gravity
Given a point charge q whose motion up until time t is known, the formulae for its electric and magnetic fields (at points other than that occupied by the charge itself) are well known and specified in reference 1. Thus we can theoretically specify the formulae for the gravitational and gravitomagnetic fields of a point mass by making the appropriate substitutions. For the gravitational field we have
g(r,t)=Gmρ/(ρ dot u)3[u(c2-vret2)+ρ x (u x aret)]. (1)
In this equation ρ is the retarded displacement from m to the point (r). Its magnitude is
ρ = c(t-tr). (1a)
u is a utility velocity vector
u = c<ρ> – v, (1b)
where <ρ> is the unit vector of ρ.
vret and aret are m’s retarded velocity and acceleration respectively.
Note that ρ points from the retarded position of m to point r.
The formula for m’s gravitomagnetic field is
O(r,t) = <ρ>/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)
m, m2, g and O are mathematically imaginary. Thus the total force, experienced by m2 in the fields of m, has a radial attractive component toward m’s retarded position, and a transverse component found by application of the left-hand rule to m2v2 x O.
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1. See INTRODUCTION TO ELECTRODYNAMICS, Griffiths, Second Edition, Sect. 9.2.