note 420(3)

Checking note 420(3)

This general Cartesian calculation looks to be very useful. Eqs. (15) to (21) are OK. Maybe these should be run through the computer to double check. Eqs. (15) to (17) are definitions, and Eq. (18) differentiates Eq. (17) on both sides. Eq. (19) is a change of variable. Eq. (20) is dimensionally correct, inverse seconds on both sides. The dimensions of Eq. (21) are correct, kgm m squared per second.

For cartesian coordinates the equations of motion can be derived for a fully general m(X,Y). It is to be considered however that this was derived from the spherically symmetric spacetime so it will make sense only for a form

m(r) = m(sqrt(X^2+Y^2)).

Several approximations are possible.
Case m(X,Y)=mu including original gamma (with mu):

Case gamma=1/sqrt(mu):

Case m(X,Y)=1/gamma^2:

The second case is the effective mass approximation with relativistic terms. The equations can be written in vector format, will do this for section 3 of the paper.


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