Alpha Institute for Advanced Studies

I thought about this quite a lot over the weekend, and I summed up my thoughts so far in note 189(8), i.e. m is a function of a function, (eq. (1) of note 189(8)) so the chain rules apply as in note 189(8). The key equation is (4), where R is not a function of r, so no chain rule applies. I agree that t and r are linearly independent but as you know, the orbital equation is found as in standard general relativity by the chain rule:

dr / dt = (dr / d tau) (d tau / dr)

It would be very helpful to plot m(r, t) (eq. (1) of note 189(7)) against r, and compare it with a plot of

f(r) = 1 – r0 / r

to see if the two curves can be superimposed by variation of parameters.

The great advantage of this approach is that all orbits can be described in theory by eq. (9) of note 189(7). Naturally I will check all my calculations again today.