Problem: Assume (don't question) you know for certain that F=maμ(aa0),
where μ is an unknown function. (I know how you feel, but put those feelings aside and think about it as a solely math problem😅)
Your goal is to determine the unknown function μ.
Another assumption is that a0 is found by the following:
In an expanding Universe ds2=−c2dt2+R2(t)dx2,
(note that as a special case, for R(t)=1 this yields the Minkowski metric)
Then a0 is just the acceleration of null rays of the above metric, meaning that if you let ds=0, then cdt=±R(t)dx so dxdt=±cR, then a0:=d2xdt2|t=0.
The simplest approach to solving this is to recall that, if we define Force as F=md2xdτ2, where dτ=dsc, by chain rule we have d2xdτ2=d2xdt2(dtdτ)2+dxdtd2tdτ2,
So by comparison with the assumption μ=(dtdτ)2,
but this shows the problem: If we use the metric above, dtdτ is not a function of acceleration.
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