I want to calculate the difference in time measured by a clock at on earths surface (r=6000km), and a geostationary satellite (r=26000km). My approach is as follows:
For simplicity, we consider curves of the form x(phi) = (t(phi), r0, pi/2, phi), where r0 is some constant (I am parametrizing the worldline using phi). I plug this into the geodesic equations for the Schwarzschild metric to solve for t(phi) in Mathematica:
In[506]:= (*define coords, path *)
x = {t[\[Phi]], r0, \[Theta]0, \[Phi]};
g = DiagonalMatrix[{-(1 - Rs/x[[2]]), (1 - Rs / x[[2]])^-1, x[[2]]^2,
x[[2]]^2 Sin[x[[3]]]^2}];
gi = Inverse[g];
christoffel = Table[Sum[(1/2*
gi[[\[Lambda]]][[\[Sigma]]]*(D[g[[\[Nu]]][[\[Sigma]]],
x[[\[Mu]]]] + D[g[[\[Sigma]]][[\[Mu]]], x[[\[Nu]]]] -
D[g[[\[Mu]]][[\[Nu]]], x[[\[Sigma]]]])), {\[Sigma], 1,
4}], {\[Lambda], 1, 4}, {\[Mu], 1, 4}, {\[Nu], 1,
4}] /. {\[Theta]0 -> \[Pi]/2} // FullSimplify;
(*geodesic equations*)
Table[D[x[[\[Mu]]], {\[Phi], 2}] +
Sum[christoffel[[\[Mu]]][[\[Sigma]]][[\[Rho]]] D[
x[[\[Sigma]]], \[Phi]] D[x[[\[Rho]]], \[Phi]], {\[Sigma], 1,
4}, {\[Rho], 1, 4}] == 0, {\[Mu], 1, 4}] // FullSimplify
Out[510]= {(t^[Prime][Prime])[[Phi]] == 0, Rs + ((r0 - Rs) Rs Derivative[1][t][[Phi]]^2)/(2 r0^3) == r0, True, True}
Solving the second equation gives t[\[Phi]] = Sqrt[2 r0^3/Rs]\[Phi] + t0. When I use this to find the time difference, I get a change that amounts to a difference of about 10hrs per day between the two. This cannot be correct. I'm not sure how to proceed, advice is much appreciated! Thanks.
