Why the Shapiro time delay is calculated with respect to the difference in coordinates and not with respect to the physical distance?(adsbygoogle = window.adsbygoogle || []).push({});

I follow the discussion of Clifford (Theory and Experiment in Gravitational Physics) which uses isotropic coordinates and PPN approximation.

To make more clear my question I consider the simple case of a light ray who travel from a point A with coordinates (t_{A},x_{A},0,0) to a point B with coordinates (t_{B},x_{B},0,0) located near a mass M, so the coordinate time difference is:

t_{B}-t_{A}=x_{A}-x_{B}+(1+gamma)*G*M*ln(x_{A}/x_{B}), if now I consider the same cordinate difference (say x_{A}-x_{B}=x_{C}-x_{D}) in a region where the gravitational field is negligible I have that the excess coordinate time delay is (t_{B}-t_{A})-(t_{D}-t_{C})=(1+gamma)*G*M*ln(x_{A}/x_{B}), that is the correct form of Shapiro time delay who take account for time dilatation and space curvature. In particular the curvature of space should should be expressed by the fact that the same coordinate difference covers different physical distances in different regions of space-time: L_{AB}=x_{A}-x_{B}+gamma*G*M*ln(x_{A}/x_{B}) and L_{CD}=x_{C}-x_{D}=x_{A}-x_{B}. But, if I have no a priori knowledge of General Relativity, how can I build an experiment in which I have the condition x_{A}-x_{B}=x_{C}-x_{D}?

Put in another way if I have a rod of length L_{AB}(i don't think to propagation in an optical fiber but a geodesic path between the ends of the rod) and put it in different regions of space-time I can't put [x][/B]-[x][/A]=[x][/D]-[x][/C] but I must have L_{AB}=L_{CD}so the excess coordinate time delay results (t_{B}-t_{A})-(t_{D}-t_{C})=G*M*ln(x_{A}/x_{B}) so measuring only time dilatation. What is wrong?

I think the answer is that we can not measure the curvature of space using a single rod (it is right?), so we look at the real experiment of Shapiro time delay with planets, Clifford says:

"Since one does not have access to a "Newtonian" signal against which to compare the round trip travel time of the observed signal, it is necessary to do a differential measurement of the variations in round trip travel times as the target passes through superior conjunction, and to look for the logarithmic behavior. To achieve this accurately however, one must take into account the variations in round trip travel time due to the orbital motion of the

target relative to the Earth. This is done by using radar-ranging (and possibly other) data on the target taken when it is far from superior conjunction (i.e., when the time delay term is negligible)to determine an accurate ephemeris for the target, using the ephemeris to predict the PPN coordinate trajectory near superior conjunction, then combining that trajectory with the trajectory of the Earth to determine the quantity |xand the logarithmic term in the equation."_{pianeta}-x|_{terra}

But, again, if I have no a priori knowledge of General Relativity, how is possible predict the PPN coordinate trajectory? If I use General Relativity to predict it, why the measure of curvature is effective and not "redundant"?

Thanks and sorry for my english...

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# Why the Shapiro delay is calculated with respect to the...?

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