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

[Chhhiral]

Why the Shapiro time delay is calculated with respect to the difference in coordinates and not with respect to the physical distance?
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 |x_pianeta-x_terra| and the logarithmic term in the equation."
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...

 

[Chhhiral]

Sorry, I wrote: "... 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..."
I wanted write: "... and put it in different regions of space-time I can't have x_A-x_B=x_C-x_D but I must have..."

 

[harrylin]

In order to help this thread start, here's my 2 cts (I only have a qualitative/approximate understanding of GR and I don't know what PPN means).

I think that you more or less answered your own question.Locally measured (with a local ruler and a local clock), the speed of light is always c.

The most obvious thing that we can measure is differences in transit times along different trajectories. It's similar to the bending of light near the Sun, which is also determined by comparing light that closely passes near the heavy mass with light that passes at a great distance from that same mass.

 

[Chhhiral]

Thanks for your answer!
You are right, but this doesn't answer to my question. When you consider the bending of light you can compare (for example) the angular separation between two stars when their positions in the sky are far from the sun with the angular separation when theirs are near to it. So that you can measure the fact that light bending exists and that his value is (1+gamma)/2 times the value predicted using only the equivalence principle. The point is that you can compare two values of a local misurable quantity: the angular separation of the stars. You need no a priori knowledge of General relativity to measure this values!
What about the Shapiro time-delay experiment? If I have no a priori knowledge of General Relativity, how is possible to predict the coordinate trajectory?
If I try to answer to this I say: I can't because coordinates have no meanings at all, the only things that have physical meanings are physical distances. So I say: I can compare physical distances, but this is wrong because in doing so the rsults of measurament of Shapiro time delay should be time dilation only and not space curvature.
The only reasonable answer is: if I performed measurement (for example) on the solar system using only local measurable quantity without the help of General Relativity I mistake the value of the physical distances by an amount that is exactly the difference between physicall distances and coordinate distance when I use isotropic coordinates. The question is: where does the physical meaning of this apparent coincidence arise when I use isotropic coordinates?

P.s.: you are right again, I have not specified the meaning of PPN.
PPN means Parametrized-Post-Newtonian approximation.
"The PPN formalism constitues a method for obtaining corrections to the Newtonian motions of a system, resulting from the metric theory in question, to higher orders in G*M/r." (Phy.Rev. D 91, 064041 (2015).
In order to describe the motion of light ray in the PPN approximation you can write the line element as ds² =-(1-G*M/r)dt²+(1+gamma*G*M/r)dx*dx,(where I use isotropic coordinates and c=1).
The gamma factor takes account for space curvature, different metric theories predicts different value of gamma,General Relativity is gamma=1, no curvature is gamma=0.

 

[Chhhiral]

Sorry again, I wrote: "...his value is (1+gamma)/2 times the value predicted using only the equivalence principle."
I wanted write: "...his value is (1+gamma) times the value predicted using only the equivalence principle."

 

[WannabeNewton]

Read the paragraph at the top of p.174; it addresses your concern.

To summarize: at the PN order that Shapiro delay is calculated, the difference between the round-trip coordinate time and the proper time as measured by a local clock is a trivial effect that can easily be accounted for in PN theory.

 

[Chhhiral]

Read the paragraph at the top of p.174; it addresses your concern.

To summarize: at the PN order that Shapiro delay is calculated, the difference between the round-trip coordinate time and the proper time as measured by a local clock is a trivial effect that can easily be accounted for in PN theory.

It is not my question... As you said this is a trivial effect.
Thanks for answered me, however.

 

[WannabeNewton]

It is not my question...

Your question was how can we compute the round trip coordinate time interval (which is itself defined in terms of the coordinate distance) without already knowing the details of coordinates/space-time metrics in PN theory i.e. how can we compute it using only local measurements; the point is we don't. The Shapiro delay is computed using the proper time interval measured by a local clock on the Earth. Cliff Will is simply ignoring the difference between the proper time interval and the coordinate time interval for clarity in the calculation.

 

[Chhhiral]

Your question was how can we compute the round trip coordinate time interval (which is itself defined in terms of the coordinate distance) without already knowing the details of coordinates in PN theory i.e. how can we compute it using only local measurements. The point is we don't. The Shapiro delay is computed using the proper time interval measured by a local clock on the Earth. Will is simply ignoring the difference between the proper time interval and the coordinate time interval for clarity in the calculation.

You are right but

Your question was how can we compute the round trip coordinate time interval (which is itself defined in terms of the coordinate distance) without already knowing the details of coordinates/space-time metrics in PN theory i.e. how can we compute it using only local measurements; the point is we don't. The Shapiro delay is computed using the proper time interval measured by a local clock on the Earth. Cliff Will is simply ignoring the difference between the proper time interval and the coordinate time interval for clarity in the calculation.

What you say is: RoundTripProperTime=(1-G*M/x_A)*RoundTripCoordinateTime, as you said this is trivial. your answer about the difference between proper time and coordinate time difference. My question is about Shapiro time delay and geometrical meaning of isotropic coordinates.

 

[Chhhiral]

Maybe my question is not clear, but my English does not allow me to rephrase it in a clearer way. But I can do another question that can help me to another way:
If you try to measure the position of Venus without the help of general relativity, only being on Earth, how wrong?

I find that the error is exactly the difference between the value of the coordinates diference and the value of the physical distance.

Is this correct?

If it is, of course, this does not happen with all coordinate systems but only with isotropic coordinates.

Can someone tell me a paper or a book in which is discussed the geometrical meaning of isotropic coordinates?

 

[Chhhiral]

Maybe my question is not clear, but my English does not allow me to rephrase it in a clearer way. But I can do another question that can help me to another way:
If you try to measure the position of Venus without the help of general relativity, only being on Earth, how wrong?

I find that the error is exactly the difference between the value of the coordinates diference and the value of the physical distance.

Is this correct?

If it is, of course, this does not happen with all coordinate systems but only with isotropic coordinates.

Can someone tell me a paper or a book in which is discussed the geometrical meaning of isotropic coordinates?

Sorry: I find that the error is exactly two times the difference between the value of the coordinates diference and the value of the physical distance.

 

[CycoFin]

I have noticed there are at least two ways to calculate Shapiro delay

1) delay = time_from_null_geodesic - (spacelike path / c)
2) delay = time_from_null_geodesic - (Newtonian path / c)

Addition to these are Earth time vs coordinate time and geometric (ligth bends) terms but we can assume these zero on solar system.

IMO case 1 only shows that light does not follow spacelike path and case 2 is correct way to calculate Shapiro delay.