Shapiro delay

It's not just the spatial distortion (more distance), but also gravitational time dilation (slower light according to distant clock).

 

No, it is sort of either/or, take your pick. The invariant observation is the delay. Pick one set of coordinates, and you have pure distance distortion; pick another and you have pure coordinate lightspeed change; or pick a mixture with funky coordinates.

 

Posts [edit: removed individual post #'s]

Posts here also give some nice insights:

Energy shift in a photon passing by a gravitational well
https://www.physicsforums.com/showthread.php?t=690006&highlight=shapiro+delay

Like Pallen just posted here in #7 while I was typing....

 

What about Schwarzschild coordinates?

 

MTW works out the shapiro delay on pg 1107 in PPN coordinates. These are basically the same as isotropic coordinates (not the usual schwarzschild) - though the surfaces of time simultaneity will be the same for isotropic and schwarzschild coordinates.

They use the cartesian form of the isotropic / ppn coordinates (dt, dx, dy, dz).

MTW's derivation is approximate - they spend some time explaining why assuming light travels in a straight line in PPN coordinates (rather than the actual curved and not time-invariant path it actually follows) gives small errors, due to Fermat's principle.

The basic answer they get with these approximations is:

[tex]
t_{TR} = \int_{-a_t}^{a_r} \left( \frac{g_{xx}}{-g_{tt}} \right) ^ {\frac{1}{2}} \, dx
[/tex]

where ##-a_t## and ##a_r## are the x-coordinate positions in the ppn system. (There is a parameter b in the result which is the constant y-coordinate in the ppn system as well). These are hard to observe, a later formula eliminates them by considering the rate of change of the Shapiro delay.

I would say that the answer that bests fits the above derivation is "both". Certainly we see ##g_{tt}## there, we also see the spatial terms ##g_{xx}##. It's a bit hard to summarize this in the simplistic way the thread is being discussed :-(.

[add]
Probably the best simple answer is not an answer, but another question, i.e. to ask "what do you really mean by distance? in GR? How are you measuring it?".

There is certainly no conflict with the notion that the notion of proper distance is based on a constant speed of light though, which seems to be the thrust of the question. But the details of carrying out the calcuation (even with an approximation) typically involve setting up a coordinate system, using changes in coordinates (dx - which isn't a distance), and calcuating the proper time by using the coordinate change dx and the non-constant coordinate speed of light (the number in the square root).

MTW makes a few more approximations, and subtract out a Newtonian part, and get an answer that depends only on the ppn parameter ##\gamma##

There may or may not be other ways of calculating it, this is what my textbook does though.

 

That was my point.

 

In SC coords, coordinate light speed plays a role. But in radar coordinates, the same prediction is purely from spatial distortion.

 

Personally, I think of distance informally as the length of the shortest curve connecting two points - more formally this mutates into "the greatest lower bound" or infimum of the set whose members are the lengths of all curves connecting two points. (We replace the concept of minimum for a finite set with the concept of the infimum which applies to finite and infinite sets, the two concept are equivalent for finite sets). In the context of physics, we also require that the curves be in a hypersurface of constant time, this construction then gives "distance as a function of time".

When one computes this distance by means of coordinates, we can see from the above example that the coordinate expression of gravitational time dilation and the coordinate expression of spatial curvature both play a role in computing the distance in that coordinate system. However, this should not be interpreted as the local speed of light varying - the local speed of light is always equal to c. Furthermore we note that the distance (by this definition) really does increase when the connecting curve passes near a massive object - again the point is not to "throw away" the fundamental notion that the speed of light when measured locally is a constant.

 

The horizontal coordinate velocity of light (travelling at constant radius from the gravitational centre) is slower than the local horizontal speed of light, by a factor of sqrt(1-2m/r) in the Schwarzschild metric, which is purely due to gravitational time dilation. The vertical component of the coordinate speed of light that is slower by a factor of (1-2m/r), which you can think of as a slow down due to gravitational time dilation by a factor of sqrt(1-2m/r) and an additional factor of sqrt(1-2m/r) due to spatial distortion.

 

Yes, according to a local clock. But the whole point of the Shapiro delay is what duration a distant observer measures with his clock.