"Uh oh, something is wrong. The average little tick t=1/c is completely immune to time dilation. An individual clock is ticking always at its nominal rate!" This only holds in the proper frame of each clock. This doesn't hold far as every other clock is concerned. 

Thank you for your attempt at an answer.  It would be a decent answer for a beginner question, but this is a deeper question.

I understand the puzzle is hard to read through and get all the parts in one's head, and it's always possible I have made a simple mistake somewhere (for which I would be grateful for it to be pointed out).  I made a few while writing it.  But as far as I know I caught them all.

WHAT THIS IS NOT ... this is not a beginner relativity puzzle like the Twins problem or something like that.  It is not posed by a beginner, and it is not obvious to a beginner.  It is a very deep puzzle that would only occur to someone very familiar with all the platitudes who stumbled across it while looking for something else.  So please pay close attention to the following paragraph.

I'm quite aware that an individual clock SHOULD tick at its nominal rate ONLY in its proper frame (or actually, any frame of equal potential, but that's not an issue here).  There are several ways to calculate this rate.  The PUZZLE is that I have shown what appears to be a legitimate calculation by another method than the usual direct ones that gives the nominal ticking rate in a different frame of different potential. 

To answer the puzzle, I expect 8 (brief) calculations to be consistent, i.e. to give answers that agree when appropriate transformations are applied.  Let me call the two reference frames E for ephemeral and P for proper.

• The view of (1) total path time and (2) individual clock time from the view of E and P.  We should then be able to transform either view into the other to check the calculation.
• Check that the total time from adding individual clock times matches the directly calculated total path time, and continues to match when the component times are transformed and added.  Remember that the number of physical meter sticks is an objective quantity and thus cannot be transformed.
• Repeat all of the above with lengths.

If it were a special relativity problem, there would be motion involved, and clock skew, and it would be resolved by changes in simultaneity.  The only objective measurements in SR are those done at a point with both observers present.  All remote measurements get transformed by the Minkowski metric.  (The Minkowski length is constant but not  directly measured, so it isn't actually an "objective measurement").

But in GR we can make the whole thing static, creating the possibility of objective measurement which involves both observers traveling around at their leisure counting things which are not changing.

What I have suspected for a long time is that length contraction (or dilation) MUST be associated with time skew, i.e. with loss of simultaneity.  This is something very different than GR clock ticking rates.  It is easy to calibrate a clock's ticking rate and if nothing is moving there are no arguments over simultaneity in GR. 

I just now, however, connected this suspicion about the link between time skew and length contraction with my puzzle, when responding to your answer.  So thank you for contributing to that insight.  Apparently I have discovered an example of my suspicion?

"Let me call the two reference frames E for ephemeral and P for proper." As far as I know, there is no global proper reference frames in curved space-time there are only local ones. Robert, one of the things I learned in getting a degree in physics is that I am and will always will be a beginner. 

Yes, surely about the beginner thing.

Of course there is no global proper frame.  It is an instantaneous frame of an object and proper measurements are integrated over the path.  I just needed something to call it.  Suggest something?  "The integrated results of measurements made in the frame of the object?"  It is kind of a "framework for making measurements" but not a collection of rods and clocks.  Actually, statically it may be?  However, how about if I call them observer E and P.  E remains at +r1.  P travels (slowly) from +r1 to -r1 counting meter sticks. Let R=+r1 - -r1 = 2r1 = R (slightly new definition).

I am gradually simplifying the dilemma and maybe soon will come across the error.  Kind of a "zero in" approach.

E sends light to -r1 and back, so ds=0 and Te=KR/c where K is an integral which is loosely the average time dilation factor.  Exact numerical values are not too important for the puzzle.

E calculates the coordinate velocity of light as CO=R/Te=R/(KR/c)=c/K.  That seems right.

E calculates P's measured distance assuming dt=0 and finds ∫ds=S=KR.  This also seems right.  Hmm, I think I see where I mixed reference frames.

It follows that as P traverses the path, P counts KR meter sticks.  E can observe optically from a distance and must agree with this number.

One can't get a meaningful number by following a light ray in proper time (i.e. Tp_light=0).  But one can add the local stationary observer P times for light to traverse each meter stick.  This is by definition 1/c for each meter stick, so the total is Tp=S/c=KR/c.  This the time that would pass in the collective system of the many P's (one each meter).  It does not look right.  This is also the external interval Te, they are numerically equal.  But more time should pass for observers in the field.

If we take the time interval Te and broadcast it to an observer Pi on the i'th meter stick, where the local time dilation factor is ki, This observer should see Te/ki as the time interval.  If we pick an "average" i then ki=K and this observer sees Te/K. 

Here is the thing.  If the number of meter sticks were NOT KR, but R, then everything would work out.  It so  happens R was the number I got using inertia theory (developed between 2008 and 2011 when it was published in December that year).

As long as the number is KR, then the sum of the parts is greater than the whole.

What I think is that the length of meter sticks is determined by what's called "constructive relativity" not by "spacetime."  In constructive relativity, this would be a wavelength in an atomic bond, or an electromagnetic timing arrangement between two radio stations.  It is easier to analyze the radio stations.  Suppose they are 1 meter apart.  Do everything in E's coordinates.  They communicate using light which is slower by K using clocks slower by K and thus the distance per tick of their local clocks is unchanged between E and P coordinates.

Now there is no DIRECT argument between that and a meter on a GR path.  However, there is a relationship between the times of the parts and the time of the whole path, which is like another equation constraining the flexibility of what geometric "scale" can be chosen.  This constraint forces R=S and we may not really choose the Schwarzschild coordinates. 

Since they were rigorously derived from GR which was rigorously derived from its assumptions, then an assumption is incorrect.  It isn't likely equivalence because 100 years of scrutiny have upheld it.  However, the geodesic assumption has not had much scrutiny.  Are there any other assumptions in GR?

"Uh oh, something is wrong. The average little tick t=1/c is completely immune to time dilation. An individual clock is ticking always at its nominal rate!" I'm not certain I understand this. A clock will always tick at it's nominal rate in it's proper frame. However, if observed by an observer at a different R it will appear to tick faster or slower relative to the observers clock personal clock depending on relative R of the observer. Same holds for meter sticks. In their proper frame they are always a meter long. Relative to an observer at a different R they appear shorter of longer depending on R. This leaves me wondering about the correctness of adding up the "proper" measurements as you seem to be doing. I think using proper times and length globally leads to puzzles and perhaps should within GR. 

Yeah, obviously something is wrong.  You are still looking at the original question, but in my last post I simplified it a lot, and I think now I can do MUCH better.  Try this ...

Suppose the coordinate distance in A's frame from A to B and back is R.  A's radius is greater than B's.  Then ∫ds=S=KR where K>1.  So KR  meter sticks can be laid from A to B and both parties will agree.

The coordinate velocity of light through each of A's coordinate meters is reduced by K in A's frame.  This can be determined by dividing the coordinate distance R by the transit time in A's frame KR/c giving R/(KR/c)=c/K. 

Time dilation for a proper 1 meter light clock is increased by K as viewed by A.  This follows from time dilation alone without reference to length changes.  But this is also the time dilation for A's coordinate meters, so A's coordinate meters must equal the proper meters.

Oh my ...

I think that the scale of local coordinates is arbitrary if one considers only one path segment, like a meter.  The geodesic assumption removes this flexibility by requiring a scale choice that points the curvature tangent in the direction of motion derived from equivalence.

However, requiring that the whole equals the sum of the parts places also another constraint on the scale.  So it is the geodesic assumption that is wrong!

Robert:   I believe that I have identified the basis of the problem that you explain. Since this involves some equations, the answer is contained in the PDF attached below. Please excuse the fact that I did not use the same symbols as you. The answer was drawn from an earlier work and I did not take the time to change symbols.

John's answer is very interesting.  It can be quickly summarized.  He calculates the coordinate velocity of light as being proportional to K2 rather than K.  He is using Γ=1/K and r=s and dR = my dr.

So basically he is saying my first proposition is incorrect: E sends light to -r1 and back, so ds=0 and Te=KR/c should be Te=K2R/c.  It is usual for a mistake to be in the early assumptions as they are closest to the imprecise intuitive portion of the idea.

What John is pointing out is basically SHAPIRO DELAY.  The tests have been done many times in the solar system with ever more refined accuracy.  Radar sent near the sun should be delayed by K2 instead of K.  Given K is a number close to 1, and the effect we are looking for is the difference between K and 1, then the effect is doubled.  Not easy to miss.

BUT ... the formulas for Shapiro delay are obscure, i.e. hard to trace back to basic relativity parameters. It should provide direct verification of curvature, since it is proportional to Γ squared (i.e. curvature + time dilation). I saw somewhere an estimate of 200 microseconds for the deviation from Newtonian calculated round trip delay, but when I calculate it I get values in the range of 90-270 nanoseconds depending on whether I use full GR or only time dilation. I must be doing something wrong (by three orders of magnitude) or else there was a typo in the number I found.

I have Cliff Will's book on verification of metric gravity theories. It only compares one metric theory to another. It is hard to follow the meaning of all his "parameters." By comparing his analysis of bending to delay, I it looks like maybe the square factor is missing, but I really cannot determine that for sure, and it is hard to believe anyway since some random person caught a similar error in some calculations I posted this week (i.e. someone would have caught it). These tests tend to generate a lot of controversy and naysayers even within the establishment.

Charles, thanks.  After adjusting to your notation, I see 1/Γ2 in the integral.  That was buried too far back to trace in Will's book. 

Also this morning I am thinking clearly for the first time about double bending, which is very embarrassing because I have been thinking about it for a long time.  It seems the double bending also implies coordinate velocity c/Γ2.  So both Shapiro and double bending verify "curvature" when seen this way.