In general relativity the local speed of light is a constant and has the usual value $c$, but the speed of light that we measure from here for a part of space over there (called the coordinate speed) may differ from the accepted value.

This is one way to structure arguments about gravitational red/blue shift and the curvature of light paths relative coordinate systems fixed to a particular observer. It is a common way of explaining the Shapiro delay.

Indeed in that kind of context this point of view is successful enough that it is tempting to take it as definitive. To say

Not being a relativist in any serious way myself (my sole course on general relativity is more than twenty years in the past!) I've been wondering about how that notion gets along with the equivalence principle.

Two (identical) space craft are commanded from their geometric centers and also feature a pair of transverse light clocks a height $L/2$ "above and "below" the cockpit. While accelerating the "high" clock should accumulate more time than the "low" clock just as they would if the ship was grounded upright on Earth.

Now we imaging these two craft hurtling toward each other while thrusting to reduce their closing velocity in such a way that they arrive at relative rest just as their cockpits come alongside one other.

The two cockpits momentarily share a single co-moving frame of reference. However, each will report a different expectation on the relative timing of the clocks; this has been true all along, but now the high clock in craft $A$ is beside the low clock in craft $B$ and vice versa so the comparison is between the same two regions in space.¹

If we subscribe to the point of view that the variation of the coordinate speed of light has some absolute meaning then this appears to represent a paradox, while viewing differences in the coordinate speed of light as having only relative consequence would seem to have no problem with the described scenario.

Do scenarios like this provide a reason to prefer one ontology of the coordinate speed of light to another? Or is there something that I am missing (that is: is there a correct I'm failing to make that prevent the "paradox" from coming up in the first place)?

In general relativity the local speed of light is a constant and has the usual value $c$, but the speed of light that we measure from here for a part of space over there (called the coordinate speed) may differ from the accepted value.

This is one way to structure arguments about gravitational red/blue shift and the curvature of light paths relative coordinate systems fixed to a particular observer. It is a common way of explaining the Shapiro delay.

Indeed in that kind of context this point of view is successful enough that it is tempting to take it as definitive. To say

Not being a relativist in any serious way myself (my sole course on general relativity is more than twenty years in the past!) I've been wondering about how that notion gets along with the equivalence principle.

Two (identical) space craft are commanded from their geometric centers and also feature a pair of transverse light clocks a height $L/2$ "above" and "below" the cockpit. While accelerating the "high" clock should accumulate more time than the "low" clock just as they would if the ship was grounded upright on Earth.

Now we imaging these two craft hurtling toward each other in deep space while they both employ a stead thrust to reduce their closing velocity in such a way that they arrive at relative rest just as their cockpits come alongside one other. At this instant, the two occupants of the two craft momentarily share a single co-moving frame of reference.

Naive "paradox"

However, occupants of each craft will report a different expectation for the relative timing of the clocks. In particular occupants of craft $A$ see clocks $A_\text{high}$ and $B_\text{low}$ as above them and therefore running fast while clocks $A_\text{low}$ and $B_\text{high}$ are below them and therefore running slow. Occupants of craft $B$ of course have the opposite expectations.

Inertial view

Of course the occupants of both crafts are in non-inertial frames, an observer floating freely nearby will report that both ships exist in a flat space-time for which the coordinate speed of light is everywhere equal to the local speed of light.

What's the point

In my naive view this scenario demolishes claims that the coordinate speed has some absolute significance because

• It arranges a paradox if we believe in absolute significance of the coordinate speed.

• Viewing differences in the coordinate speed of light as having only relative consequence would seem to have no problem with the described scenario.

Is this a sustainable conclusion or is there something that I am missing (that is: is there a correction I'm failing to make that prevent the "paradox" from coming up in the first place thereby leaving the ontological question unresolved)?