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Suppose there are 2 ships which keep time using atomic clocks. The atomic clocks are the same build so it is know that the two keeps keep proper time at the same rate. Suppose that the two ships travel on arbitrary paths (that is they travel in all 3 dimensions and with continuous but arbitrary velocities and accelerations) relative to each other. But the ships are able to communicate with each other (of course with communication delays due to speed of light propagation). I suppose it should also be possible that the two ships can make measurements of the position of the other ship using, e.g. visible light emitted by the ship or lidar type measurements. Of course these measurements are also limited by the speed of light so there are challenges here also.

My question is, is it possible for the two ships to build reliable records of the time recorded on the other ship.

Suppose for the sake of simplicity that the two ships start together and synchronize their clocks before going out on their arbitrary trajectories. The question is, if they meet again, at a non-premeditated location, would it be possible for ship A to correctly predict the time on ship B at their next meeting and vice-versa using the measurements and communications described above?

What if the two ships do not start out together and never synchronize their clocks but only broadcast their current times when they come in communication range?

What extra wrenches does general relativity throw into this? What if one of the ships travels close to a black hole? Is there a way for the other ship to properly track those effects?

This question is motivated by wondering how a relativistic multi-planetary civilization could agree on something like a "universal coordinated time" in spite of some of the challenges posed by relativistic travel.

Jagerber48
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    If your question boils down to the last paragraph then it's answered by Can you calculate unix timestamps (universal clock) from any planet? – John Rennie Feb 01 '23 at 05:57
  • I'm not totally sure what you're asking here. The ships are (carefully) exchanging time signals, so they know what each other's clock says, once they account for the various time delays (& compensating for time dilation) in the signal exchange. OTOH, predicting the other ship's clock reading without such signal exchanges would be extremely difficult, requiring high precision data on the masses and trajectories of all significant gravity sources near the paths of both ships. – PM 2Ring Feb 01 '23 at 09:41
  • What kind of atomic clock are the ships using? The uncertainty of a standard caesium beam is ~1e-13, but a caesium fountain is ~1e-16, and state of the art clocks are ~1e-18 and better. I have a table here. For info on how JPL convert Earthbound UTC timestamps to Barycentric Dynamical Time (of the comoving frame of the Solar System Barycentre) see section 2.3 of The JPL Planetary and Lunar Ephemerides DE440 and DE441. – PM 2Ring Feb 01 '23 at 09:56
  • @PM2Ring yes if each ship broadcasts it’s time that helps buts it’s still challenging because I don’t think the time of flight delay for each time transmission is trivial to calculate since the distance measurements are also delayed. And what if they only transmit one synchronization signal at the beginning? Can one ship predict the other’s clock just by observing (and making measurements of) the other ship? – Jagerber48 Feb 01 '23 at 13:57
  • I think I. Absence of gravitational sources it’s not necessary for each ship to broadcast its time since time dilation can be determined from the relative position measurements. But if gravity is around then each ship needs to use the excess time dilation broadcast by the other ships clock as a local probe of space time curvature. – Jagerber48 Feb 01 '23 at 15:17
  • Oh, it's highly non-trivial. Even if you're just using caesium beam clocks. You might like to read about the synchronisation protocol they used in the Tokyo Skytree experiment, which measured gravitational time dilation over a vertical distance of ~450 m using Sr-87 atomic lattice clocks, which have uncertainty of ~1e-18. – PM 2Ring Feb 01 '23 at 16:08
  • @PM2Ring yes, gravitational redshifts have been measured at much smaller length scales with better clocks now (https://www.nature.com/articles/s41586-021-04349-7). So I guess my response to "I'm not totally sure what you're asking here" is I'm asking about all these non-trivialities. – Jagerber48 Feb 01 '23 at 16:24

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Yes. When when $A$ meets $B$ after any amount of travel, all of $B$'s past exists in $A$'s past light cone, and thus, all $B$'s travels are available for calculating the proper time over his path, and likewise for $B$ v. $A$.

Now when you say "Universal Coordinate Time", that is a much more complicated thing that involves multiple atomic clocks with their different Allan variances and errors being "coordinated" to minimize total error... not the stuff of thought experiments.

JEB
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  • But what protocol does A use to track B’s relative position as a function of A’s time? This seems like a tricky problem because A’s measurements of B’s positions are going to be delayed by an amount of time depending on the displacement at the start and end of A’s displacement measurement. – Jagerber48 Feb 01 '23 at 14:00
  • For me UTC is just a broadcasted time that everyone agrees on. If everyone on Earth agreed to use one atomic clock that never had downtime that would be fine. UTC uses many clocks to increase stability and ensure zero downtime. But “in principle”, multiple clocks aren’t required, just “in practice”. But in the relativistic scenario I think each body traveling at relativistic speeds relative to other bodies needs to carry its own clock because of time dilations and thus clock needs to be compared to other clocks. – Jagerber48 Feb 01 '23 at 14:03
  • $A$ can us a lidar, a radar..or any functioning speed-of-light target interrogating system available. Then $\Delta\tau = \int{dt/\gamma(t)}$. – JEB Feb 01 '23 at 14:22
  • but how does lidar work when the source and target are (a) very far from eachother and (b) moving at relativistic velocities relative to each other? Suppose in A emits lidar pulses at $t^0_i$ and $t^1_i$. Suppose these pulses reflect off the target at time $t^0_r$ and $t^1_r$ and arrive back at the source at $t^0_f$ and $t^1_f$. Suppose (in A's frame) the distances between the two ships is $d^0_i$, $d^1_i$, $d^0_r$, $d^1_r$, $d^0_f$ and $d^1_f$, but A doesn't know any of these distance a-priori. What would A record for the distances between the two ships at which times as a result? – Jagerber48 Feb 01 '23 at 14:44
  • I guess my question boils down to: do range-finding techniques like Lidar and Radar rely on the time of flight on the out and back trips being equal? If so I think it breaks down when the source and target have relativistic velocity relative to each other. – Jagerber48 Feb 01 '23 at 14:46
  • Maybe the LiDAR pulse will be red/blue shifted or expanded/compressed due to the relative velocity of the second ship. Then this velocity information combine with timing can be used to reconstruct position at some time. – Jagerber48 Feb 01 '23 at 15:12
  • @Jagerber48. as far as I recall, A & B start out together (event 0: E0), B goes off, and the A& B rejoin (event 1, E1). B's entire world line between E0 and E1 is visible to A at $t^A_1$: the whole path is visible, you just integrate d-tau along the world line. There is really no mystery nor no paradoxes here. – JEB Feb 02 '23 at 14:48
  • @Jagerber48 Radar measures range. Why to does the velocity of the target matter to time-of-flight? I spent 20 years in NASA radar, and we never interrogated a relativistic target. However, I have shot an IR laser head on into a 28 GeV positron beam and got back hard gamma rays...a big Doppler shift that did not affect the time-of-flight. – JEB Feb 02 '23 at 14:52
  • regarding the distance comments. A knows $d^0_i=d^0_r=d^0_f = 0$ because A is the 0-th ship. $d^1_i$ and $d^1_f$ are irrelevant, and $2d^1_r = c t_{\rm of flight}$. The last equation being fundamental to radars and lidars. – JEB Feb 02 '23 at 14:57
  • no, $d_i^0, d_r^0, d_f^0$ are the distances between A and B and transmission, reflection, and detection time. With non-relativistic targets separated by "small" distances light travels so fast that the distance between the source and the target remains pretty much constant during the entire interrogation. But if the target is very far away (like light minutes or more) (so that time of flight delay is very long) and the target is moving fast then it can't be assumed that the target has a constant range from the source during the duration of the interrogation. – Jagerber48 Feb 02 '23 at 17:38
  • you can't assume the time of flight from source to target equals the time of flight from target to source... – Jagerber48 Feb 02 '23 at 17:39