Two observations are simultaneous only relative to a specified timing procedure and, for separated events, a specified reference frame. Equal arrival times, equal displayed timestamps and equal estimated event times are different statements.
Separate event time from arrival time
A remote flash happens before its signal reaches a detector. The detector's timestamp can include propagation delay, electronic delay and an offset in the clock used to label the record. A comparison of event times must account for those contributions.
In a simple fixed-frame model, write a = t + d + b. Here a is the recorded arrival timestamp, t is the event time in the chosen coordinate convention, d is the total calibrated signal delay and b is the receiving clock's offset. An estimate of t subtracts the delay and offset from a.
This equation is deliberately modest. It assumes that the relevant delays and clock behaviour can be represented by these terms over the interval considered. Moving clocks, changing propagation conditions and relativistic corrections require a more complete model.
Equal arrivals can mean unequal events
Suppose two channels reach the same recorder with timestamps of 120 ns. Channel A has a calibrated delay of 10 ns; channel B has a delay of 30 ns. With a zero recorder offset in the chosen convention, the inferred event times are 110 ns and 90 ns.
The arrivals coincide, but the inferred events are separated by 20 ns. Conversely, events both occurring at 100 ns would arrive at 110 ns and 130 ns through those channels. Equal event times need not produce equal arrival times.
The numerical example concerns calibrated channel delays, so it does not require guessing a geometric distance from a signal speed. In a physical experiment, cable propagation, optical paths and electronics can all contribute. Replacing the whole delay with distance divided by c is justified only when that model fits the channel.
A common offset can cancel
If both timestamps use the same recorder clock with an unknown but constant offset b, their difference eliminates b. The estimated event-time difference is (aA − aB) − (dA − dB). The absolute time labels may remain uncertain while the difference is well constrained.
This cancellation fails for separate unknown clock offsets, and it need not remove a changing clock error between observations. Delay uncertainties also remain. A shared timing reference helps only to the extent that its errors enter the comparison in the same way.
Consequently, uncertainty should be propagated with its correlations. Treating a fully shared offset as two independent errors overstates its effect on a difference. Ignoring independent channel-delay errors understates the uncertainty. The apparatus determines which contributions are shared.
Overlap is not an equality proof
Suppose corrected event times are reported as intervals [98,102] ns and [101,105] ns. The overlap means the data permit a common time under that interval interpretation. They also permit unequal times. Overlapping intervals alone do not establish exact simultaneity.
If the experiment needs a coincidence criterion, define a tolerance in advance. For example, classify two estimates as coincident when their difference lies within a stated window, then report the window and the timing uncertainty. This is a reproducible operational rule rather than a claim of infinitely precise equality.
Changing the coincidence window can change measured coincidence rates. The window is therefore part of the analysis, especially when interpreting correlations between independently detected events.
Remote clock comparison needs a procedure
Two-way timing exchanges can estimate propagation delays by comparing signals sent in both directions. Taking half a round-trip delay as a one-way delay relies on the relevant symmetry assumptions and on accounting for equipment delays. NIST's treatment explains why asymmetries limit this cancellation.
For spatially separated events, relativity adds the dependence of simultaneity on the chosen inertial frame. Correcting an instrumental delay within one frame does not establish a frame-independent present. The book's clocks and radar coordinates connects local clock readings with a stated coordinate construction.
For SMM/DTT, the observational requirement remains concrete: explain the clocks, signal propagation, completion and timing comparison with the same physical dictionary. A proposed inner order is not identified merely by finding equal or unequal recorded timestamps. The mapping from generation to observable timing must be specified.
Sources and further reading
- NIST, Two-way time transfer, for delay symmetry and equipment contributions.
- Mohamed Haj Yousef, Clocks and radar coordinates and Records without a global clock. The timestamp and interval calculations are teaching examples developed here.