A received signal tells us when something reached a detector. Inferring where and when it began requires a propagation model and calibrated clocks. Two receivers are enough to make that distinction visible in a simple one-dimensional reconstruction.
Two receivers, one flash
Place stationary receivers at positions 0 and L in one inertial frame. A source between them emits a flash at position x and time t₀. Assume vacuum propagation in both directions, synchronized coordinate-time readings and no detector latency.
The two arrival times are tL = t₀ + x/c and tR = t₀ + (L − x)/c. Adding and subtracting these equations gives
t₀ = (tL + tR − L/c)/2,
x = [L + c(tL − tR)]/2.
The sum identifies emission time; the difference identifies position. These conclusions depend on knowing L and c in compatible units, placing the source between the receivers and assigning both records to the same flash. Without those assumptions, the formulas need not solve the intended physical problem.
Reconstruct the event
Move the emission; compare the clock readings. Use model units with c = 1 and L = 10. A right-clock offset changes its displayed reading, not the propagation.
Left clock: 7.0. Right clock: 11.0. Ignoring the offset gives emission time 4.0 and position 3.0. Correcting b = 0.0 recovers time 4.0 and position 3.0.
Solid teal: light to the left receiver. Dashed brown: light to the right. The vertical axis is shared coordinate time; position and time use different display scales. Clock offsets affect the readouts only.
What a clock offset does
Let the right clock read TR = tR + b, while the left clock is correctly calibrated. Substituting the raw reading as if b were zero biases the inferred time by +b/2 and the position by −cb/2. Correcting the right reading before reconstruction removes these biases in the ideal model.
At x = 3, t₀ = 4 and b = 2, the clocks read 7 and 13. Ignoring the offset gives an apparently consistent event at x = 2, t₀ = 5. Both inferred coordinates are wrong even though the arithmetic is exact. Precision in solving equations cannot compensate for a missing calibration parameter.
If b is unknown, two readings must constrain three unknown quantities: x, t₀ and b. A family of events can fit. Calibration, another independent observation or a justified prior restriction is needed. The network synchronization essay explains why time transfer also depends on assumptions about delays.
Uncertainty and model failure
Small independent timing errors δtL and δtR affect the reconstructed time through half their sum, and position through c/2 times their difference. Equal positive errors shift the inferred time; opposite errors can shift position without changing that time. Reporting only one combined “timing accuracy” may hide this difference.
Moving receivers, refraction, reflection, unequal detector latencies or a source outside the interval change the forward model. In three dimensions, two arrival times do not generally identify a unique event. The one-dimensional example is deliberately constrained to make the inverse calculation exact and inspectable.
From reconstruction to the books
NASA's SPICE light-time routine addresses the practical task of relating transmission and reception epochs for specified observer and target trajectories. Our example is much simpler, but it illustrates why both events and the model connecting them must be named.
For inner generation and outer completion, reconstruction asks what correlations completed records retain about earlier events. The published completion-principles framework provides a mathematical route into observable descriptions; physical application still requires the calibration and propagation inputs made explicit here.
Sources and further reading
- NASA/JPL NAIF: transmission and reception light-time calculations. The demonstration is an independent teaching construction, not SPICE output.
- Why seeing means receiving the past; what makes a distance?; Duality of Time: book contents.