Two clocks held at different heights can accumulate different elapsed times. To interpret the comparison, we need to say where the clocks are, how they move, how their readings are exchanged and which physical model relates those readings. “Time runs differently” becomes precise through that procedure.
Compare defined worldlines
Each ideal clock measures proper time along its own trajectory. Comparing two remote clocks adds a communication procedure: signals can transfer frequency information, or clocks can later be brought together. Signal travel, motion and the transport procedure must be included in the interpretation.
For stationary clocks in a weak, nearly static gravitational field, the leading fractional rate difference is approximately ΔΦ/c², where ΔΦ is the difference in gravitational potential. Near Earth's surface, for a modest vertical separation h, the approximation ΔΦ ≈ gh gives
The higher clock accumulates more proper time per common comparison interval in this arrangement. The statement compares clocks in a specified geometry; it does not require either observer to experience an unusual local second.
A height calculation
Use g = 9.81 m/s² and c = 299,792,458 m/s for a teaching estimate. For a height difference of one metre, gh/c² is approximately 1.09 × 10⁻¹⁶. Multiplying by 86,400 seconds gives an elapsed-time difference of approximately 9.43 picoseconds over a day in this ideal stationary comparison.
| Height difference | Fractional rate difference | Difference over one day |
|---|---|---|
| 1 centimetre | 1.09 × 10⁻¹⁸ | 0.0943 picoseconds |
| 1 metre | 1.09 × 10⁻¹⁶ | 9.43 picoseconds |
| 100 metres | 1.09 × 10⁻¹⁴ | 943 picoseconds |
This table neglects rotation, motion, variation of g and instrumental effects. It is a scale calculation, not a prediction for every pair of clocks with the same geographical height difference. Precision work uses a suitable gravitational potential and a full comparison model.
Signals and signs
The rate comparison can initially seem to conflict with gravitational redshift. A light signal sent upward is received at a lower frequency relative to the upper observer's local standard than its emission frequency relative to the lower standard. Meanwhile, the upper stationary clock accumulates more proper time over the common coordinate interval.
There is no contradiction. The phrases compare different ratios. A sound calculation names the emission event, reception event, clock standards and frequency ratio before assigning a sign. A schematic drawing of “faster time” is not a substitute for those definitions.
An experimental anchor
Bothwell and colleagues reported in 2022 a spatial frequency gradient consistent with gravitational redshift within a millimetre-scale ultracold strontium sample. Their paper describes differential measurements and corrections for instrumental effects, including atomic-density and field-related shifts. It provides a concrete example of how a tiny gravitational signal is separated from other causes of frequency change.
The scientific result concerns a specified apparatus and comparison, with stated uncertainty. It should not be turned into an unsupported claim that every biological or psychological process has been directly measured at that precision.
What a new clock law must do
A proposed theory of emergent time must recover these comparisons or predict a controlled, testable difference. Giving several clocks the same update label does not ensure that their material processes respond identically to motion and gravitational potential. Their dynamics must supply that agreement.
This is the role of the universal material-clock problem in the SMM/DTT programme. A source model should identify physical clock carriers, derive their couplings and compare their readings under the same conditions. Different clock mechanisms provide opportunities to test whether a proposed effect is universal or depends on the device.
The equivalence-principle essay explains another related requirement: a shared account of free fall. A successful local free-fall description and a successful remote clock comparison support different parts of the gravitational theory.
Sources and further reading
- T. Bothwell et al., “Resolving the gravitational redshift across a millimetre-scale atomic sample,” Nature 602 (2022), 420–424. Published paper; institutional copy.
- The time measured along a journey; the principle of an atomic clock.
- DoT book contents; relativity and physical clocks.