A clock comparison can reveal a potential difference, not a height by itself
Two excellent clocks can follow the same local atomic transition while accumulating different amounts of proper time. Comparing their rates can therefore supply information about gravity’s potential. Turning that information into metres requires another model. A frequency ratio is not a hidden ruler that automatically returns every kind of height.
This guide follows the inference in small steps: define the compared rates, choose a potential sign, calculate an original near-Earth example, account for motion and transfer, then ask which height claim the evidence supports. The numerical cases use fictional clocks and simplified fields; no commercial instrument capability or operational accuracy is implied.
The aim is practical understanding rather than a full relativistic time-transfer procedure. The IERS conventions provide the specialist coordinate-time framework, while the NIST report on the 2022 JILA experiment provides a concrete research example of gravitational redshift measurement. Our worked cases are independent teaching calculations, not reconstructions of that experiment.
Geodesy subject guide · Learning that transfers into life · How X Works hub
Choose your reading route
Route 1 · Chapters 1–2: Define the rate comparison
Route 2 · Chapters 3–5: Calculate the potential signal
Work a ten-metre constant-gravity example from the beginning
Route 3 · Chapters 6–8: Keep motion and transfer separate
Keep the motion term when the clocks do not share a stationary setup
Route 4 · Chapters 9–10: Convert and propagate carefully
Convert a potential difference to height only under a stated model
Route 5 · Chapters 11–12: Test and report the inference
Use changed conditions to test the proposed potential inference
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Define the rate comparison
Separate local clock operation from a remote comparison
Choose a potential convention before writing a redshift formula
Calculate the potential signal
Work a ten-metre constant-gravity example from the beginning
Distinguish gravitational potential from gravitational acceleration
Translate to Earth-fixed effective geopotential without losing the sign
Keep motion and transfer separate
Keep the motion term when the clocks do not share a stationary setup
Treat the transfer link as part of the measurement
Separate clock shifts, statistical instability and a potential signal
CHAPTER 1 OF 12 · Define the rate comparison
1. Separate local clock operation from a remote comparison
An ideal clock records proper time along its own path. Denote the two readings by τ_A and τ_B. If both use an ideal transition with local frequency ν0, their phases can be written φ_A = 2πν0τ_A and φ_B = 2πν0τ_B. Locally, each clock follows the same transition; neither needs to be “broken” for their remotely compared rates to differ.
A comparison needs a way to relate events and transfer frequency information. To explain the mathematics, introduce one coordinate-time parameter t and compare dτ_B/dt with dτ_A/dt after the relevant transfer effects are handled. The small fractional difference is y = (dτ_B/dt)/(dτ_A/dt) − 1. This definition fixes which clock is in the numerator.
Exchanging A and B reverses the leading small difference. More exactly, the reversed fractional result is 1/(1 + y) − 1. At the tiny scales in our examples, that is approximately −y. Keeping the exact ratio in mind helps prevent sign mistakes when reading a report that uses the opposite comparison direction.
The distinction matters beyond vocabulary. A change in the local atomic transition caused by an instrument-related shift is not automatically the gravitational signal. A real comparison must assess clock-specific shifts as well as the transfer and relativistic model. The equations here describe ideal rates first, so that later uncertainty terms have identifiable places to enter.
CHAPTER 2 OF 12 · Define the rate comparison
2. Choose a potential convention before writing a redshift formula
For an initial stationary, non-rotating weak-field teaching model, define U as a Newtonian potential that is negative near Earth and increases upward toward zero. Then dτ/dt ≈ 1 + U/c² for a stationary ideal clock, within the approximation being used. A higher U gives a faster proper-time rate relative to the common coordinate time.
Subtracting the two rates to first order gives y ≈ (U_B − U_A)/c² = ΔU/c². A positive y means B runs faster in this declared comparison. The quantity U has units of metres squared per second squared, so dividing by c² produces a dimensionless fractional frequency difference.
Some geodetic conventions use a positive gravitational potential and an effective geopotential W that includes rotation. Their formulas have different-looking signs. The IERS chapter uses a positive Earth potential in its proper-time relation. We have deliberately introduced U with the opposite gravitational sign for the first example; do not substitute symbols between conventions without translating their definitions.
A reliable quick check asks for a physical prediction under the specified conditions: raising an otherwise comparable stationary clock in the simple model should make y positive when B is the raised clock. If the algebra predicts the reverse, inspect the potential sign, comparison order and included motion terms. Do not fix the discrepancy by silently changing only the final sign.
CHAPTER 3 OF 12 · Calculate the potential signal
3. Work a ten-metre constant-gravity example from the beginning
Choose a fictional vertical separation Δh = 10 m and treat the relevant gravity magnitude as a constant g = 9.8 m/s² over that interval. In the simple model, ΔU = gΔh = 98 m²/s². Use c = 299,792,458 m/s. The predicted fractional rate difference is y = 98/c², approximately 1.0904 × 10⁻¹⁵.
To make that fraction tangible, choose an ideal transition frequency ν0 = 5 × 10¹⁴ Hz. The corresponding first-order compared frequency difference is ν0y, approximately 0.5452 Hz. The large optical frequency does not create the fractional gravitational effect; it expresses the same fraction as a number of cycles per second.
Check the scale with a one-metre separation under the same assumptions. It gives ΔU = 9.8 m²/s² and y ≈ 1.0904 × 10⁻¹⁶. Multiplying the height by ten multiplied the potential difference and fractional rate difference by ten. This proportionality follows from the chosen constant-gravity approximation, not from a universal equivalence between any frequency ratio and a height.
The fictional 0.5452 Hz is a comparison quantity after a stated model, not a prediction that a handheld counter placed beside either clock reads a locally changed atomic frequency. Locally, each ideal clock still realises ν0 per unit of its own proper time. The distinction between local operation and compared rate is the central idea that prevents the example from becoming misleading.
CHAPTER 4 OF 12 · Calculate the potential signal
4. Distinguish gravitational potential from gravitational acceleration
Acceleration describes a local spatial rate of change of potential; potential difference accumulates that change along a path. Near the vertical in the chosen simple field, ΔU = ∫ g(h) dh. Replacing the integral by gΔh is justified only to the accuracy for which constant or appropriately averaged gravity is adequate.
Create two fictional ten-metre paths. Along one, the appropriate average gravity is 9.8 m/s²; along the other it is 9.7 m/s². Their potential differences are 98 and 97 m²/s², so their ideal clock ratios differ. The same geometric separation did not uniquely determine the potential difference because the gravity fields differed.
Reverse the question. A potential difference of 98 m²/s² corresponds to 10 m under the first constant-gravity model and about 10.1031 m under the second. Neither conversion is a universal output of the clocks. Each introduces a different gravity-to-height relationship. This is why the separate guide on geopotential numbers and physical heights remains necessary.
Nor does a single local gravity magnitude determine a remote potential difference. Knowing one derivative at one point does not specify an integral over an unknown interval. A clock comparison and an absolute-gravity measurement can therefore supply complementary information without being interchangeable instruments. Keep the local acceleration, integrated potential and adopted height definition as three distinct quantities.
CHAPTER 5 OF 12 · Calculate the potential signal
5. Translate to Earth-fixed effective geopotential without losing the sign
For clocks fixed to a rotating Earth, it is useful to combine the gravitational and rotational contributions consistently. In the simplified stationary Earth-fixed account, write W for the positive effective geopotential. Then dτ/dt has a leading dependence 1 − W/c², apart from a common reference-rate term and other contributions being set aside.
With that convention, y ≈ (W_A − W_B)/c². A higher point along the local upward direction has lower W in the ordinary near-surface case, so B can still run faster. This agrees with the earlier example because the potential sign changed together with the formula. The physical prediction did not change.
A geopotential number relative to a chosen reference surface is C_P = W0 − W_P. Therefore C_B − C_A = W_A − W_B. In the stated first-order model, the clock comparison constrains this difference divided by c². Changing W0 shifts both geopotential numbers equally and leaves their difference unchanged.
This provides a valuable reference test. Add the same constant to both effective potentials and the predicted comparison is unchanged. A pairwise frequency ratio cannot independently determine the arbitrary potential zero. Connecting the result to an adopted vertical reference requires that reference information separately. Do not mistake an insensitive difference for evidence that the absolute reference has been measured.
CHAPTER 6 OF 12 · Keep motion and transfer separate
6. Keep the motion term when the clocks do not share a stationary setup
In our negative gravitational-potential convention, a simple weak-field moving-clock expression is dτ/dt ≈ 1 + U/c² − v²/(2c²). Both location and coordinate velocity matter. A comparison between moving clocks cannot be interpreted as only a gravitational potential difference unless the kinematic contribution has been handled appropriately.
For a fictional illustration, let B have 100 m²/s² more squared speed than A while their U values are equal in the chosen frame. The kinematic contribution to y is −50/c², about −5.5633 × 10⁻¹⁶. A gravity-only interpretation would assign a spurious negative potential difference of 50 m²/s².
Do not turn this isolated number into a transportable-clock correction recipe. Real paths, reference systems, timing and transfer require a consistent relativistic treatment. The IERS chapter explicitly relates proper time to coordinate time and motion. Its conventions, rather than a mixture of memorised signs from different frames, provide the appropriate specialist starting point.
There is a common double-counting trap. If Earth-fixed W already incorporates the rotational contribution for the adopted model, adding the same stationary rotational effect again as a separate speed correction is inconsistent. Either keep the gravitational and kinematic terms separately in one declared frame or use a consistent effective-potential representation. Record which account you chose before combining results.
CHAPTER 7 OF 12 · Keep motion and transfer separate
7. Treat the transfer link as part of the measurement
Two clocks do not compare themselves merely because both are accurate. Frequency information must be transferred and related to the chosen events. A link can introduce effects that resemble a rate difference. The ideal relation y = ΔU/c² describes the target after appropriate handling of transfer and clock-specific contributions, not an unprocessed remote signal.
A simple phase model makes the issue visible. Let a transferred carrier have phase 2πν[t − d(t)], where d(t) is a propagation delay in seconds. Differentiating gives a received rate proportional to 1 − d′(t) in this teaching description. A constant delay shifts phase but not frequency; a changing delay contributes a frequency shift.
If the fictional delay increases at 10⁻¹⁶ seconds per second, its first-order fractional contribution is −10⁻¹⁶. Interpreted incorrectly as a gravitational signal, its magnitude would correspond to about 0.917 m under g = 9.8 m/s². This number illustrates sensitivity to an unmodelled link rate; it is not a performance claim for any fibre, satellite or comparison system.
A useful evidence question asks which observations and model constrain that transfer contribution, and whether the comparison interval matches the relevant corrections. Agreement between two processing routes using the same link correction does not independently validate the shared correction. Preserve the distinction between the physical clock difference and the pathway used to observe it.
CHAPTER 8 OF 12 · Keep motion and transfer separate
8. Separate clock shifts, statistical instability and a potential signal
Write a deliberately simple observation equation y_obs = ΔU/c² + b_B − b_A + l + e. Here b_A and b_B represent residual fractional clock shifts, l a residual link contribution, and e a random comparison error. This is a conceptual account, not a complete uncertainty budget for an actual optical standard.
One observed number cannot determine every term independently. A positive y_obs could reflect a positive potential difference, a clock-related offset, a link effect or a mixture. Additional evidence is needed to constrain the nuisance contributions. Longer observation can help characterise some random behaviour without automatically identifying a constant bias.
For example, suppose the true teaching potential signal is 10⁻¹⁶ and a common comparison bias adds another 10⁻¹⁶. Averaging many independent e terms toward zero would leave an apparent 2 × 10⁻¹⁶ result. Its repeatability would improve while the potential interpretation remained biased. A smaller statistical error bar is not proof that the environmental inference is now correct.
Use an evidence hierarchy suited to the claim: characterise the individual clocks, assess transfer, inspect comparison stability, establish the relativistic model and connect the reference. Independent comparisons can help, but only to the extent that they do not reuse the same unknown contributions. State what is shared when interpreting agreement among several reported potential differences.
CHAPTER 9 OF 12 · Convert and propagate carefully
9. Convert a potential difference to height only under a stated model
If a local constant-gravity approximation is appropriate, the teaching height difference is Δh ≈ c²y/g after the relevant comparison corrections. Each symbol has a role: c² converts fractional rate to potential difference, and division by g converts that potential difference to a conditional length. The second step introduces the gravity model.
Set y = 10⁻¹⁸ and g = 9.8 m/s². The resulting height-equivalent scale is about 0.00917 m, or 9.17 mm. This is sensitivity arithmetic. It does not mean that every clock reporting an 18-digit ratio measures a real site’s height to nine millimetres. Clock, link, model, reference and temporal uncertainties still matter.
Different height systems also need their own definitions. Ellipsoidal height is geometric relative to an ellipsoid; gravity-related heights connect to potential and reference gravity in specified ways. The constant-g teaching length is not automatically an ellipsoidal, orthometric or normal height. Naming the output “height” without qualification hides the inference that produced it.
A good changed-context question asks whether the intended comparison is between two clocks’ active reference locations, two ground marks or two adopted height-datum points. Their connecting offsets and epochs may differ. A clock can constrain a potential difference between its relevant locations without directly locating every nearby benchmark. Preserve those endpoint connections just as carefully as in an antenna survey.
CHAPTER 10 OF 12 · Convert and propagate carefully
10. Propagate uncertainty and shared references through the conversion
For D = c²y, the standard uncertainty from y alone is σ_D = c²σ_y. For h = D/g, small perturbations give δh ≈ δD/g − Dδg/g². If D and g are independent, their variance contributions add after applying these sensitivities. If they are correlated, include the cross term with its sign.
For an isolated teaching case, take σ_y = 2 × 10⁻¹⁸ and hold g = 9.8 m/s² exact. The height-equivalent standard uncertainty from that rate uncertainty alone is about 18.34 mm. It is not a total site-height uncertainty unless all other contributions really are negligible or already included. State that scope beside the number.
A shared reference clock creates dependence among comparisons. Let estimates be y_BA = b − a and y_CA = c − a for independent underlying rate errors a, b and c. Their covariance includes Var(a). Forming y_CA − y_BA cancels a algebraically; treating the two comparisons as independent would miss that cancellation.
The same reasoning applies to common link models and reference-potential information, but only when the shared contribution and its sensitivity really match. Do not assume cancellation merely because two reports cite the same reference. Document timing, definitions and covariance, and distinguish a relative difference that is well constrained from an absolute reference connection that remains uncertain.
CHAPTER 11 OF 12 · Test and report the inference
11. Use changed conditions to test the proposed potential inference
Start with internal tests. Swap the comparison order and check the ratio transformation. Add a common potential constant and confirm that the prediction is unchanged. Change the nominal transition frequency and confirm that y remains the same while the difference expressed in hertz scales accordingly. These tests detect different mistakes.
Then test the physics under changed conditions. If B is raised in the simple stationary model, predict the sign and magnitude before examining the result. If motion is introduced, include its contribution consistently. If the link’s delay changes, ask whether its rate could imitate the expected signal. An unexplained agreement in one configuration is less informative than predictions that survive deliberately different conditions.
Research demonstrates that the gravitational redshift can be resolved at very small spatial scales: the 2022 JILA experiment compared regions of a strontium-atom sample, as reported by NIST. That is not the same setup as two independently transported clocks at distant benchmarks. Cite the actual experimental arrangement rather than treating its scale as a universal surveying guarantee.
For a real height-related claim, ask what external information would challenge the interpretation: a compatible potential or levelling comparison, independent clock characterisation, transfer validation and reference-location surveys. The point is not to demand every conceivable measurement, but to choose checks capable of exposing the specific nuisance terms that could otherwise masquerade as potential.
CHAPTER 12 OF 12 · Test and report the inference
12. Deliver the rate result before the derived height claim
A reusable report gives the comparison direction, interval, clock locations and reference events, the fractional result and its uncertainty, and the treatment of clock and transfer contributions. It identifies the relativistic reference system and potential convention. These details let another reader reconstruct why a positive number means what the report says it means.
Report the inferred potential difference separately from any height conversion. If a local constant-g approximation is used, state g and the approximation’s scope. If a formal height system is intended, provide the compatible reference model and endpoint connection. Preserve which quantities were observed, estimated or supplied externally.
Avoid performance language that the evidence does not establish. A height-equivalent sensitivity, a statistical comparison uncertainty, an independently validated potential difference and a certified benchmark height are distinct achievements. Explain the one actually supported. A reader should not have to reverse-engineer that distinction from a footnote after seeing an impressive number of decimal places.
The transferable lesson is to keep the inference chain intact. A clock rate can become a potential difference through relativity, and that difference can become a particular height statement through additional definitions and evidence. Each step is useful. None should disappear merely because the final unit, metres, feels more familiar than the measurement that made the inference possible.
Independent worked case and evidence checks
Take a new stationary teaching case: ΔU = 49 m²/s² and g = 9.8 m/s². Predict y and the conditional height difference. The results are y = 49/c² ≈ 5.4520 × 10⁻¹⁶ and Δh = 5 m. An ideal 5 × 10¹⁴ Hz transition gives a compared difference of about 0.2726 Hz.
Change the ideal transition to 4 × 10¹⁴ Hz while leaving the physical setup unchanged. The fractional y stays the same, while the difference in hertz becomes about 0.2181 Hz. If a calculation changes the inferred potential solely because the nominal transition changed, it has confused the absolute frequency difference with the fractional comparison.
Now keep the 49 m²/s² potential difference but use a fictional constant gravity of 9.7 m/s². The conditional height becomes about 5.05155 m. This changed-context result demonstrates exactly where the height model enters. Finally reverse A and B: the leading fractional signal changes sign, while its magnitude remains the same to the approximation used.
Evidence check: the arithmetic assumes ideal stationary clocks and a corrected comparison. It does not supply clock-shift assessments, a transfer model, a national reference potential or surveyed connections to ground marks. Name those missing inputs before turning a fictional rate calculation into a claim about an actual site.
Worked comparison
| Quantity or change | Fictional result | Interpretation |
|---|---|---|
| 10 m at constant 9.8 m/s² | ΔU = 98 m²/s² | The gravity-to-height model supplies this step |
| ΔU = 98 m²/s² | y ≈ 1.0904 × 10⁻¹⁵ | First-order ideal stationary comparison |
| Same y; transition 5 × 10¹⁴ Hz | Difference ≈ 0.5452 Hz | Absolute frequency difference depends on ν0 |
| y = 10⁻¹⁸; g = 9.8 m/s² | Height equivalent ≈ 9.17 mm | Sensitivity, not a total accuracy claim |
| Common potential zero changed | No pairwise-rate change | The ratio does not identify an absolute zero |
Questions and practical checks
Does a higher clock have a different local atomic transition?
Not in the ideal model used here. Each clock follows the same ν0 per unit of its own proper time. The gravitational information appears when their rates are compared consistently across locations. Real clock-related frequency shifts are separate contributions to assess, not automatically part of the gravitational signal.
Do clocks measure gravity or height?
A suitably modelled comparison constrains a potential difference through the relativistic rate relation. Local acceleration and geometric height are different quantities. Converting the result to a particular height requires a gravity model, reference definition and endpoint information.
Can I quote the nine-millimetre number as accuracy?
No. It is the height-equivalent scale of 10⁻¹⁸ under the fictional constant gravity 9.8 m/s². It omits other contributions unless explicitly included. The number demonstrates sensitivity; it does not establish an instrument’s total uncertainty or a real survey result.
Why are potential signs different in different references?
Some formulas use a negative Newtonian potential increasing upward; others use a positive gravitational or effective geopotential. The accompanying rate formula changes consistently. Define the symbol, frame and comparison direction, and test the physical prediction before transferring a formula.
What would be a convincing practical check?
Use an independent check that constrains the suspected confounder: clock characterisation for clock shifts, link validation for transfer, a changed-height prediction for the potential response, and compatible reference surveys for the height interpretation. Repeating one configuration alone cannot isolate every shared bias.
Continue with the connected subjects
Understand the physical height and reference
Follow geopotential numbers into normal and orthometric height distinctions
Distinguish shared geoid uncertainty from a height difference
Connect instrument locations and nearby survey marks
Keep clock-related observation models distinct
Compare a GNSS orbit-related clock correction without duplicating terms
Separate clock drift from geometric range-rate information
Build the broader relativity concepts behind clocks, motion and gravity
Sources and scope
IERS Conventions, Chapter 10: proper time, coordinate time and relativistic models near Earth
NIST (2022): the actual JILA millimetre-scale gravitational-redshift experiment and its scope
The numerical cases in this guide are original fictional teaching examples, not measurements from a real site. This is an educational explanation, not a field safety instruction, engineering approval or an official certification. For a real project, use qualified practitioners, current product documentation and site-specific evidence. Sources were checked on 1 November 2026.
Compare the physical quantities without merging them
A station’s support can change position, a falling body can reveal acceleration, clocks can constrain a potential difference, and a plumb line can establish direction. Read these as complementary measurements with different references, not interchangeable ways to report one height.
Follow the support and the acceleration
Thermal monument motion: connect temperature, support geometry and the moving point
Absolute gravimetry: recover acceleration from a calibrated free-fall trajectory
Follow potential and direction
Deflection of the vertical: distinguish local plumb up from an ellipsoid normal