# Kinematic time dilation in gravitational fields

Kinematic time dilation in gravitational fields is the velocity-dependent part of the rate at which a moving clock ticks when it also sits in a gravitational potential. In the weak fields near Earth, the rate of proper time (time measured by a clock moving along with the object) τ of a clock moving at speed v in a Newtonian potential U, taken negative, relative to coordinate time t is dτ/dt ≈ 1 + U/c² − v²/(2c²), where the first correction is the gravitational redshift and the second is the special-relativistic Doppler (kinematic) correction.<sup>[1](https://ar5iv.labs.arxiv.org/html/1812.09161)</sup> Both effects act on every orbiting or airborne clock at once, so tests of gravitation by moving clocks always measure their combination unless the geometry separates them.

| Key fact | Value |
|---|---|
| Weak-field combined rate | dτ/dt ≈ 1 + U/c² − v²/(2c²), with U the negative Newtonian potential<sup>[1](https://ar5iv.labs.arxiv.org/html/1812.09161)</sup> |
| Kinematic vs gravitational clock comparisons | Both Doppler effects (special relativity) and gravitational redshift (general relativity) govern comparisons<sup>[2](https://arxiv.org/html/2310.11576)</sup> |
| Exact circular-orbit Schwarzschild rate | \(\sqrt{(1 - 3r_{\mathrm{g}}/2r)/(1 - r_{\mathrm{g}}/r_1)}\), with 3/2 encoding the combined terms<sup>[3](https://arxiv.org/html/2302.11146)</sup><sup> • </sup><sup>[9](https://www.scielo.br/j/rbef/a/gYmz8wxpVN8w76sLNv9XLnB/?lang=en)</sup> |
| Apollo 12/13 net clock gains | 570.3 ± 0.3 µs and 327.6 ± 0.2 µs<sup>[4](https://ntrs.nasa.gov/api/citations/19720022040/downloads/19720022040.pdf?attachment=true)</sup> |
| Galileo eccentric-orbit modulation | ~370 ns peak, peak-to-peak Δf/f ≈ 1×10⁻¹⁰<sup>[1](https://ar5iv.labs.arxiv.org/html/1812.09161)</sup> |
| GNSS satellite net offset | about +38.6 µs/day, pre-corrected as 10.22999999543 MHz<sup>[5](http://urn.kb.se/resolve?urn=urn%3Anbn%3Ase%3Akth%3Adiva-365873)</sup> |
| Redshift test status | Tested at 10⁻⁵ level; ACES targets 10⁻⁶<sup>[2](https://arxiv.org/html/2310.11576)</sup> |

## Deriving the combined rate from the metric

In Schwarzschild spacetime, the rate ratio of a clock on a circular orbit of radius \(r\) to a static clock at radius \(r_1\) is \((1 - 3r_{\mathrm{g}}/(2r))/(1 - r_{\mathrm{g}}/r_1)\), where \(r_{\mathrm{g}}\) is the gravitational radius, with the 3/2 coefficient on \(r_{\mathrm{g}}/r\) encoding the combination of gravitational and kinematic terms.<sup>[3](https://arxiv.org/html/2302.11146)</sup>

Whether the combined rate factorizes into a purely gravitational part and a purely kinematic part is itself geometry-dependent. In a study of decoupling, the gravitational factor \(\gamma_g = 1/\sqrt{g_{tt}}\), and the kinematic factor \(\gamma_s = 1/\sqrt{1 - v_O^2}\) multiply exactly for particular symmetries or particular types of motion, such as radial free fall observed by a distant inertial observer, with \(v_O\) measured by the local static observer. Such a factorization is not a universal feature of all coordinate systems and motions; a necessary and sufficient criterion for decoupling can be defined.<sup>[6](https://ar5iv.labs.arxiv.org/html/0712.2359)</sup>

## How the two terms combine

In bound orbits the two terms have opposite signs: the gravitational potential makes the satellite clock run faster, and its orbital speed makes it run slower. For a satellite in a circular orbit of radius R* compared with an Earthbound clock at radius R, the net rate is dτ*/dτ_E ≈ 1 − (GM/2c²)(3/R* − 2/R), and the predicted effects are about 100 times larger than those encountered for jet-borne clocks.<sup>[7](https://doi.org/10.1038/241263a0)</sup> For the GNSS constellations, the combined offset is a large net speedup of about +38.620 µs/day in the weak-field approximation and +38.619 µs/day in an exact Schwarzschild-plus-Kepler model.<sup>[5](http://urn.kb.se/resolve?urn=urn%3Anbn%3Ase%3Akth%3Adiva-365873)</sup>

For clocks flying in Earth's atmosphere, the kinematic term carries a directional component. Hafele's result for equatorial flight is dτ/dτ₀ ≈ 1 + gh/c² − (2ΩRu + u²)/2c², where g is gravitational acceleration, h altitude, Ω Earth's rotation rate, R Earth's radius, and u the aircraft's ground speed. The term linear in u, positive for eastward flights, and negative for westward ones, arises because u is measured in Earth's rotating frame.<sup>[7](https://doi.org/10.1038/241263a0)</sup>

For a clock moving at speed \(V\) at the same field point as a resting clock, the elapsed-time ratio is \(\Delta T'/\Delta T = \sqrt{1 - V^2}\): clocks in motion tick slowly, independently of their position in the field. In this local comparison the transverse Doppler shift is kinetic time dilation; the unified general-relativistic treatment of the redshift, [Doppler effect](https://www.edgepedia.org/doppler-effect), and time dilation treats it as one contribution among position- and motion-dependent terms rather than as an effect requiring separate curvature corrections.<sup>[3](https://arxiv.org/html/2302.11146)</sup> Conversely, a distant observer watching a freely falling emitter sees the received frequency split into one gravitational blue-shift factor \(1/\sqrt{g_{tt}}\) and two kinematic red-shift factors, \(\omega_{\mathrm{IFO}}/\omega_{\mathrm{IO}} = (1/\sqrt{g_{tt}}) \cdot \sqrt{1 - V_{\mathrm{IFO}}^2}/(1 + V_{\mathrm{IFO}})\), so the apparent separation into gravitational and kinematic pieces depends on who observes and from where.<sup>[6](https://ar5iv.labs.arxiv.org/html/0712.2359)</sup>

## Clock experiments testing both effects together

**Apollo.** Trajectory-based calculations for the lunar missions show the second-order Doppler effect and the gravitational redshift giving corrections of opposite sign. Using the refined trajectory data, [Apollo 12](https://www.edgepedia.org/apollo-12) clocks gained 570.3 ± 0.3 µs and [Apollo 13](https://www.edgepedia.org/apollo-13) clocks gained 327.6 ± 0.2 µs relative to a ground clock (a preliminary computation gave 560 ± 1.5 µs and 326 ± 1.3 µs); the report judges these gains large enough that rubidium atomic frequency standards could measure them to about ±0.33 percent.<sup>[4](https://ntrs.nasa.gov/api/citations/19720022040/downloads/19720022040.pdf?attachment=true)</sup>

**Galileo eccentric satellites.** The satellites GSAT-0201/0202, in orbits of eccentricity \(e \approx 0.16\), swing through varying gravitational potential and speed, producing a relativistic eccentricity correction that peaks at approximately 370 ns, a peak-to-peak relative frequency modulation \(\Delta f / f \approx 1 \times 10^{-10}\) over the 12.94 h orbital period.<sup>[1](https://ar5iv.labs.arxiv.org/html/1812.09161)</sup> The redshift-only violation parameter came out \(\alpha_{\mathrm{rs}} = (4.5 \pm 3.1) \times 10^{-5}\), a fourfold reduction in uncertainty compared with [Gravity Probe A](https://www.edgepedia.org/gravity-probe-a)'s \(\alpha_{\mathrm{rs}} < 1.4 \times 10^{-4}\), and the combined result improved on Gravity Probe A fivefold.<sup>[1](https://ar5iv.labs.arxiv.org/html/1812.09161)</sup> These satellite figures sit about two orders of magnitude above the jet-flight effects, as expected from the circular-orbit comparison of about 100 times.<sup>[7](https://doi.org/10.1038/241263a0)</sup>

## Current precision and what has changed recently

[Gravitational redshift](https://www.edgepedia.org/gravitational-redshift) is tested at the 10⁻⁵ level (Herrmann et al. 2018; Delva et al. 2018), and the upcoming [Atomic Clock Ensemble in Space](https://www.edgepedia.org/atomic-clock-ensemble-in-space) (ACES) proposes to test it at the 10⁻⁶ level, while modern clock campaigns reach frequency-comparison sensitivities around 10⁻¹⁶.<sup>[2](https://arxiv.org/html/2310.11576)</sup>

## Conceptual issues: kinematic or gravitational?

Textbook treatments disagree on how to attribute orbital time dilation. One research line holds that the split into gravitational and kinematic factors is not coordinate-invariant: it appears exactly only in particular geometries and motions, so calling an orbiting clock's dilation "really" gravitational or "really" kinematic depends on the description chosen.<sup>[6](https://ar5iv.labs.arxiv.org/html/0712.2359)</sup> Another line, following Narlikar's unified approach, treats the redshift, Doppler effect, and time dilation in a single general-relativistic formalism applicable to moving observers generally, without assigning the split fundamental status.<sup>[3](https://arxiv.org/html/2302.11146)</sup> These are recorded here as an unresolved interpretational difference, not a numerical contradiction.

The equivalence-principle framing distinguishes local from global descriptions. Locally, two clocks at the same field point, one moving at speed V relative to the other, differ by the pure kinetic factor √(1 − V²), regardless of gravity.<sup>[3](https://arxiv.org/html/2302.11146)</sup> Globally, the gravitational Doppler effect is described by a position-dependent rate of proper time in the field, with the energy or frequency of a freely falling photon a [constant of motion](https://www.edgepedia.org/constant-of-motion); a proposed nonlocal experiment using a geostationary satellite would measure the gravitational Doppler effect and the position-dependent rate of proper time simultaneously, making the local-versus-global distinction observable in principle.<sup>[8](https://link.springer.com/article/10.1007/BF00715040)</sup>

## References

1. [Test of the Gravitational Redshift with Galileo Satellites in an Eccentric Orbit](https://ar5iv.labs.arxiv.org/html/1812.09161)
2. [General Relativistic Chronometry with Clocks on Ground and in Space](https://arxiv.org/html/2310.11576)
3. [A unified treatment of the redshift, the Doppler effect, and the time dilation in general relativity](https://arxiv.org/html/2302.11146)
4. [Relativistic Time Corrections for Apollo 12 and Apollo 13 (NASA)](https://ntrs.nasa.gov/api/citations/19720022040/downloads/19720022040.pdf?attachment=true)
5. [Relativistic corrections to satellite navigation systems (KTH report)](http://urn.kb.se/resolve?urn=urn%3Anbn%3Ase%3Akth%3Adiva-365873)
6. [Decoupling of kinematical time dilation and gravitational time dilation in particular geometries](https://ar5iv.labs.arxiv.org/html/0712.2359)
7. [Relativistic Behaviour of Circumnavigating Clocks (Nature 241, 1973)](https://doi.org/10.1038/241263a0)
8. [The gravitational Doppler effect explored by means of a geostationary satellite (Foundations of Physics)](https://link.springer.com/article/10.1007/BF00715040)
9. [scielo.br](https://www.scielo.br/j/rbef/a/gYmz8wxpVN8w76sLNv9XLnB/?lang=en)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Relativity and gravitation › General relativity and curved spacetime › Tests and observable effects › Gravitational time dilation and clock tests › Kinematic time dilation in gravitational contexts*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026; Oct 11, 2026; Oct 21, 2026; Oct 25, 2026 · Last review: Sep 30, 2026*

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