Frequency / Clock Shift Calculator

First-order Doppler, gravitational redshift, and special-relativistic (transverse) Doppler — combined fractional frequency shift between an emitter and receiver.

📊 Sample calculation — GPS satellite clock offset: lab agrees with Ashby (2003) within stated tolerance — published +38.575 μs/day to 0.01 % (full precision)

Inputs

Carrier frequency $f_0$ (Hz) Default ≈ 1550 nm optical carrier.

Line-of-sight velocity $v_\parallel$ (m/s) Positive = emitter receding from receiver. Default 0 (transverse-only, canonical GPS configuration).

Emitter speed $v_e$ (m/s) 0 for ground clocks (the "Earth geoid" preset already includes Earth rotation). Use √(GM/r) for a circular-orbit satellite.

Receiver speed $v_r$ (m/s) Default = √(GM_⊕/a_gps) (GPS orbital speed). Set to 0 for stationary receivers.

Emitter location "Geoid" is the full effective potential (gravitational + J2 + centrifugal) — use this for any Earth-surface clock at full precision.

Receiver location

Integration time $T$ (s) Used for accumulated cycle count.

Tolerance (fractional Δf/f)

Tolerance (Hz) Ranking uses the looser of fractional or Hz check.

Result

Total Δf/f

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Total Δf

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Cycles over $T$

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Tolerance

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Formula registry: frequency_shift_ashby_v1 (equation, assumptions, not-included, source IDs)

Loading ../shared/formula_registry.json…

Term-by-term breakdown

Term	Δf/f (fractional)	Δf (Hz)	% of total
Compute to populate.

Formula

Combined first-order frequency shift $$\frac{\Delta f}{f} \;\approx\; -\frac{v_\parallel}{c} \;+\; \frac{U_e - U_r}{c^2} \;-\; \frac{v_e^2 - v_r^2}{2c^2}$$ Sign convention

Positive $\Delta f$ = received frequency higher than emitted. Newtonian potential $U = -GM/r$ (so $U_e - U_r > 0$ when the emitter is at the higher potential / "less deep" well).

Assumptions

Low-order weak-field approximation: $|U|/c^2 \ll 1$ and $v/c \ll 1$.
First-order Doppler only ($-v_\parallel/c$); higher-order Doppler terms not retained.
Special-relativistic time-dilation term is the standard "transverse Doppler" piece, $-v^2/(2c^2)$, expanded to leading order.
Newtonian potential $U = -GM/r$ at the emitter and receiver positions; no tidal or J2 corrections.
Sign convention: positive $\Delta f$ means received frequency is higher than emitted.
Ignores frame-consistent relativistic time-scale transformations (TT, TCB, TDB).

What this would need for mission-grade use

Show requirements

Full IERS Earth-orientation parameters
Solid-Earth tide + ocean-loading potentials
Topocentric station coordinates + plate motion
Tropospheric / ionospheric clock-rate effects
Spacecraft on-board clock model + drift
Higher-order GR (frame-dragging, gravito-magnetic) for orbital regime
Mission-specific clock calibration

Warnings

Public engineering surrogate clock comparisons require full TT/TCB/TDB transformation, including J2, Earth-tides, and ephemeris-frame effects. This is a first-order back-of-envelope tool only.
For relative velocities approaching the speed of light, retain the full relativistic Doppler factor; the first-order approximation here breaks down.
The displayed gravitational term uses Newtonian potentials; for compact objects or near-horizon scenarios use the full $\sqrt{g_{00}}$ ratio.

Source notes

Standard weak-field clock comparison; see e.g. Misner, Thorne, Wheeler, Gravitation §6 and §25; or Will, Theory and Experiment in Gravitational Physics (2018), §6.
Constants from CODATA 2018 / IAU; full table on the Validation page.

Certificate

Each computed result can be exported as a JSON or Markdown certificate with a SHA-256 hash of the canonical content.