---
title: Gravitational Redshift and the Shapiro Delay
module: Tests of General Relativity
moduleNumber: 7
lessonNumber: 3
order: 703
summary: >
  A clock deeper in a gravitational well ticks slower, and a photon climbing out
  loses frequency by the ratio of the metric's time-time components. Pound and
  Rebka measured the 2.5×10⁻¹⁵ shift over a 22.5-metre tower. Radar signals grazing
  the Sun return late by about 250 microseconds, the Shapiro delay. Both probe the
  time part of the metric directly.
topics: [Tests of General Relativity]
draft: false
sources:
  - book: Hartle
    ref: "Gravity, Ch. 6 — Gravity as Geometry (gravitational redshift); Ch. 10 — Solar-System Tests (the time delay of light)"
  - book: Carroll
    ref: "Lecture Notes on General Relativity, §7 — The Schwarzschild Solution (redshift, radar echoes), arXiv:gr-qc/9712019"
---

Two of the classical tests measure the metric's time-time component $g_{tt}$
directly, without reference to how space is curved. The gravitational redshift
compares the rates of two clocks at different depths in a gravitational potential:
the deeper clock runs slow, so a photon climbing from it arrives with reduced
frequency. The Shapiro delay measures the extra coordinate time a light signal
takes when its path dips into the potential well of the Sun. Neither depends on
the spatial curvature that doubles the deflection of light, so together with the
deflection they separate the two parts of the weak-field metric.

Both effects follow from the weak-field limit worked out with the [equivalence
principle](/relativity/curved-spacetime/the-equivalence-principle-formalized) and
the [Schwarzschild metric](/relativity/the-schwarzschild-solution/the-schwarzschild-metric).
The signature is $(-,+,+,+)$.

## Gravitational redshift from the metric

A static observer at radius $r$ in the Schwarzschild geometry measures proper time

$$
\d\tau = \sqrt{-g_{tt}}\ \d t = \sqrt{1 - \frac{2GM}{c^2 r}}\ \d t,
$$

where $t$ is the Schwarzschild coordinate time, common to all static observers.
Two static clocks at radii $r_1 < r_2$ therefore accumulate proper time at
different rates for the same $\Delta t$:

$$
\frac{\d\tau_1}{\d\tau_2}
= \sqrt{\frac{1 - 2GM/(c^2 r_1)}{1 - 2GM/(c^2 r_2)}} < 1 .
$$

The lower clock (smaller $r_1$, deeper in the well) runs slow relative to the
higher one. A signal of proper period $\d\tau_1$ emitted at $r_1$ spans a fixed
coordinate interval $\d t$; received at $r_2$ it is measured against $\d\tau_2 >
\d\tau_1$, so its period is longer and its frequency lower. In terms of frequency,

$$
\frac{\nu_2}{\nu_1} = \frac{\d\tau_1}{\d\tau_2}
= \sqrt{\frac{1 - 2GM/(c^2 r_1)}{1 - 2GM/(c^2 r_2)}} .
$$

Light climbing out of the well is redshifted; light falling in is blueshifted. In
the weak field, expanding to first order in $GM/(c^2 r)$ and writing the Newtonian
potential $\Phi = -GM/r$,

$$
\frac{\Delta\nu}{\nu} = \frac{\nu_2 - \nu_1}{\nu_1}
\approx \frac{\Phi_1 - \Phi_2}{c^2} = -\frac{\Delta\Phi}{c^2} .
$$

For a height difference $h$ in the nearly uniform field $g$ near the Earth's
surface, $\Delta\Phi = g h$ and

$$
\frac{\Delta\nu}{\nu} = -\frac{g h}{c^2} .
$$

> **Result.** A photon rising through a potential difference $\Delta\Phi$ shifts
> in frequency by $\Delta\nu/\nu = -\Delta\Phi/c^2$. The redshift measures $g_{tt}$
> alone; it follows from the equivalence principle and energy conservation and
> needs no assumption about spatial curvature.

$$
% caption: A photon emitted at the bottom of a potential well arrives at the top
% with lower frequency; the fractional loss equals the potential difference
% divided by c squared. The wave is drawn stretching as it climbs.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % potential well curve
  \draw[black] (0,3.3) .. controls (1.6,3.3) and (2.2,0.4) .. (3.0,0.4)
                        .. controls (3.8,0.4) and (4.4,3.3) .. (6.0,3.3);
  \node[black, left] at (0,3.3) {high};
  \node[black] at (1.75,0.35) {deep well};
  % emitter at bottom, receiver at top
  \fill (3.0,0.4) circle (2pt);
  \node[below] at (3.05,0.45) {emit};
  \fill[black!75] (5.7,3.3) circle (1.8pt);
  \node[black!70, above] at (5.7,3.4) {receive};
  % climbing wave: short wavelength near bottom, long near top
  \draw[acc, very thick] (3.15,0.7)
    sin (3.45,0.95) cos (3.75,0.7) sin (4.05,0.45) cos (4.35,0.7)
    sin (4.75,1.6) cos (5.15,1.35) sin (5.55,2.6) cos (5.95,2.35);
  % annotation
  \node[acc, right] at (4.15,1.55) {redshifted climbing out};
\end{tikzpicture}
$$

### The Pound–Rebka experiment

Pound and Rebka measured this shift in 1959 over the $22.5\ \text{m}$ height of a
tower at Harvard. The predicted fractional shift is

$$
\frac{\Delta\nu}{\nu} = \frac{g h}{c^2}
= \frac{(9.81)(22.5)}{(3.00\times10^8)^2}
= 2.46\times10^{-15},
$$

a shift of two parts in $10^{15}$. Resolving it required the Mössbauer effect: the
$14.4\ \text{keV}$ gamma ray from $^{57}\text{Fe}$ embedded in a crystal lattice is
emitted with no recoil, giving a line sharp enough that a shift of $10^{-15}$ is a
measurable fraction of its width. Moving the source at a few millimetres per second
introduced a compensating first-order Doppler shift; the velocity that nulled the
gravitational shift measured it. The experiment confirmed $\Delta\nu/\nu = gh/c^2$
to about $10\%$, later improved to $1\%$. Modern optical-clock comparisons detect
the gravitational redshift over height differences of a few centimetres.

$$
% caption: A gamma ray from an iron-57 source climbs the 22.5-metre tower to an
% absorber at the top and arrives redshifted by g h over c squared, about 2.5
% parts in ten to the fifteen; the recoilless Mossbauer line is sharp enough to
% resolve it, and a slow source motion supplies a compensating Doppler shift.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % tower shaft
  \draw[black] (0,0) rectangle (1.3,4.2);
  \draw[black] (0.65,0.1) -- (0.65,4.1);
  \node[black, below] at (0.65,-0.12) {tower};
  % source at the bottom
  \fill (0.65,0.35) circle (2.6pt);
  \node[right] at (1.5,0.35) {source (Fe-57)};
  % absorber/detector at the top
  \draw[very thick] (0.35,3.95) rectangle (0.95,4.2);
  \node[right] at (1.5,4.05) {absorber};
  % climbing gamma ray
  \draw[->, acc, very thick] (0.65,0.6) -- (0.65,3.85);
  \node[acc, left] at (0.55,2.25) {gamma ray};
  % height marker
  \draw[<->, black] (2.95,0.35) -- (2.95,4.05);
  \node[black, right] at (3.0,2.2) {height h};
\end{tikzpicture}
$$

## The Shapiro time delay

A light signal passing near a mass takes longer, in coordinate time, than the same
path would take in flat spacetime. This is the Shapiro delay, and it probes
$g_{tt}$ over an extended path rather than at two endpoints. The coordinate speed
of light in the weak-field metric is not $c$: for a radial ray,

$$
\d s^2 = 0
\ \Rightarrow\
\frac{\d r}{\d t}
= c\left(1 + \frac{2\Phi}{c^2}\right)
= c\left(1 - \frac{2GM}{c^2 r}\right),
$$

slightly less than $c$ where the potential is deep. A signal spends extra
coordinate time wherever its path dips into the well. Integrating the slowdown
along a nearly straight path from $r_1$ to $r_2$ with impact parameter $b$ gives
the one-way excess

$$
\Delta t \approx \frac{2GM}{c^3}\,
\ln\!\left(\frac{4 r_1 r_2}{b^2}\right).
$$

The logarithm makes the delay largest for a grazing path (small $b$) and grow only
slowly with the endpoint distances. For a radar signal sent from Earth, reflected
off a planet on the far side of the Sun, and returned, the geometry doubles the
path and the total excess round-trip delay is

$$
\Delta t_{\text{round}} \approx \frac{4GM}{c^3}\,
\ln\!\left(\frac{4 r_\oplus r_p}{b^2}\right).
$$

$$
% caption: A radar pulse from Earth to a planet near superior conjunction skims
% the Sun; where the path enters the potential well the coordinate light speed
% drops, and the round trip returns late by the Shapiro delay.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw (3.1,0) circle (0.4);
  \node at (3.1,0) {Sun};
  \fill[black!75] (0.2,1.15) circle (1.8pt);
  \node[black!70, left] at (0.1,1.15) {Earth};
  \fill[black!75] (6.2,-1.05) circle (1.8pt);
  \node[black!70, right] at (6.3,-1.05) {planet};
  % grazing path
  \draw[acc, very thick] (0.2,1.15) .. controls (2.4,0.72) and (3.9,0.35) .. (6.2,-1.05);
  % impact parameter marker
  \draw[black, dashed] (3.1,0) -- (2.85,0.62);
  \node[black, right] at (2.9,0.42) {$b$};
  % slowed segment note
  \node[black, below] at (3.1,-0.95) {slowed near the Sun};
\end{tikzpicture}
$$

> **Worked example.** Estimate the round-trip Shapiro delay for a radar pulse
> grazing the Sun on its way to Mars near superior conjunction. Take $b = R_\odot =
> 6.96\times10^8\ \text{m}$, $r_\oplus = 1.50\times10^{11}\ \text{m}$, and $r_p =
> 2.28\times10^{11}\ \text{m}$. The prefactor is
> $$4GM_\odot/c^3 = 4(1.477\times10^3)/(3.00\times10^8) = 1.97\times10^{-5}\ \text{s}.$$
> The logarithm is
> $$
> \ln\!\left(\frac{4 r_\oplus r_p}{b^2}\right)
> = \ln\!\left(\frac{4(1.50\times10^{11})(2.28\times10^{11})}{(6.96\times10^8)^2}\right)
> = \ln(2.82\times10^{5}) = 12.5.
> $$
> The delay is $\Delta t_{\text{round}} \approx (1.97\times10^{-5})(12.5) =
> 2.5\times10^{-4}\ \text{s}$, about $250\ \mu\text{s}$. Shapiro proposed the test
> in 1964; radar ranging to Mercury and Venus, then transponders on the Mariner and
> Viking spacecraft, confirmed the predicted delay, and the Cassini mission measured
> the coefficient to about $10^{-5}$.

$$
% caption: The excess round-trip delay is largest when the ray grazes the Sun
% (small impact parameter) and falls off logarithmically as the path clears the
% limb; the vertical scale is the delay in microseconds.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (0,3.7) node[above] {delay (microsec)};
  \draw[->, black] (0,0) -- (5.6,0) node[right] {impact parameter (solar radii)};
  % logarithmic decay curve: delay ~ A ln(C/b^2); use samples, b from 1 to 5
  \draw[acc, very thick, domain=1:5, samples=120, variable=\x]
    plot ({\x},{3.0 - 1.55*ln(\x)});
  % solar limb marker
  \draw[black, dashed] (1,0) -- (1,3.05);
  \fill (1,3.0) circle (1.6pt);
  \node[above right] at (1.0,3.0) {grazing};
  \foreach \x in {1,2,3,4,5} \draw[black] (\x,0.05)--(\x,-0.05) node[below]{\x};
  \node[black, right] at (3.3,{3.0-1.55*ln(3.4)+0.28}) {logarithmic decay};
\end{tikzpicture}
$$

## What the two tests isolate

The deflection of light samples both the time and space parts of the metric and
returns their sum, the factor $4GM/(c^2 b)$. The redshift and the Shapiro delay
sample the time part alone.

- **Redshift** — a local rate comparison, fixed by $g_{tt}$ at the two endpoints
  through $\Delta\nu/\nu = -\Delta\Phi/c^2$. It tests the equivalence principle and
  the metric's time component without probing curvature of space at all.
- **Shapiro delay** — an integrated coordinate-time excess along a path, fixed by
  $g_{tt}$ over the whole trajectory. In the standard parametrization of weak-field
  metrics its coefficient carries the same combination that sets the redshift.

Measuring the deflection (space $+$ time), the redshift (time only), and the
Shapiro delay (time only) overdetermines the weak-field metric and pins both of its
independent functions to the general-relativistic values. The one everyday
application that relies on the redshift piece is satellite navigation, taken up
next.[^hartle][^carroll]

[^hartle]: Hartle, _Gravity: An Introduction to Einstein's General Relativity_,
§6.2–§6.3 (gravitational redshift and the Pound–Rebka experiment) and §10.4 (the
time delay of light).
[^carroll]: Carroll, _Lecture Notes on General Relativity_, §7 (the Schwarzschild
solution: gravitational redshift and radar echoes), arXiv:gr-qc/9712019.