Gravitational Redshift and the Shapiro Delay

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 and 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 τ = - g_{tt} d t = 1 - \frac{2 GM}{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 $Δ t$ :

\frac{d τ _{1}}{d τ _{2}} = \frac{1 - 2 GM / ( c ^{2} r _{1} )}{1 - 2 GM / ( 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 $d τ_{1}$ emitted at $r_{1}$ spans a fixed coordinate interval $d t$ ; received at $r_{2}$ it is measured against $d τ_{2} > d τ_{1}$ , so its period is longer and its frequency lower. In terms of frequency,

\frac{ν _{2}}{ν _{1}} = \frac{d τ _{1}}{d τ _{2}} = \frac{1 - 2 GM / ( c ^{2} r _{1} )}{1 - 2 GM / ( 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 $Φ = - GM / r$ ,

\frac{Δ ν}{ν} = \frac{ν _{2} - ν _{1}}{ν _{1}} \approx \frac{Φ _{1} - Φ _{2}}{c ^{2}} = - \frac{ΔΦ}{c ^{2}} .

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

\frac{Δ ν}{ν} = - \frac{g h}{c ^{2}} .

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.

The Pound–Rebka experiment

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

\frac{Δ ν}{ν} = \frac{g h}{c ^{2}} = \frac{( 9.81 ) ( 22.5 )}{( 3.00 \times 1 0 ^{8} ) ^{2}} = 2.46 \times 1 0^{- 15},

a shift of two parts in $1 0^{15}$ . Resolving it required the Mössbauer effect: the $14.4 keV$ gamma ray from $^{57} Fe$ embedded in a crystal lattice is emitted with no recoil, giving a line sharp enough that a shift of $1 0^{- 15}$ 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 $Δ ν / ν = g h / c^{2}$ to about $10%$ , later improved to $1%$ . Modern optical-clock comparisons detect the gravitational redshift over height differences of a few centimetres.

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.

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 (1 + \frac{2Φ}{c ^{2}}) = c (1 - \frac{2 GM}{c ^{2} r}),

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

Δ t \approx \frac{2 GM}{c ^{3}} ln (\frac{4 r _{1} r _{2}}{b ^{2}}) .

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

Δ t_{round} \approx \frac{4 GM}{c ^{3}} ln (\frac{4 r _{\oplus} r _{p}}{b ^{2}}) .

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.

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.

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 $4 GM / (c^{2} b)$ . 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 $Δ ν / ν = - ΔΦ/ c^{2}$ . 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.¹²

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, Lecture Notes on General Relativity, §7 (the Schwarzschild solution: gravitational redshift and radar echoes), arXiv:gr-qc/9712019. ↩

Gravitational redshift from the metric

The Pound–Rebka experiment

The Shapiro time delay

What the two tests isolate

Footnotes