Math 2.0

Relativity and gravitational theory Classical tests and observational relativity Proposition

Shapiro time delay

Builds on Schwarzschild metricGeodesics in general relativityParameterized post-Newtonian formalism

Statement

Let MM be a mass, GG the gravitational constant and cc the speed of light. Consider the weak field of the Schwarzschild metric of mass MM. Let a light signal travel along a null geodesic between two points at distances r1r_1 and r2r_2 from the mass. Let dd be the Euclidean distance between the two points. Then, to first order in GM/c2GM/c^2, the coordinate travel time exceeds the flat-space value d/cd/c by Δt=2GMc3ln⁡(r1+r2+dr1+r2−d).\Delta t = \frac{2 G M}{c^3} \ln\left( \frac{r_1 + r_2 + d}{r_1 + r_2 - d} \right). If the ray passes the mass at closest distance b≪r1,r2b \ll r_1, r_2, this becomes Δt≈2GMc3ln⁡4r1r2b2\Delta t \approx \frac{2GM}{c^3} \ln \frac{4 r_1 r_2}{b^2}, and a radar echo that goes and returns has twice this delay.

Remarks

In the weak field the metric is ds2≈−(1−2GM/(c2r))c2dt2+(1+2GM/(c2r))∣d𝐱∣2ds^2 \approx -(1 - 2GM/(c^2 r)) c^2 dt^2 + (1 + 2GM/(c^2 r)) \abs{d\mathbf{x}}^2 in isotropic coordinates. Along a null ray, c dt≈(1+2GM/(c2r))∣d𝐱∣c \, dt \approx (1 + 2GM/(c^2 r)) \abs{d\mathbf{x}}, so the delay is (2GM/c3)∫dl/r(2GM/c^3) \int dl / r along the straight line, and this integral is the logarithm above. Shapiro proposed the effect in 1964 as a new test of general relativity, using radar echoes from Venus and Mercury. In the parameterized post-Newtonian formalism the delay is multiplied by (1+γ)/2(1 + \gamma)/2; tracking of the Cassini spacecraft in 2002 found γ\gamma equal to 11 within about 2×10−52 \times 10^{-5}.

Going deeper

For a student

The Shapiro delay is the time-domain partner of the deflection of light: both come from the same weak-field metric and both measure the parameter γ\gamma. It is also seen in binary pulsars, where the pulses from a pulsar are delayed as they pass its companion, and this gives a way to measure the companion's mass. The delay depends on the path through its logarithm, so it is largest for rays that pass close to the mass.

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  1. Geodesics in general relativity concept
  2. Parameterized post-Newtonian formalism concept
  3. Ricci curvature concept
  4. Schwarzschild metric example
  5. Shapiro time delay statement

Where this leads

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Further reading

  1. Web https://en.wikipedia.org/wiki/Shapiro_time_delay
About this page
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Kindstatement, proposition
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MSC83C10, 83B05
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AboutGeodesics in general relativity
UsesSchwarzschild metric, Geodesics in general relativity, Parameterized post-Newtonian formalism
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Changed3 November 2026

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