Shapiro delay

The Shapiro delay is the additional time, measured by a distant observer, that a light or radio signal takes to travel a given path when that path passes close to a massive body. In the Schwarzschild geometry the coordinate speed of light, dr/dt = c(1 − r_s/r), is reduced below c by a fractional amount r_s/r, where r_s = 2GM/c² is the Schwarzschild radius. Integrating this slowdown along a ray that skirts the Sun at impact parameter b yields an extra one-way delay Δt ≈ (2GM/c³) ln[(r₁+x₁)(r₂+x₂)/b²], on the order of a hundred microseconds for a signal grazing the solar limb.

Crucially, nothing local is slowed: an observer riding alongside the signal always measures the speed of light to be exactly c. The delay is a feature of the relationship between the distant observer's coordinate time and the path through curved geometry — the gravitational time dilation and spatial stretching of the metric, not any force or medium acting on the light. The effect is therefore a clean demonstration that gravity is geometry. Because the impact parameter enters only through a logarithm, the delay spikes sharply but never diverges as the line of sight closes on the mass.

Predicted by Irwin Shapiro in 1964 as the 'fourth test' of general relativity and first measured by bouncing radar off Venus and Mercury in 1966–67, the effect has since been confirmed to one part in 10^5 by the Cassini spacecraft in 2002, providing the tightest single constraint on the PPN curvature parameter gamma. Today it is routinely corrected for in deep-space navigation and pulsar timing rather than tested.

Irwin Shapiro proposed the test in a two-page Physical Review Letters paper in December 1964 and confirmed it with the Haystack and Millstone Hill radars over the following years. Planetary-surface echoes limited early precision to ~20%; switching to spacecraft transponders (Viking, 1976) reached 0.1%, and the multi-frequency Cassini radio link (2002) reached γ − 1 = (2.1 ± 2.3) × 10⁻⁵.