Send a radio pulse from the Earth to Venus and time how long it takes to bounce back. When the Earth and Venus line up almost straight across the Sun, the pulse passes just beside the edge of the Sun.
The round trip then takes longer than the round-trip distance divided by the speed of light. In Shapiro worked out the difference from general relativity and showed that it was large enough to measure. The extra time is called the Shapiro delay.
The classical tests of general relativity were three: the perihelion shift of Mercury, the bending of light and the gravitational redshift. The Shapiro delay, called the fourth test, measures the travel time of the signal itself.
Where gravity is strong, light makes slower progress as seen from far away. Measured with the clocks and rulers on the spot, the speed of light is still .
What differs is the clocks and the distances. Near the Sun clocks run slow, and because space is curved the path is also longer than it would be in flat space. Seen from far away, the two hold the light back by equal amounts.
The round-trip delay is . Here and are the distances from the Sun to the Earth and to Venus, is the distance of closest approach of the path to the centre of the Sun, and is the natural logarithm.
The smaller , the larger the delay, which is greatest when the path grazes the edge. Because sits inside the logarithm, making ten times larger reduces the delay by only about per cent.
Putting in the Sun's m³/s² and m/s makes equal to microseconds.
With m, m and, for a grazing path, equal to the Sun's radius of m, the quantity inside the logarithm is and the logarithm is . The round trip is delayed by , about microseconds.
In distance that is about km, against a round trip of some million km.
Radar measurements bouncing signals off Venus and Mercury ran from the late s, and by about they agreed with general relativity to within per cent. In the transponders on the Viking landers on Mars confirmed the agreement to one part in a thousand.
Seen as a wave, light makes slower progress on the side nearer the Sun, so the wavefront lags on that side.
The direction of travel is perpendicular to the wavefront, so when the front tilts the direction tilts too, and the light bends toward the slower side, toward the Sun.
Starlight grazing the edge of the Sun during an eclipse is bent by arcseconds. That bending and the microsecond delay are both calculated from the slower progress of light near the Sun.
The delay is the slowness added up along the path, while the bending is set by the difference in slowness between the two sides of the path.
In , while the Cassini spacecraft on its way to Saturn passed behind the Sun, the frequency of the radio signals travelling between the Earth and the spacecraft was measured. As the path moved closer to the Sun and away again day by day, the changing delay showed up as a shift in frequency.
The result agreed with general relativity to about two parts in a hundred thousand. Many other theories of gravity have been proposed, and those that fail to match at this precision are ruled out.
A neutron star that sends out radio pulses at regular intervals is called a pulsar. The same delay has been measured for pulsars orbiting another star.
When the pulsar goes round behind its companion, the pulses are delayed only while they pass close to the companion. The shape of that delay gives the mass of the companion.