Shapiro time delay
In a 1964 article entitled Fourth Test of General Relativity, Irwin Shapiro wrote:[1] Because, according to the general theory, the speed of a light wave depends on the strength of the gravitational potential along its path, these time delays should thereby be increased by almost 2×10−4 sec when the radar pulses pass near the sun.
Such a change, equivalent to 60 km in distance, could now be measured over the required path length to within about 5 to 10% with presently obtainable equipment.
Throughout this article discussing the time delay, Shapiro uses c as the speed of light and calculates the time delay of the passage of light waves or rays over finite coordinate distance according to a Schwarzschild solution to the Einstein field equations.
Shapiro proposed an observational test of his prediction: bounce radar beams off the surface of Venus and Mercury and measure the round-trip travel time.
When the Earth, Sun, and Venus are most favorably aligned, Shapiro showed that the expected time delay, due to the presence of the Sun, of a radar signal traveling from the Earth to Venus and back, would be about 200 microseconds,[1] well within the limitations of 1960s-era technology.
The first tests, performed in 1966 and 1967 using the MIT Haystack radar antenna, were successful, matching the predicted amount of time delay.
Shapiro's original formulation was derived from the Schwarzschild solution and included terms to the first order in solar mass (
) for a proposed Earth-based radar pulse bouncing off an inner planet and returning passing close to the Sun:[1] where
is the distance along the line of flight from the Earth-based antenna to the point of closest approach to the Sun, and
The right-hand side of this equation is primarily due to the variable speed of the light ray; the contribution from the change in path, being of second order in
[4] Shapiro delay must be considered along with ranging data when trying to accurately determine the distance to interplanetary probes such as the Voyager and Pioneer spacecraft.
After the direct detection of gravitational waves in 2016, the one-way Shapiro delay was calculated by two groups and is about 1800 days.
In general relativity and other metric theories of gravity, though, the Shapiro delay for gravitational waves is expected to be the same as that for light and neutrinos.
