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TheShapiro time delay effect, orgravitational time delay effect, is one of the four classic Solar Systemtests of general relativity.Radar signals passing near a massive object take slightly longer to travel to a target and longer to return than they would if the mass of the object were not present. The time delay is caused bytime dilation, which increases the time it takes light to travel a given distance from the perspective of an outside observer. In a 1964 article entitledFourth 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 almost2×10⁻⁴ 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 usesc as the speed of light and calculates the time delay of the passage of light waves or rays over finite coordinate distance according to aSchwarzschild solution to theEinstein field equations.

The time delay effect was first predicted in 1964, byIrwin Shapiro. 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 theMITHaystack radar antenna, were successful, matching the predicted amount of time delay.^[2] The experiments have been repeated many times since then, with increasing accuracy.

In a nearly static gravitational field of moderate strength (say, of stars and planets, but not one of a black hole or close binary system of neutron stars) the effect may be considered as a special case ofgravitational time dilation. The measured elapsed time of a light signal in a gravitational field is longer than it would be without the field, and for moderate-strength nearly static fields the difference is directly proportional to the classicalgravitational potential, precisely as given by standard gravitational time dilation formulas.

Shapiro's original formulation was derived from the Schwarzschild solution and included terms to the first order in solar mass (${\displaystyle M}$) for a proposed Earth-based radar pulse bouncing off an inner planet and returning passing close to the Sun:^[1]

${\displaystyle \mathrm{\Delta }t\approx \frac{4GM}{c^3}\left(\ln \left[\frac{x_p+(x_p^2+d^2{)}^{1/2}}{-x_e+(x_e^2+d^2{)}^{1/2}}\right]-\frac{1}{2}\left[\frac{x_p}{(x_p^2+d^2{)}^{1/2}}+\frac{2x_e+x_p}{(x_e^2+d^2{)}^{1/2}}\right]\right)+\mathcal{O}\left(\frac{G^2{M}^2}{c^{5}d}\right),}$

where${\displaystyle d}$ is the distance of closest approach of the radar wave to the center of the Sun,${\displaystyle x_e}$ is the distance along the line of flight from the Earth-based antenna to the point of closest approach to the Sun, and${\displaystyle x_p}$ represents the distance along the path from this point to the planet. 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${\displaystyle M}$, is negligible.${\displaystyle \mathcal{O}}$ is theLandau symbol of order of error.

For a signal going around a massive object, the time delay can be calculated as the following:^[3]

${\displaystyle \mathrm{\Delta }t=-\frac{2GM}{c^3}\ln (1-\mathbf{R}\cdot \mathbf{x}).}$

Here${\displaystyle \mathbf{R}}$ is theunit vector pointing from the observer to the source, and${\displaystyle \mathbf{x}}$ is the unit vector pointing from the observer to the gravitating mass${\displaystyle M}$. The dot denotes the usual Euclideandot product.

Using${\displaystyle \mathrm{\Delta }x=c\mathrm{\Delta }t}$, this formula can also be written as

${\displaystyle \mathrm{\Delta }x=-R_{\text{s}}\ln (1-\mathbf{R}\cdot \mathbf{x}),}$

which is a fictive extra distance the light has to travel. Here${\displaystyle R_{\text{s}}=\frac{2GM}{c^2}}$ is theSchwarzschild radius.

InPPN parameters,

${\displaystyle \mathrm{\Delta }t=-(1+\gamma )\frac{R_{\text{s}}}{2c}\ln (1-\mathbf{R}\cdot \mathbf{x}),}$

which is twice the Newtonian prediction (with${\displaystyle \gamma =0}$).

The doubling of the Shapiro factor can be explained by the fact that there is not only the gravitational time dilation, but also the radial stretching of space, both of which contribute equally in general relativity for the time delay as they also do for the deflection of light.^[4]

${\displaystyle \tau =t\sqrt{1-\frac{R_{\text{s}}}{r}}}$

${\displaystyle c^{\prime }=c\sqrt{1-\frac{R_{\text{s}}}{r}}}$

${\displaystyle s^{\prime }=\frac{s}{\sqrt{1-\frac{R_{\text{s}}}{r}}}}$

${\displaystyle T=\frac{s^{\prime }}{c^{\prime }}=\frac{s}{c\left(1-\frac{R_{\text{s}}}{r}\right)}}$

Shapiro delay must be considered along with ranging data when trying to accurately determine the distance to interplanetary probes such as theVoyager andPioneer spacecraft.^{[citation needed]}

From the nearly simultaneous observations ofneutrinos andphotons fromSN 1987A, the Shapiro delay for high-energy neutrinos must be the same as that for photons to within 10%, consistent with recent estimates of theneutrino mass, which imply that those neutrinos were moving at very close to thespeed of light. After the direct detection ofgravitational waves in 2016, the one-way Shapiro delay was calculated by two groups and is about 1800 days. Ingeneral 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. However, in theories such astensor–vector–scalar gravity and other modified GR theories, which reproduceMilgrom's law and avoid the need fordark matter, the Shapiro delay for gravitational waves is much smaller than that for neutrinos or photons. The observed 1.7-second difference in arrival times seen between gravitational wave and gamma ray arrivals from neutron star mergerGW170817 was far less than the estimated Shapiro delay of about 1000 days. This rules out a class ofmodified models of gravity that dispense with the need fordark matter.^[5]

• Gravitational redshift andblueshift
• Gravitational lens
• Proper time
• VSOP (planets)
• Gravitomagnetic time delay

• van Straten W; Bailes M; Britton M; et al. (12 July 2001)."Boost for General Relativity".Nature.412 (6843):158–60.arXiv:astro-ph/0108254.Bibcode:2001Natur.412..158V.doi:10.1038/35084015.hdl:1959.3/1820.PMID 11449265.S2CID 4363384.
• d'Inverno, Ray (1992).Introducing Einstein's Relativity.Clarendon Press.ISBN 978-0-19-859686-8. SeeSection 15.6 for an excellent advanced undergraduate level introduction to the Shapiro effect.
• Will, Clifford M. (2014)."The Confrontation between General Relativity and Experiment".Living Reviews in Relativity.17 (1):4–107.arXiv:1403.7377.Bibcode:2014LRR....17....4W.doi:10.12942/lrr-2014-4.PMC 5255900.PMID 28179848. A graduate level survey of the solar system tests, and more.
• John C. Baez; Emory F. Bunn (2005). "The Meaning of Einstein's Equation".American Journal of Physics.73 (7):644–652.arXiv:gr-qc/0103044.Bibcode:2005AmJPh..73..644B.doi:10.1119/1.1852541.S2CID 119456465.
• Michael J. Longo (January 18, 1988)."New Precision Tests of the Einstein Equivalence Principle from Sn1987a".Physical Review Letters.60 (3):173–175.Bibcode:1988PhRvL..60..173L.doi:10.1103/PhysRevLett.60.173.PMID 10038466.
• Lawrence M. Krauss; Scott Tremaine (January 18, 1988). "Test of the Weak Equivalence Principle for Neutrinos and Photons".Physical Review Letters.60 (3):176–177.Bibcode:1988PhRvL..60..176K.doi:10.1103/PhysRevLett.60.176.PMID 10038467.
• S. Desai; E. Kahya; R. P. Woodard (2008). "Reduced time delay for gravitational waves with dark matter emulators".Physical Review D.77 (12): 124041.arXiv:0804.3804.Bibcode:2008PhRvD..77l4041D.doi:10.1103/PhysRevD.77.124041.S2CID 118785933.
• E. Kahya; S. Desai (2016). "Constraints on frequency-dependent violations of Shapiro delay from GW150914".Physics Letters B.756:265–267.arXiv:1602.04779.Bibcode:2016PhLB..756..265K.doi:10.1016/j.physletb.2016.03.033.S2CID 54657234.

• Effects of gravity
• Tests of general relativity

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