Shapiro delay

Shapiro delay

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The Shapiro time delay effect, or gravitational time delay effect, is one of the four classic solar system tests of general relativity
Tests of general relativity
At its introduction in 1915, the general theory of relativity did not have a solid empirical foundation. It was known that it correctly accounted for the "anomalous" precession of the perihelion of Mercury and on philosophical grounds it was considered satisfying that it was able to unify Newton's...

. Radar
Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...

 signals passing near a massive object take slightly longer to travel to a target and longer to return than it would if the mass of the object were not present.


The time delay effect was first noticed in 1964, by Irwin I. Shapiro
Irwin I. Shapiro
Irwin I. Shapiro is an American astrophysicist. Since 1982, he has been a professor at Harvard University. Shapiro was director of the Harvard-Smithsonian Center for Astrophysics from 1982 to 2004.- Biography :Irwin Shapiro was born in New York City in 1929...

. 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, well within the limitations of 1960s era technology.

The first tests, performed in 1966 and 1967 using the MIT
Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...

 Haystack radar antenna
Haystack Observatory
Haystack Observatory is a group of astronomical observatories owned and operated by Massachusetts Institute of Technology. It is located in Westford, Massachusetts . It is the home of the Millstone Hill Observatory....

, were successful, matching the predicted amount of time delay. The experiments have been repeated many times since then, with increasing accuracy.

Calculating time delay

In a near-statical 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 of gravitational time dilation
Gravitational time dilation
Gravitational time dilation is the effect of time passing at different rates in regions of different gravitational potential; the lower the gravitational potential, the more slowly time passes...

. The speed of light in meters per given interval of "local time" (calculated by the metric tensor
Metric tensor (general relativity)
In general relativity, the metric tensor is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational field familiar from Newtonian gravitation...

) is a constant, however the travel time of any electromagnetic wave, or signal, moving at 299,792,458 meters per "second"
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

 is affected by the time dilation in regions of the space through which it travels. This is because the coordinate time
Coordinate time
In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. In many coordinate systems, an event is specified by one time coordinate and three spatial coordinates...

 and locally calculated time diverge as the gravitational field potential increases (by absolute value).

Time delay due to light traveling around a single mass

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

Here is the unit vector  pointing from the observer to the source, and
is the unit vector pointing from the observer to the gravitating mass M.

For the dot in , see Dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...


The above formula can be rearranged like this:

which is the extra distance the light has to travel, where is the Schwarzschild radius
Schwarzschild radius
The Schwarzschild radius is the distance from the center of an object such that, if all the mass of the object were compressed within that sphere, the escape speed from the surface would equal the speed of light...


This is the same as:

Interplanetary probes

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 (see the Voyager program
Voyager program
The Voyager program is a U.S program that launched two unmanned space missions, scientific probes Voyager 1 and Voyager 2. They were launched in 1977 to take advantage of a favorable planetary alignment of the late 1970s...

, the Pioneer program
Pioneer program
The Pioneer program is a series of United States unmanned space missions that was designed for planetary exploration. There were a number of such missions in the program, but the most notable were Pioneer 10 and Pioneer 11, which explored the outer planets and left the solar system...

, and the Pioneer anomaly
Pioneer anomaly
The Pioneer anomaly or Pioneer effect is the observed deviation from predicted accelerations of the Pioneer 10 and Pioneer 11 spacecraft after they passed about on their trajectories out of the Solar System....


Quote by Einstein

"In the second place our result shows that, according to the general theory of relativity, the law of the constancy of the velocity of light in vacuum, which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position. Now we might think that as a consequence of this, the special theory of relativity and with it the whole theory of relativity would be laid in the dust. But in reality this is not the case. We can only conclude that the special theory of relativity cannot claim an unlimited domain of validity ; its results hold only so long as we are able to disregard the influences of gravitational fields on the phenomena (e.g. of light)." – Albert Einstein (The General Theory of Relativity: Chapter 22 – A Few Inferences from the General Principle of Relativity)

Shapiro delay of neutrinos and gravitational waves

From the near-simultaneous observations of neutrinos and photons from SN 1987A
SN 1987A
SN 1987A was a supernova in the outskirts of the Tarantula Nebula in the Large Magellanic Cloud, a nearby dwarf galaxy. It occurred approximately 51.4 kiloparsecs from Earth, approximately 168,000 light-years, close enough that it was visible to the naked eye. It could be seen from the Southern...

, we know that the Shapiro delay for high-energy neutrinos is the same as that for photons to within 10% (consistent with recent estimates of the neutrino mass which imply that those neutrinos were moving at very close to the speed of light). Since gravitational waves have not been directly detected, we don't have any data on the Shapiro delay for gravitational waves. In general relativity and other metric theories of gravity, the Shapiro delay for gravitational waves is expected to be same as that for light (and neutrinos). However in theories such as Teves
Tensor-vector-scalar gravity
Tensor–vector–scalar gravity , developed by Jacob Bekenstein, is a relativistic generalization of Mordehai Milgrom's MOdified Newtonian Dynamics paradigm.The main features of TeVeS can be summarized as follows:...

 and other modified GR theories which reproduce Milgrom's law and avoid the need for dark matter, the Shapiro delay for gravitational waves is much smaller than that for neutrinos or photons.

See also

  • Gravitational redshift
    Gravitational redshift
    In astrophysics, gravitational redshift or Einstein shift describes light or other forms of electromagnetic radiation of certain wavelengths that originate from a source that is in a region of a stronger gravitational field that appear to be of longer wavelength, or redshifted, when seen or...

     and Blueshift
  • Proper time
    Proper time
    In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...

  • Secular variations of the planetary orbits
    Secular variations of the planetary orbits
    The secular variations of the planetary orbits is a concept describing long-term changes in the orbits of the planets Mercury to Neptune. If one ignores the gravitational attraction between the planets and only models the attraction between the Sun and the planets, then with some further...

Further reading

See Section 15.6 for an excellent advanced undergraduate level introduction to the Shapiro effect. A graduate level survey of the solar system tests, and more.