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 serve to establish observational evidence for the theory of general relativity. The first three tests, proposed by Albert Einstein in 1915, concerned the "anomalous" precession of the perihelion of Mercury, the ben ...
.
Radar
Radar is a detection system that uses radio waves to determine the distance ('' ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, we ...
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 by spacetime dilation, which increases the time it takes light to travel a given distance from the perspective of an outside observer. In a 1964 article entitled ''Fourth Test of General Relativity'', Shapiro wrote:
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
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an
exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assump ...
to the
Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
.
History
The time delay effect was first predicted in 1964, by
Irwin 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,
[ well within the limitations of 1960s-era technology.
The first tests, performed in 1966 and 1967 using the ]MIT
The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...
Haystack radar antenna, 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 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 of gravitational time dilation
Gravitational time dilation is a form of time dilation, an actual difference of elapsed time between two events as measured by observers situated at varying distances from a gravitating mass. The lower the gravitational potential (the closer ...
. 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 classical gravitational potential
In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...
, precisely as given by standard gravitational time dilation formulas.
Time delay due to light traveling around a single mass
Shapiro's original formulation was derived from the Schwarzschild solution and included terms to the first order in solar mass (''M'') for a proposed Earth-based radar pulse bouncing off an inner planet and returning passing close to the Sun:
:
where ''d'' is the distance of closest approach of the radar wave to the center of the Sun, ''xe'' is the distance along the line of flight from the Earth-based antenna to the point of closest approach to the Sun, and ''xp'' 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 ''M'', is negligible. ''O'' is the Landau symbol of order of error.
For a signal going around a massive object, the time delay can be calculated as the following:
:
Here R is the unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction v ...
pointing from the observer to the source, and x is the unit vector pointing from the observer to the gravitating mass ''M''. The dot denotes the usual Euclidean dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
.
Using Δ''x'' = ''c''Δ''t'', this formula can also be written as
:
which is a fictive extra distance the light has to travel. Here is the Schwarzschild radius
The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteris ...
.
In PPN parameters,
:
which is twice the Newtonian prediction (with ).
The doubling of the Shapiro factor can be explained by the fact that there is not only the gravitational time dilation, but also the stretching of space, both of which contribute equally in general relativity for the time delay as they also do for the deflection of light.
:
:
:
[ Elena V. Pitjevabr> Tests of General Relativity from observations of planets and spacecraft]
(slides undated).
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
Pioneer commonly refers to a settler who migrates to previously uninhabited or sparsely inhabited land.
In the United States pioneer commonly refers to an American pioneer, a person in American history who migrated west to join in settling and de ...
spacecraft.
Shapiro delay of neutrinos and gravitational waves
From the nearly simultaneous observations of neutrinos
A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass is ...
and photons from SN 1987A
SN 1987A was a type II supernova in the Large Magellanic Cloud, a dwarf satellite galaxy of the Milky Way. It occurred approximately from Earth and was the closest observed supernova since Kepler's Supernova. 1987A's light reached Earth on ...
, the Shapiro delay for high-energy neutrinos must be the same as that for photons to within 10%, consistent with recent estimates of the neutrino mass
A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass i ...
, which imply that those neutrinos were moving at very close to the speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
. After the direct detection of gravitational waves
Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
in 2016, the one-way Shapiro delay was calculated by two groups and is about 1800 days. In general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
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 as tensor–vector–scalar gravity
Tensor–vector–scalar gravity (TeVeS),
developed by Jacob Bekenstein in 2004, is a relativistic generalization of Mordehai Milgrom's Modified Newtonian dynamics (MOND) paradigm.
The main features of TeVeS can be summarized as follows:
* As i ...
and other modified GR theories, which reproduce Milgrom's law and avoid the need for dark matter
Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not a ...
, 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 merger GW170817
GW 170817 was a gravitational wave (GW) signal observed by the LIGO and Virgo detectors on 17 August 2017, originating from the shell elliptical galaxy . The signal was produced by the last minutes of a binary pair of neutron stars' insp ...
was far less than the estimated Shapiro delay of about 1000 days. This rules out a class of modified models of gravity that dispense with the need for dark matter
Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not a ...
.
See also
* Gravitational redshift
In physics and general relativity, gravitational redshift (known as Einstein shift in older literature) is the phenomenon that electromagnetic waves or photons travelling out of a gravitational well (seem to) lose energy. This loss of energy ...
and blueshift
In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in f ...
* Gravitational lens
A gravitational lens is a distribution of matter (such as a cluster of galaxies) between a distant light source and an observer that is capable of bending the light from the source as the light travels toward the observer. This effect is known ...
* Proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...
* VSOP (planets)
The semi-analytic planetary theory VSOP (French: ''Variations Séculaires des Orbites Planétaires'') is a mathematical model describing long-term changes (secular variation) in the orbits of the planets Mercury to Neptune. The earliest modern sci ...
* Gravitomagnetic time delay
References
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.
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{{Relativity
Effects of gravitation
Tests of general relativity