Shapiro time 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. 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 by time 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, 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

The first tests, performed in 1966 and 1967 using the MIT Haystack 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 of gravitational 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 classical gravitational 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]

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

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 the Landau symbol of order of error.

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

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

Here ${\displaystyle \mathbf {R} }$ is the unit 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 Euclidean dot product.

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

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

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

In PPN parameters,

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

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]

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

${\displaystyle c'=c{\sqrt {1-{\tfrac {R_{\text{s}}}{r}}}}}$

${\displaystyle s'={\frac {s}{\sqrt {1-{\tfrac {R_{\text{s}}}{r}}}}}}$

${\displaystyle T={\frac {s'}{c'}}={\frac {s}{c\left(1-{\tfrac {R_{\text{s}}}{r}}\right)}}}$

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

From the nearly simultaneous observations of

• blueshift
• Gravitational lens
• Proper time
• VSOP (planets)
• Gravitomagnetic time delay