Editing Shapiro time delay (section)

== Calculating time delay ==
[[File:Shapiro delay thumbnail.png|thumb|right|320px|link=File:Shapiro_delay.gif|Left: unperturbed lightrays in a flat spacetime, right: Shapiro-delayed and deflected lightrays in the vicinity of a gravitating mass (click to start the animation)]]
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.

=== 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 (<math>M</math>) for a proposed Earth-based radar pulse bouncing off an inner planet and returning passing close to the Sun:<ref name="Shapiro1964" />
: <math>\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^2M^2}{c^5 d}\right),</math>
where <math>d</math> is the distance of closest approach of the radar wave to the center of the Sun, <math>x_e</math> is the distance along the line of flight from the Earth-based antenna to the point of closest approach to the Sun, and <math>x_p</math> 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 <math>M</math>, is negligible. <math>\mathcal{O}</math> is the [[Big O notation|Landau symbol]] of order of error.

For a signal going around a massive object, the time delay can be calculated as the following:<ref>{{Cite journal |last=Desai |first=S. |last2=Kahya |first2=E. O. |date=2016-04-30 |title=Galactic one-way Shapiro delay to PSR B1937+21 |url=https://www.worldscientific.com/doi/abs/10.1142/S0217732316500838 |journal=Modern Physics Letters A |language=en |volume=31 |issue=13 |pages=1650083 |doi=10.1142/S0217732316500838 |issn=0217-7323|arxiv=1510.08228 }}</ref>
: <math>\Delta t = -\frac{2GM}{c^3} \ln(1 - \mathbf{R}\cdot\mathbf{x}).</math>

Here <math>\mathbf{R}</math> is the [[unit vector]]  pointing from the observer to the source, and <math>\mathbf{x}</math> is the unit vector pointing from the observer to the gravitating mass <math>M</math>. The dot denotes the usual Euclidean [[dot product]].

Using <math>\Delta x = c \Delta t</math>, this formula can also be written as
: <math>\Delta x = -R_\text{s} \ln(1 - \mathbf{R}\cdot\mathbf{x}),</math>
which is a fictive extra distance the light has to travel. Here <math>\textstyle R_\text{s} = \frac{2GM}{c^2}</math> is the [[Schwarzschild radius]].

In [[Parameterized post-Newtonian formalism|PPN parameters]],
: <math>\Delta t = -(1 + \gamma) \frac{R_\text{s}}{2c} \ln(1 - \mathbf{R}\cdot\mathbf{x}),</math>
which is twice the Newtonian prediction (with <math>\gamma = 0</math>).

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.<ref name=Pitjeva>[[Elena V. Pitjeva]]:[http://www.acfc2011.ptb.de/acfc2011/574.html?&no_cache=1&cid=849&did=649&sechash=db8ad6ad  Tests of General Relativity from observations of planets and spacecraft] {{Webarchive|url=https://web.archive.org/web/20120426052017/http://www.acfc2011.ptb.de/acfc2011/574.html?&no_cache=1&cid=849&did=649&sechash=db8ad6ad |date=2012-04-26 }} (slides undated).</ref>
: <math>\tau = t\sqrt{1-\tfrac{R_\text{s}}{r}}</math>
: <math>c' = c\sqrt{1-\tfrac{R_\text{s}}{r}}</math>
: <math>s' = \frac{s}{\sqrt{1-\tfrac{R_\text{s}}{r}}}</math>
: <math>T = \frac{s'}{c'} = \frac{s}{c\left(1-\tfrac{R_\text{s}}{r}\right)} </math>