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 (${\displaystyle M}$) 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:

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

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

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

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

In PPN parameters,

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

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.

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

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

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

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

^[3]