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 () for a proposed Earth-based radar pulse bouncing off an inner planet and returning passing close to the Sun:[1]
where is the distance of closest approach of the radar wave to the center of the Sun, is the distance along the line of flight from the Earth-based antenna to the point of closest approach to the Sun, and 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 , is negligible. 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:[citation needed]
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 . The dot denotes the usual Euclidean dot product.
Using , this formula can also be written as
which is a fictive extra distance the light has to travel. Here is the Schwarzschild radius.
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 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.
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