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 (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 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. Using Δx = cΔt, 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 stretching of space, both of which contribute equally in general relativity for the time delay as they also do for the deflection of light. 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 spacecraft. Shapiro delay of neutrinos and gravitational waves From the nearly simultaneous observations of neutrinos and photons from SN 1987A, 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, which imply that those neutrinos were moving at very close to the speed of light. After the direct detection of gravitational waves in 2016, the one-way Shapiro delay was calculated by two groups