Shapiro time delay is the effect where light passing close to a massive object will take longer to reach the observer than light travelling a similar distance in vacuum. This arises due to the curvature of space-time in General Relativity, and is considered to be one of the classical tests of the theory. Consider a photon travelling by a mass with an impact parameter . Using Schwarzschild metric with a light-like path () we get (in polar coordinates around the mass ):
The unperturbed, straight line solution, is given by: (which is not a geodesic). It follows that: which leads to: . The metric can be re-written as:
Expanding in the small parameter we get:
This expression may be integrated between two locations and to give the time it takes light to travel between these locations:
The first term is the Newtonian delay between emission at and absorption at , while the other terms are corrections due to the curvature of space-time.