The most general form of the metric that describes a static, isotropic, four-dimensional spacetime is
,
where
is time,
,
,
are coordinates in a spherical coordinate system, and
and
are some functions of the radial coordinate
.
A free falling particle in a gravitational field follows a trajectory in spacetime called a geodesic. This path is a solution to the differential equation
,
where
for
are coordinates in spacetime and
is a parameter along the geodesic. Here,
represents time while
,
, and
are the spatial coordinates. The objects
with
,
and
, each taking values 0, 1, 2, and 3 are called Christoffel symbols; they can be explicitly calculated from the metric given above. Solving the geodesic equation in the static, isotropic gravitational field yields the "bending" of the path of a photon (a ray of light) in the gravitational field of a massive spherical object like a star.
The deflection of light by the Sun was a significant early experimental verification of the general theory of relativity. The deflection was first measured in 1919 and was found to be in agreement (within experimental bounds) with the theoretical prediction of 1.75". For further details see
S. Weinberg,
Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, New York, NY: Wiley, 1972.
