Geodesics and Light Rays

In conventional geometry, the geodesic is the shortest curve between two points measured by counting how many rulers fit along the curve. When learning relativity theory, one often reads statements that conflict with intuition, such as the following: “In any space-time, with or without a gravitational field, light always moves along geodesics and traces out the geometry of space-time.” “In a space warped by a gravitational field, the light rays are curved and in general do not coincide with geodesics.” Why are these phrases, taken from the general theory of relativity (GTR), not really in conflict with each other?

Answer

The two statements are not in conflict. One must always distinguish geodesics in four-dimensional space-time from geodesics in three-dimensional space. Light rays always follow geodesics in 4-D space-time, but these paths are not necessarily geodesics in 3-D space. An analogy is helpful. Each great circle on a globe is a geodesic line on the two-dimensional surface but, being a circle, the great circle is not a geodesic line in the 3-D Euclidean space in which the globe sits.

In conventional geometry, the geodesic is the shortest curve between two points measured by counting how many rulers fit along the curve. In a flat space that is, in a space free from gravitational fields the geodesic is a straight line. In the GTR one can define the distance between two points in space as half the time it takes for light to travel from one point to the other and back, multiplied by the speed of light. In flat space, the two definitions agree.

In 4-D space-time, light always moves along geodesics and traces the geometry of space-time. In a 3-D space warped by a gravitational field, however, the light rays are curved and do not coincide with geodesics in general, so the geometry of space is not traced by light rays.