Curved spacetime as a gravitational field

Abstract

Gravitational time dilation can be interpreted as showing the warpage of spacetime in the time direction. Such considerations led Einstein to propose general relativity as a gravitational field theory with the field being curved spacetime. The geodesic equation is identified as the equation of motion of this field theory. This equation reduces to Newton’s equation of motion in the limit of particles moving with nonrelativistic velocities in a weak and static gravitational field. This clarifies the sense how Newton’s theory is extended by GR to new physical regimes. GR equations must be tensor equations with covariant derivatives in order to have proper transformation properties under position-dependent transformations. The replacement of ordinary derivatives by covariant derivatives, which bring in Christoffel symbols, automatically introduces gravity into physics equations.