Einstein had still one further clue to guide him. Classical science had assumed that at an infinite distance from attracting matter the law of inertia would hold rigorously; that is to say, a body moving freely would follow a rectilinear course with constant speed when referred to a Galilean frame. Expressed in the language of space-time, this would imply that at an infinite distance from matter, space-time would become rigorously flat, as we had assumed it to be in the special theory. This condition necessitated the vanishing of the Riemann-Christoffel tensor
, at infinity. If this condition is maintained, the curvature of space-time around matter will have to be of a type which is capable of dying down to perfect flatness at infinity, degenerating to
.
Now the law of curvature
was precisely of such a type; hence Einstein, in his original paper, found it natural to select the tensor
for the left-side term of his gravitational equations. Accordingly he wrote: