Page:Eddington A. Space Time and Gravitation. 1920.djvu/102

86 more complex character of gravitational influence which is absolute.

The question must now be put, Can every possible kind of space-time occur in an empty region in nature? Suppose we give the ten potentials perfectly arbitrary values at every point; that will specify the geometry of some mathematically possible space-time. But could that kind of space-time actually occur—by any arrangement of the matter round the region?

The answer is that only certain kinds of space-time can occur in an empty region in nature. The law which determines what kinds can occur is the law of gravitation.

It is indeed clear that, since we have reduced the theory of fields of force to a theory of the geometry of the world, if there is any law governing fields of force (including the gravitational field), that law must be of the nature of a restriction on the possible geometries of the world.

The choice of $$g$$'s in any special problem is thus arrived at by a three-fold sorting out: (1) many sets of values can be dismissed because they can never occur in nature, (2) others, while possible, do not relate to the kind of space-time present in the problem considered, (3) of those which remain, one set of values relates to the particular mesh-system that has been chosen. We have now to find the law governing the first discrimination. What is the criterion that decides what values of the $$g$$'s give a kind of space-time possible in nature?

In solving this problem Einstein had only two clues to guide him.

(1) Since it is a question of whether the kind of space-time is possible, the criterion must refer to those properties of the $$g$$'s which distinguish different kinds of space-time, not to those which distinguish different kinds of mesh-system in the same space-time. The formulae must therefore not be altered in any way, if we change the mesh-system.

(2) We know that flat space-time can occur in nature (at great distances from all gravitating matter). Hence the criterion must be satisfied by any values of the $$g$$'s belonging to flat space-time.

It is remarkable that these slender clues are sufficient to indicate almost uniquely a particular law. Afterwards the