It was by following this method that Einstein was able to anticipate a number of gravitational effects which classical science had never even suspected. Chief among these are the bending of a ray of light in a gravitational field, and the Einstein-shift effect, since observed on the companion of Sirius.
We may understand without difficulty how the bending of a ray of light was anticipated. For consider an enclosure floating in empty space far from matter. A wave of light enters by a crack in the wall and travels through the enclosure along a line parallel to the floor. But if now the enclosure be uniformly accelerated, the floor advances with accelerated motion across the ray. As referred to the enclosure, the ray of light will thus assume a bent path, just as the course of a bullet would appear bent under similar circumstances.[83] Then the postulate of equivalence allows us to assert that exactly the same results would have ensued had the enclosure been at rest in a uniform gravitational field; so that regardless of whether or not uniform gravitational fields generated by matter can exist, we are led to the general conclusion that a gravitational field will bend the course of light waves and modify their velocity.
It is this qualitative discovery that is of importance and that constitutes a fact which was unknown to classical science. What the precise course of a light ray will be in a non-uniform gravitational field generated by matter, is a question of another order.
The postulate, however, enables us to answer this question, provided we know how the gravitational field is spatially distributed around matter. We may understand this point as follows: Owing to its gradual variations from place to place, even a non-uniform field may be regarded as uniform if we restrict our attention to very small regions of space.[84] It follows that we can decompose a non-uniform field into a series of contiguous volumes of space, each one of which may be assimilated to the spaces in enclosures possessing appropriate uniform accelerations. If, therefore, the spatial distribution of a non-uniform gravitational field is given, we may deduce (according to the methods of differential geometry) the path of a ray of light from place to place over contiguous infinitesimal distances. In this way, assuming Newton’s law to be correct, Einstein deduced the curvature of a ray of light in the sun’s gravitational field. He found that for a ray grazing the sun’s limb the deflection would be 0".87, just one-half of what he was to establish subsequently, after he had recognised that Newton’s law could not be correct.
In other cases, we are not concerned with the study of a phenomenon extending over large areas; its behaviour in a small area, say in our room, suffices. In this case, of course, the earth’s gravitational field may always be regarded as uniform; so that by considering the effect arising from the introduction of a constant acceleration of suitable magnitude, the action of gravitation on our phenomenon can be anticipated.
In a general way, the postulate of equivalence led Einstein to the following conclusions: Inertial mass and gravitational mass being one and the same thing, all forms of existence which possess inertial mass must also manifest weight in a gravitational field. Hence, as all forms of energy possess inertial mass, we see that energy has weight. A hot brick weighs more than a cold one, an electrified body more than a neutral one; possibly, also, a living animal more than a dead one, and so on. Conversely, all forms of energy must develop a gravitational field.
In the following pages we shall show that the postulate of equivalence, which in the present chapter was derived as a generalisation from the equality of the two masses, can also be deduced in a purely rational way from the existence of space-time. We shall thus be in a position to understand how all these discoveries dealing with forces and gravitation can be woven into the common space-time fabric.
CHAPTER XXV
THE INCLUSION OF GRAVITATION IN THE MODEL OF SPACE-TIME
IN the preceding chapter the following results were established:
1. No essential difference exists between a field of gravitation produced by matter and a field of inertia produced by the acceleration of our frame of reference (postulate of equivalence).