Page:Eddington A. Space Time and Gravitation. 1920.djvu/114

98 We start with $$r$$ large. By-and-by we approach the point where $$r = 2m$$. But here, from its definition, \gamma is equal to 0. So that, however large the measured interval $$ds$$ may be, $$dr = 0$$. We can go on shifting the measuring-rod through its own length time after time, but $$dr$$ is zero; that is to say, we do not reduce $$r$$. There is a magic circle which no measurement can bring us inside. It is not unnatural that we should picture something obstructing our closer approach, and say that a particle of matter is filling up the interior.

The fact is that so long as we keep to space-time curved only in the first degree, we can never round off the summit of the hummock. It must end in an infinite chimney. In place of the chimney, however, we round it off with a small region of greater curvature. This region cannot be empty because the law applying to empty space does not hold. We describe it therefore as containing matter—a procedure which practically amounts to a definition of matter. Those familiar with hydrodynamics may be reminded of the problem of the irrotational rotation of a fluid; the conditions cannot be satisfied at the origin, and it is necessary to cut out a region which is filled by a vortex-filament.

A word must also be said as to the coordinates $$r$$ and $$t$$ used in (6). They correspond to our ordinary notion of radial distance and time—as well as any variables in a non-Euclidean world can correspond to words which, as ordinarily used, presuppose a Euclidean world. We shall thus call $$r$$ and $$t$$, distance and time. But to give names to coordinates does not give more information—and in this case gives considerably less information—than is already contained in the formula for $$ds^2$$. If any question arises as to the exact significance of $$r$$ and $$t$$ it must always be settled by reference to equation (6).

The want of flatness in the gravitational field is indicated by the deviation of the coefficient $$\gamma$$ from unity. If the mass $$m=0$$, $$\gamma = 1$$, and space-time is perfectly flat. Even in the most intense gravitational fields known, the deviation is extremely small. For the sun, the quantity $$m$$, called the gravitational mass, is only 1.47 kilometres, for the earth it is 5 millimetres. In any practical problem the ratio $$2m/r$$ must be exceedingly small.