Historical background

In 1915 Einstein put the finishing touches to the general theory of relativity. The Schwarzschild solution described in Chapter 9 was the first physically significant solution of the field equations of general relativity. It showed how spacetime is curved around a spherically symmetric distribution of matter. The problem solved by Schwarzschild was basically a local problem, in the sense that the deviations of spacetime geometry from the Minkowski geometry of special relativity gradually diminish to zero as we move further and further away from the gravitating sphere. This result can be easily verified from the Schwarzschild line element by letting the radial coordinate go to infinity. In technical jargon a spacetime satisfying this property is called asymptotically flat. In general any spacetime geometry generated by a local distribution of matter is expected to have this property. Even from Newtonian gravity we expect an analogous result: that the gravitational field of a local distribution of matter will die away at a large distance from the distribution. Can the Universe be approximated by a local distribution of matter?

Einstein rightly felt that the answer to the above question would be in the negative. Rather, he expected the Universe to be filled with matter, howsoever far we are able to probe it. A Schwarzschild-type solution cannot therefore provide the correct spacetime geometry of such a distribution of matter. Since we can never get away from gravitating matter, the concept of asymptotic flatness must break down.