General discussion of the Schwartzschild solution

In order to understand the physical properties of the Schwartzschild solution we first have to clarify the physical significance of the integration constant . This is best done through a comparison with Newtonian theory. For large values of the coordinate r, the line element deviates only a little from that of flat space, and from our analysis of the Newtonian limit we see that

so

We thus have to interpret the Schwarzschild solution as the gravitational field outside a spherically symmetric mass distribution whose [ Newtonian ] mass is M.

Notice that the line element is singular when . This is the Schwartzschild radius or the gravitational radius of the source. For example for the Sun km and for the Earth it is mm.

From the line element, we can immediately deduce some of the effects we derived using the equivalence principle.

• Time dilation in a gravitational field
• Length contraction in a gravitational field