The Schwarzschild geometry is described by the line element \begin{equation} ds^2 = -\left(1-\frac{2m}{r}\right)\,dt^2 + \frac{dr^2}{1-\frac{2m}{r}} \\ + r^2\,d\theta^2 + r^2\sin^2\theta\,d\phi^2 \label{schwmet} \end{equation} As we will see later, this metric is the unique spherically symmetric solution of Einstein's equation in vacuum, and describes the gravitational field of a point mass at the origin. Yes, it also describes a black hole, but this was not realized for nearly 50 years; we will study these aspects of the Schwarzschild geometry later.

The geometry which bears his name was discovered by Karl Schwarzschild in early 1916 — just a few months after Einstein proposed the theory of general relativity in late 1915. It was also independently discovered later in 1916 by Johannes Droste, a student of Hendrik Lorentz, who was the first to write the line element in the form above.

When writing the line element in the form above, we have introduced geometric units not only for space ($r$) and time ($t$), but also for mass ($m$), all of which are measured in centimeters by setting both the speed of light $c$ and the gravitational constant $G$ to $1$. If you prefer, replace $t$ by $ct$, and $m$ by $mG/c^2$.

The Schwarzschild metric is asymptotically flat, in the sense that it reduces to the Minkowski metric of special relativity as $r$ goes to infinity. But there is clearly a problem with the Schwarzschild line element (\ref{schwmet}) when \begin{equation} r = 2m \end{equation} which is called the Schwarzschild radius. However, the Schwarzschild radius of the Earth is about 1 cm, and that of the Sun is about 3 km, so the peculiar effects which might occur at or near that radius are not relevant to our solar system. For example, the Earth's orbit is some 50 million Schwarzschild radii from the (center of the) Sun. We will interpret $t$ and $r$ as the coordinates of such a “far-away” observer, and we will assume that $r>2m$ until further notice.