The basic features of the Schwarzschild geometry are readily apparent by examining the metric.

• Asymptotic Flatness:
  As already discussed, the Schwarzschild geometry reduces to that of Minkowski space as $r$ goes to infinity.
• Spherical Symmetry:
  The angular dependence of the Schwarzschild metric is precisely the same as that of a sphere; the Schwarzschild geometry is spherically symmetric. It is therefore almost always sufficient to consider the equatorial plane $\theta=\pi/2$, so that $d\theta=0$. For instance, a freely falling object must move in such a plane — by symmetry, it can not deviate one way or the other. We can therefore work with the simpler line element \begin{equation} ds^2 = -\left(1-\frac{2m}{r}\right)\,dt^2 + \frac{dr^2}{1-\frac{2m}{r}} + r^2\,d\phi^2 \end{equation} As already stated, we will assume for now that $r>2m$.
• Circumference:
  Holding $r$ and $t$ constant, the Schwarzschild metric reduces to that of a sphere of radius $r$; such surfaces are spherical shells. We can consider $r$ to be the radius of such shells, but some care must be taken with this interpretation: The statement is not that there exists a ball of radius $r$, but only the surface of the ball. A more precise statement would be that the circumference of the spherical shell is given (at the equator) by \begin{equation} \int r\,d\phi = 2\pi r \end{equation} or that the area of the shell is $4\pi r^2$. Thus, $r$ is a geometric quantity, not merely a coordinate, even though the center of the spherical shells may not exist. In this sense, “radius” refers to the dimensions of the corresponding spherical shell, rather than to “distance” from the “center”.
• Gravitational Redshift:
  If we consider an observer “standing still” on shells of constant radius (so $dr=0=d\phi$), we see that \begin{equation} d\tau = \sqrt{|ds^2|} = \sqrt{1-\frac{2m}{r}}\>dt < dt \end{equation} Thus, “shell clocks” run slower than “far-away clocks”, and the period of, say, a beam of light of a given frequency will be larger for a far-away observer than for a shell observer. This is the gravitational redshift.
• Curvature:
  Similarly, if we measure the distance between two nearby shells (so $dt=0=d\phi$), we see that \begin{equation} ds = \frac{dr}{\sqrt{1-\frac{2m}{r}}} > dr \end{equation} so that the measured distance is larger than the difference in their “radii”. This is curvature.