Schwarzschild Coordinates - A Family of Static Nested Spheres

A Family of Static Nested Spheres

In the Schwarzschild chart, the surfaces appear as round spheres (when we plot loci in polar spherical fashion), and from the form of the line element, we see that the metric restricted to any of these surfaces is

That is, these nested coordinate spheres do in fact represent geometric spheres with

1. surface area
2. Gaussian curvature

That is, they are geometric round spheres. Moreover, the angular coordinates are exactly the usual polar spherical angular coordinates: is sometimes called the colatitude and is usually called the longitude. This is essentially the defining geometric feature of the Schwarzschild chart.

It may help to add that the four Killing fields given above, considered as abstract vector fields on our Lorentzian manifold, give the truest expression of both the symmetries of a static spherically symmetric spacetime, while the particular trigonometric form which they take in our chart is the truest expression of the meaning of the term Schwarzschild chart. In particular, the three spatial Killing vector fields have exactly the same form as the three nontranslational Killing vector fields in a spherically symmetric chart on E3; that is, they exhibit the notion of arbitrary Euclidean rotation about the origin or spherical symmetry.

However, note well: in general, the Schwarzschild radial coordinate does not accurately represent radial distances, i.e. distances taken along the spacelike geodesic congruence which arise as the integral curves of . Rather, to find a suitable notion of 'spatial distance' between two of our nested spheres, we should integrate along some coordinate ray from the origin:

Similarly, we can regard each sphere as the locus of a spherical cloud of idealized observers, who must (in general) use rocket engines to accelerate radially outward in order to maintain their position. These are static observers, and they have world lines of form, which of course have the form of vertical coordinate lines in the Schwarzschild chart.

In order to compute the proper time interval between two events on the world line of one of these observers, we must integrate along the appropriate coordinate line: