Schwarzschild Coordinates - Visualizing The Static Hyperslices

Visualizing The Static Hyperslices

To better understand the significance of the Schwarzschild radial coordinate, it may help to embed one of the spatial hyperslices (they are of course all isometric to one another) in a flat Euclidean space. Humans who find it difficult to visualize four dimensional Euclidean space will be glad to observe that we can take advantage of the spherical symmetry to suppress one coordinate. This may be conveniently achieved by setting . Now we have a two-dimensional Riemannian manifold with a local radial coordinate chart,

To embed this surface (or at an annular ring) in E3, we adopt a frame field in E3 which

1. is defined on a parameterized surface, which will inherit the desired metric from the embedding space,
2. is adapted to our radial chart,
3. features an undetermined function h(r).

To wit, consider the parameterized surface

The coordinate vector fields on this surface are

The induced metric inherited when we restrict the Euclidean metric on E3 to our parameterized surface is

To identify this with the metric of our hyperslice, we should evidently choose h(r) so that

To take a somewhat silly example, we might have .

This works for surfaces in which true distances between two radially separated points are larger than the difference between their radial coordinates. If the true distances are smaller, we should embed our Riemannian manifold as a spacelike surface in E1,2 instead. For example, we might have . Sometimes we might need two or more local embeddings of annular rings (for regions of positive or negative Gaussian curvature). In general, we should not expect to obtain a global embedding in any one flat space (with vanishing Riemann tensor).

The point is that the defining characteristic of a Schwarzschild chart in terms of the geometric interpretation of the radial coordinate is just what we need to carry out (in principle) this kind of spherically symmetric embedding of the spatial hyperslices.