Gaussian polar coordinates
In the theory of Lorentzian manifolds, spherically [symmetric spacetime]s admit a family of nested round spheres. In each of these spheres, every point can be carried to any other by an appropriate rotation about the centre of symmetry.
There are several different types of coordinate chart that are adapted to this family of nested spheres, each introducing a different kind of distortion. The best known alternative is the Schwarzschild chart, which correctly represents distances within each sphere, but distorts radial distances and angles. Another popular choice is the isotropic chart, which correctly represents angles. A third choice is the Gaussian polar chart, which correctly represents radial distances, but distorts transverse distances and angles. There are other possible charts; the article on spherically symmetric spacetime describes a coordinate system with intuitively appealing features for studying infalling matter. In all cases, the nested geometric spheres are represented by coordinate spheres, so we can say that their roundness is correctly represented.
Definition
In a Gaussian polar chart, the metric takes the formDepending on context, it may be appropriate to regard and as undetermined functions of the radial coordinate. Alternatively, we can plug in specific functions to obtain an isotropic coordinate chart on a specific Lorentzian spacetime.