In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres.
This means that (except in the trivial case of a locally flat manifold), the angular isotropic coordinates do not faithfully represent distances within the nested spheres, nor does the radial coordinate faithfully represent radial distances.
On the other hand, angles in the constant time hyperslices are represented without distortion, hence the name of the chart.
Isotropic charts are most often applied to static spherically symmetric spacetimes in metric theories of gravitation such as general relativity, but they can also be used in modeling a spherically pulsating fluid ball, for example.
For isolated spherically symmetric solutions of the Einstein field equation, at large distances, the isotropic and Schwarzschild charts become increasingly similar to the usual polar spherical chart on Minkowski spacetime.
In an isotropic chart (on a static spherically symmetric spacetime), the metric (aka line element) takes the form Depending on context, it may be appropriate to regard
as undetermined functions of the radial coordinate (for example, in deriving an exact static spherically symmetric solution of the Einstein field equation).
is irrotational means that the vorticity tensor of the corresponding timelike congruence vanishes; thus, this Killing vector field is hypersurface orthogonal.
form a family of (isometric) spatial hyperslices (spacelike hypersurfaces).
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 where
shows that the radial coordinate do not correspond to area in the same way as for spheres in ordinary euclidean space.
The line element given above, with f,g, regarded as undetermined functions of the isotropic coordinate r, is often used as a metric Ansatz in deriving static spherically symmetric solutions in general relativity (or other metric theories of gravitation).
As an illustration, we will sketch how to compute the connection and curvature using Cartan's exterior calculus method.
(The fact that our spacetime admits a frame having this particular trigonometric form is yet another equivalent expression of the notion of an isotropic chart in a static, spherically symmetric Lorentzian manifold).
Taking the exterior derivatives and using the first Cartan structural equation, we find the nonvanishing connection one-forms Taking exterior derivatives again and plugging into the second Cartan structural equation, we find the curvature two-forms.