Isotropic coordinates
In the theory of
Isotropic charts are most often applied to
Definition
In an isotropic chart (on a static spherically symmetric spacetime), the
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
Killing vector fields
The
and three spacelike Killing vector fields
Here, saying that is irrotational means that the vorticity tensor of the corresponding timelike congruence vanishes; thus, this Killing vector field is hypersurface orthogonal. The fact that the spacetime admits an irrotational timelike Killing vector field is in fact the defining characteristic of a static spacetime. One immediate consequence is that the constant time coordinate surfaces form a family of (isometric) spatial hyperslices (spacelike hypersurfaces).
Unlike the Schwarzschild chart, the isotropic chart is not well suited for constructing embedding diagrams of these hyperslices.
A family of static nested spheres
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
where are coordinates and is the Riemannian metric on the 2 sphere of unit radius. That is, these nested coordinate spheres do in fact represent geometric spheres, but the appearance of rather than shows that the radial coordinate do not correspond to area in the same way as for spheres in ordinary euclidean space. Compare Schwarzschild coordinates, where the radial coordinate does have its natural interpretation in terms of the nested spheres.
Coordinate singularities
The loci mark the boundaries of the isotropic chart, and just as in the Schwarzschild chart, we tacitly assume that these two loci are identified, so that our putative round spheres are indeed topological spheres.
Just as for the Schwarzschild chart, the range of the radial coordinate may be limited if the metric or its inverse blows up for some value(s) of this coordinate.
A metric Ansatz
The line element given above, with f,g, regarded as undetermined functions of the isotropic coordinate r, is often used as a metric
As an illustration, we will sketch how to compute the connection and curvature using Cartan's exterior calculus method. First, we read off the line element a
where we regard as undetermined smooth functions of . (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.
See also
- static spacetime,
- spherically symmetric spacetime,
- static spherically symmetric perfect fluids,
- Schwarzschild coordinates, another popular chart for static spherically symmetric spacetimes,
- frame fields in general relativity, for more about frame fields and coframe fields.
References
- Misner, Thorne, Wheeler (1973). Gravitation. W H Freeman and Company. ISBN 0-7167-0344-0.)
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