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In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres.In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical coordinate chart on a static and spherically symmetric spacetime, which is adapted to these nested round spheres.
The defining characteristic of an isotropic chart is that its radial coordinate (which is different from the radial coordinate of a Schwarzschild chart) is defined so that light cones appear round. This means that (except in the trivial case of a locally flat manifold), the angular isotropic coordinates do not faithfully represent distances ...
The Schwarzschild solution, taken to be valid for all r > 0, is called a Schwarzschild black hole. It is a perfectly valid solution of the Einstein field equations, although (like other black holes) it has rather bizarre properties. For r < r s the Schwarzschild radial coordinate r becomes timelike and the time coordinate t becomes spacelike. [22]
Gullstrand–Painlevé coordinates are a particular set of coordinates for the Schwarzschild metric – a solution to the Einstein field equations which describes a black hole. The ingoing coordinates are such that the time coordinate follows the proper time of a free-falling observer who starts from far away at zero velocity, and the spatial ...
An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations = and = (but these may not hold at other points in the frame). These coordinates are also called normal coordinates.
The transformation between Schwarzschild coordinates and Kruskal–Szekeres coordinates defined for r > 2GM and < < can be extended, as an analytic function, at least to the first singularity which occurs at =. Thus the above metric is a solution of Einstein's equations throughout this region.
Working in a coordinate chart with coordinates (,,,) labelled 1 to 4 respectively, we begin with the metric in its most general form (10 independent components, each of which is a smooth function of 4 variables). The solution is assumed to be spherically symmetric, static and vacuum.
The time coordinate used in the Lemaître coordinates is identical to the "raindrop" time coordinate used in the Gullstrand–Painlevé coordinates. The other three: the radial and angular coordinates r , θ , ϕ {\displaystyle r,\theta ,\phi } of the Gullstrand–Painlevé coordinates are identical to those of the Schwarzschild chart.