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The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteristic radius associated with any quantity of mass.
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English: The Schwarzschild interior (blue) and exterior (black) solutions. r s : Schwarzschild radius; r g : r -coordinate on the body's surface; ℛ 2 = r g 3 / r s ; sin η g = r g /ℛ Date
This is just an artifact of how Schwarzschild coordinates are defined; a free-falling particle will only take a finite proper time (time as measured by its own clock) to pass between an outside observer and an event horizon, and if the particle's world line is drawn in the Kruskal–Szekeres diagram this will also only take a finite coordinate ...
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 other three: the radial and angular coordinates ,, of the Gullstrand–Painlevé coordinates are identical to those of the Schwarzschild chart. That is, Gullstrand–Painlevé applies one coordinate transform to go from the Schwarzschild time t {\displaystyle t} to the raindrop coordinate t r = τ {\displaystyle t_{r}=\tau } .
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 ...
In these coordinates, the horizon is the black hole horizon (nothing can come out). The diagram for u-r coordinates is the same diagram turned upside down and with u and v interchanged on the diagram. In that case the horizon is the white hole horizon, which matter and light can come out of, but nothing can go in.