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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.
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero.
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 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 simplest example of a Lorentzian manifold is flat spacetime, which can be given as R 4 with coordinates (,,,) and the metric = + + + =. These coordinates actually cover all of R 4 . The flat space metric (or Minkowski metric ) is often denoted by the symbol η and is the metric used in special relativity .
Schwarzschild solution in Schwarzschild coordinates, with two space dimensions suppressed, leaving just the time t and the distance from the center r. In red the incoming null geodesics. In blue outcoming null geodesics. In green the null light cones on which borders light moves, while massive objects move inside the cones.
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.
This is unfounded because that law has relativistic corrections. For example, the meaning of "r" is physical distance in that classical law, and merely a coordinate in General Relativity.] The Schwarzschild metric can also be derived using the known physics for a circular orbit and a temporarily stationary point mass. [1]