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In deriving the Schwarzschild metric, it was assumed that the metric was vacuum, spherically symmetric and static. The static assumption is unneeded, as Birkhoff's theorem states that any spherically symmetric vacuum solution of Einstein's field equations is stationary ; the Schwarzschild solution thus follows.
Snap, [6] or jounce, [2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. [4] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions: = ȷ = = =.
Christoffel symbols satisfy the symmetry relations = or, respectively, =, the second of which is equivalent to the torsion-freeness of the Levi-Civita connection. The contracting relations on the Christoffel symbols are given by
Lemaître coordinates are a particular set of coordinates for the Schwarzschild metric—a spherically symmetric solution to the Einstein field equations in vacuum—introduced by Georges Lemaître in 1932. [1] Changing from Schwarzschild to Lemaître coordinates removes the coordinate singularity at the Schwarzschild radius.
where the comma indicates a partial derivative with respect to the coordinates: g a b , c = ∂ g a b ∂ x c {\displaystyle g_{ab,c}={\frac {\partial {g_{ab}}}{\partial {x^{c}}}}} As the manifold has dimension n {\displaystyle n} , the geodesic equations are a system of n {\displaystyle n} ordinary differential equations for the n ...
Free falling worldlines in classic Schwarzschild-Droste coordinates. A Schwarzschild observer is a far observer or a bookkeeper. He does not directly make measurements of events that occur in different places. Instead, he is far away from the black hole and the events.
where (,) and (,) are two metric potentials dependent on Weyl's canonical coordinates {,}.The coordinate system {,,,} serves best for symmetries of Weyl's spacetime (with two Killing vector fields being = and =) and often acts like cylindrical coordinates, [2] but is incomplete when describing a black hole as {,} only cover the horizon and its exteriors.
If the derivative does not lie on the tangent space, the right expression is the projection of the derivative over the tangent space (see covariant derivative below). Symbols of the second kind decompose the change with respect to the basis, while symbols of the first kind decompose it with respect to the dual basis.