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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]
A Schwarzschild black hole is described by the Schwarzschild metric, and cannot be distinguished from any other Schwarzschild black hole except by its mass. The Schwarzschild black hole is characterized by a surrounding spherical boundary, called the event horizon , which is situated at the Schwarzschild radius ( r s {\displaystyle r_{\text{s ...
For this case only two components of the shear stress became non-zero: = ˙ and = ˙ where ˙ is the shear rate.. Thus, the upper-convected Maxwell model predicts for the simple shear that shear stress to be proportional to the shear rate and the first difference of normal stresses is proportional to the square of the shear rate, the second difference of normal stresses is always zero.
For example, in changing units from meters to millimeters the coordinate units get smaller, but the numbers in a vector become larger: 1 m becomes 1000 mm. Covariant vectors, on the other hand, have units of one-over-distance (as in a gradient) and transform in the same way as the coordinate system. For example, in changing from meters to ...
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.
Another important tensorial derivative is the Lie derivative. Unlike the covariant derivative, the Lie derivative is independent of the metric, although in general relativity one usually uses an expression that seemingly depends on the metric through the affine connection.
The extension of the exterior region of the Schwarzschild vacuum solution inside the event horizon of a spherically symmetric black hole is not static inside the horizon, and the family of (spacelike) nested spheres cannot be extended inside the horizon, so the Schwarzschild chart for this solution necessarily breaks down at the horizon.
It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, the differential is a linear map from the tangent space of M {\displaystyle M} at x {\displaystyle x} to the tangent space of N {\displaystyle N} at φ ( x ) {\displaystyle \varphi (x)} , d φ x : T x M → T φ ( x ) N {\displaystyle \mathrm {d ...