Elements of General Relativity by Piotr T. Chruściel

Author:Piotr T. Chruściel
Language: eng
Format: epub, pdf
ISBN: 9783030284169
Publisher: Springer International Publishing

(4.3.13)

In other words, h(X, Y ) coincides with g(X, Y ) when both expressions are defined, but we are only allowed to consider vectors tangent to when using h.

Some comments are in order: If g is Riemannian, then normals to are spacelike, and (4.3.13) defines a Riemannian metric on . For Lorentzian g’s, it is easy to see that h is Riemannian if and only if vectors orthogonal to are timelike, and then is called spacelike. Similarly, h is Lorentzian if and only if vectors orthogonal to are spacelike, and then is called timelike. When the normal direction to is null, then (4.3.13) defines a symmetric tensor on with signature (0, +, ⋯ , +), which is degenerate and therefore not a metric; such hypersurfaces are called null, or degenerate.

If is not degenerate, it comes equipped with a Riemannian or Lorentzian metric h. This metric defines a measure dμh, using (4.3.3) with g there replaced by h, which can be used to integrate over .

We are ready now to formulate the Stokes theorem for open bounded sets: Let Ω be a bounded open set with piecewise differentiable boundary and assume that there exists a well-defined field of exterior-pointing conormals N = Nμdxμ to Ω. Then for any differentiable vector field X it holds that

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.