Abstract
In the previous chapters, we studied non-gravitational phenomena in inertial reference frames, and often we limited our discussion to Cartesian coordinate systems. Now we want to include gravity, non-inertial reference frames, and general coordinate systems. The aim of this chapter is to introduce some mathematical tools necessary to achieve this goal. We follow quite a heuristic approach. The term Riemannian geometry is used when we deal with a differentiable manifold equipped with a metric tensor.
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- 1.
A test-particle must have a sufficiently small mass, size, etc. such that its mass does not significantly alter the background gravitational field, tidal forces can be ignored, etc.
Reference
M.H. Protter, C.B. Morrey, A First Course in Real Analysis (Springer, New York, 1991)
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Problems
Problems
5.1
Write the components of the following tensors:
5.2
Write the non-vanishing components of the Riemann tensor, the Ricci tensor, and the scalar curvature for the Minkowski spacetime in spherical coordinates.
5.3
Check that the Ricci tensor is symmetric.
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Bambi, C. (2018). Riemannian Geometry. In: Introduction to General Relativity. Undergraduate Lecture Notes in Physics. Springer, Singapore. https://doi.org/10.1007/978-981-13-1090-4_5
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DOI: https://doi.org/10.1007/978-981-13-1090-4_5
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Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-1089-8
Online ISBN: 978-981-13-1090-4
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