1. Curvilinear coordinate systems 4 Non orthogonal example 5 General curvilinear systems 8

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Freefall in curved space


1.Curvilinear coordinate systems 4

1.1.Non orthogonal example 5

1.2.General curvilinear systems 8

2.Tensors 12

3.Christoffel symbols, 1st and 2nd kind 15

3.1.Christoffel symbols as Derivatives of the Metric Tensor 16

4.Covariant derivative 18

5.The geodesic equation 19

6.Appendix A, a wakeup problem 25

7.Appendix B, index notation 30

8.Appendix C, remarks of interest 34



9.Appendix D, T-shirt from CERN 37

10.References 39


This article derives the equation for a geodesic in a curved space. It assumes that the reader has a working knowledge of partial derivatives, the chain rule and some basics about vector algebra/calculus and variational calculus. It tries to introduce the concepts and notations that are necessary to be able to read and understand the different terms of the equation. Here is the ‘layout’ of the article:

As an introduction the reader is invited to look at a simple example in Appendix A and if unfamiliar with index notation also read Appendix B. Appendix C and D contains miscellaneous information about mathematically related areas.
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