General Relativity

Notation and Units

If not stated otherwise, the system of units used are the geometrized units. The sign convention is $(-, +, +, +)$ . Latin indices are used for spatial components and greek indices are used for spacetime components.

For components of vector $\boldsymbol{v}$ , I am using $v^{\mu}$ . For coordinates, I am using $x^{\mu}$ .

Einstein summation convention is employed:

v^i e_i = \sum_i v^i e_i.

Partial derivative with respect to spacetime component notation:

\frac{\partial v^{\mu}}{\partial x^{\nu}} \equiv \partial_{\nu} v^{\mu} \equiv v^{\mu}{}_{,\nu}.

Chapters

Mathematical Preliminaries General Relativity Basics Schwarzschild Metric

Acknowledgements

I would like to thank eigenchris and his courses on tensors, tensor calculus and relativity. These notes are from his courses. I would also like to thank Sean Carroll for freely providing his Lecture Notes on General Relativity which I sometimes used for reference.

For Lie derivative and Killing vectors, I used Robert Davie's videos and also Physics LibreTexts for reference. The Wikipedia page for the Killing tensor was also used in the Killing vectors chapter.