This article introduces and describes the mathematical structures and frameworks needed to understand the modern fundamental theory of Relativistic Spacetime Physics. The self-referential and self-contained nature of Mathematics provides enough power to prescribe a rigorous language needed to formulate the building components of the standard Einstein’s General Theory of Relativity like Spacetime, Matter, and Gravity, along with their behaviors and interactions. These notes introduce these abstract components, starting with defining the arena of smooth manifolds and then adding the necessary and suffcient differential geometric structures needed to build the primers to the General Theory of Relativity.
The complete notes can be found here.
These notes were made while I had been doing the Central Lecture course of the International Winter School on Gravity and Light 2015. Many thanks to the excellent teachings by Prof. Frederic P. Schuller in these lectures and all other online available lectures on the Mathamatical anatomy of Theoretical Physics.