Swooning Over Relativity Understanding The Math Behind General Relativity

Special Theory of Relativity and the General Theory of Relativity are arguably the most significant physical theories ever invented. They have helped thousands of physicists understand the nature of the universe. However, to a high-school student like me, these theories may seem like a never-ending labyrinth of intricate math formulae and equations that make the journey of learning these theories unpalatable.

Entering this labyrinth to produce a well-researched paper on naked singularities that I was working on, I was required to devour Einstein's General Theory of Relativity. Eventhough my foundational knowledge from various academic resources, summer programs, workshops and interactions with active researchers helped me to construct the building blocks, understanding this theory in totality posed a challenge. After my initial struggle, I concluded that the best way to approach it would be to first familiarize myself with the mathematics behind the theory, a poetry of logic in Einstein's words, and then take it further. It worked for me, and I began to appreciate the poetry.

This book grew out of a set of notes that I had prepared while I was studying the theory to write my paper. The purpose of this book is to simplify the convoluted math behind General Theory of Relativity and ease the transition from high-school mathematics to advanced mathematics - a pre-requisite to learn the theory. I did not find enough resources that helped with this transition when I began my journey. Therefore, I decided to build on my notes and write this book with the aim to help more high school students make this transition without being afraid of getting lost in the labyrinth, and eventually see more of them appreciate this poetry of logic as much I as I do today. This book assumes that its readers have a strong foundation in high-school mathematics.

Many people have contributed greatly to my understanding of General Theory of Relativity, and this book would not have existed if it were not for them. To start with, I would like to specially mention Dr Pankaj Joshi and Dr Rucha Desai, who gave me an opportunity to attend the international workshop at International Center of Cosmology, Charusat. The workshop introduced me to the mathematical formulae behind naked singularities that fuelled my desire to write an independent research paper. It was this desire that made me embrace General Theory of Relativity. Ritesh Jha sir held my hand at a time when I was struggling to make any sense of the notations and symbols used in various academic resources that I had scoured through. It is because of his guidance that I could traverse through the labyrinth. Dr Yuri Deshko, my mentor at Columbia University, USA, also deserves a special mention here, because it was he who strengthened my base on which I could build on.

So, before we begin our journey, let us first take quick notes on what Special Relativity and General Relativity deal with. While Special Relativity focuses on how the universe looks to two different inertial observers moving at constant velocity with respect to each other, General Relativity tries to understand how the universe looks to two observers moving at variable velocity with respect to each other. Putting these concepts together, Einstein has tried to use these two theories to understand what gravity is.

In summary, the two theories deal with the following points:
1) Speed of light in a vacuum is constant for all uniformly moving observers
2)Mass and energy curve spacetime.
3)From a mathematical point of view, spacetime can be portrayed as a curved space, which is locally flat.
4)A metric is used to describe the curvature of spacetime.
5)Light and all free particles follow a geodesic path.
6)The laws of physics must take the same form in all coordinate systems.
7)Laws of physics are the same for all the co-ordinate systems.

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