Part I – Introduction and Fundamental Structure
1 Introduction
The general theory of relativity, published in 1915 by Albert Einstein, forms one of the cornerstones of our modern understanding of space, time, and gravity. The theory replaces Newton’s classical description of gravity with a geometric description of spacetime: matter and energy determine the geometry of spacetime, while freely falling objects follow geodesics in that geometry. This provides a unified framework for describing gravity, spacetime, and their profound implications for both fundamental physics and cosmology.
1.1 Purpose of this document
The purpose of this document is to provide an overview of general relativity, with emphasis on:
- A careful derivation of the mathematical structure underlying the theory.
- An examination of practical applications and experiments that support the theory.
- A clear distinction between exact results, approximations, coordinate-dependent quantities, and experimentally measured quantities.
- Addressing frequently asked questions about concepts and formulas within general relativity.
1.2 Approach
The approach in this document differs from popular descriptions. We focus on:
- Thorough derivations of tensor-analytic equations.
- A careful treatment of coordinate transformations and the distinction between coordinate-dependent and locally measurable quantities.
- A clear distinction between exact results, approximations, and numerical estimates, with the assumptions underlying each approximation stated explicitly.
- Application of the Schwarzschild solution to classical experiments such as the Hafele–Keating experiment, the deflection of light by gravity, and the precession of Mercury, with careful attention to the physical model and approximations used.
We also demonstrate that the Schwarzschild solution satisfies Einstein’s field equations, and show how the geometry of spacetime is described mathematically by the metric and its associated connection, represented in coordinates by the Christoffel symbols. Spacetime curvature is then expressed invariantly through the Riemann curvature tensor and its contractions.
1.3 Intended audience
This document is written for:
- Physicists and mathematicians interested in the underlying structure of general relativity.
- Physics students who want to go beyond standard textbooks.
- Anyone who wants to understand why the equations are the way they are, not just how they work.
1.4 Final remarks
This document is structured so that each chapter builds on the previous one. Where possible, individual sections can also be read independently, although some mathematical definitions, notation, and conventions introduced earlier are assumed in later chapters. The appendices provide additional explanations, alternative derivations, and applications in the context of special relativity and nuclear physics.