Classical Physics: Spacetime and Fields

TL;DR

This work provides a self-contained, geometric formulation of classical spacetime and fields based on general covariance and the principle of least action. It develops spacetime via tensors, affine connections, curvature, and metrics, then introduces the tetrad formalism to connect coordinate and Lorentz structures, culminating in a framework in which particles are viewed as field configurations in spacetime. The text derives covariant differentiation, geodesic motion, curvature, and Bianchi identities, and clarifies how metric compatibility and torsion shape spacetime geometry, enabling a rigorous basis for classical gravity and field dynamics. The approach offers a clear, coordinate-independent path from fundamental geometric principles to the dynamics of matter and fields, highlighting the interplay between geometry (via $g_{ik}$, $R^i_{ ext{ }mjk}$, $ abla_i$) and physical laws, and setting the stage for extensions to spinors and gauge fields within a unified classical framework.

Abstract

We present a self-contained introduction to the classical theory of spacetime and fields. This exposition is based on the most general principles: the principle of general covariance (relativity) and the principle of least action. The order of the exposition is: 1. Spacetime (principle of general covariance and tensors, affine connection, curvature, metric, space and time, tetrad and spin connection, Lorentz group, spinors); 2. Fields (principle of least action, gravitational field, matter, symmetries and conservation laws, particle limit of field, gravitational field equations, spinor fields, electromagnetic field). In this order, a particle is a special case of a field existing in spacetime, and classical mechanics can be derived from field theory.

Figures (3)

• Figure 1: Contravariant and covariant components of a vector.
• Figure 2: Light cone.
• Figure 3: Distance.