Classical Covariance

Classical mechanics, from point particles through rigid objects and continuum mechanics is reviewed based on the notions of tensors, transformations, and the metric, as developed in the first two chapters. The geodesic equation on flat and curved spaces is introduced and solved in a classical setting. Motion in a potential, particularly a gravitational potential, is discussed. Galilean covariance and transformations are introduced. Time as a fourth dimension is shown to arise even in a classical setting, even if not as rigorous as it would be in relativity theory.

Special Covariance

In this chapter we study the special theory of relativity. We begin with the metric and construct all consequences such as the kinematical quantities, 4-vectors and tensors, Lorentz transformations, geometric interpretations, conservation of 4-momentum and collision problems. We conclude with a discussion of electrodynamics in covariant form.

Least Action and Classical Fields

We present the principle of least action and see how it is used in non-relativistic point particle mechanics, relativistic point particle mechanics, general relativity, derivation of field equations for scalar, vector and tensor fields as well as the energy momentum tensor. Towards the end we present examples of solutions of Einstein-Maxwell fields: The Reissner-Nordstrom metric, Kerr metric, and Kerr- Newman metric.

Physics in Curved Spacetime

We discuss mechanics in curved spacetime backgrounds, gravitational time dilation, the motion of free particles, geodesics. We use the Schwarzschild metric as a case study and solve for motion along radial and orbital geodesics. This includes the strange behaviour around the event horizons of a Schwarzschild black hole. Isometries and Killing vector fields are explained and applied. Finally a brief presentation of generally covariant electrodynamics is given.

Generalizing General Relativity

In this chapter we review 5 popular models of modifying and/or generalizing our current understanding of spacetime. These are: Brans-Dicke theory, f(R) theory, Gauss-Bonnet theory, Kaluza-Klein theory, and finally Cartan’s theory of gravity.

Riemann and Einstein

In this chapter we make notions of spatial and spacetime curvature more precise by presenting Riemann’s theory of non-Euclidean spaces. We then construct Einstein’s gravitational field equations and see how they can be solved under certain symmetries, filling in mathematical gaps of previous chapters. We also discuss the cosmological constant and its consequences.

Tensors

In this chapter we develop the concept of tensors, their meaning, and how they arise from vectors. Emphasis is placed on tensor transformations, covariance between coordinate systems, and relation to the metric. The concept of metric connection and the Christoffel symbols is introduced in three dimensions via the easily visualizable idea of parallel transport. Derivatives and intergrals in covariant form are discussed. The first two chapters are designed to familiarize the reader with the language that is the bread and butter of the general theory of relativity and other higher geometric theories.

General Covariance

The general theory of relativity is introduced based on the principle of equivalence. Gravity is shown to arise dues to spacetime curvature. Specific examples of curved spacetimes are presented from the approximate but more intuitive to the complex: Uniform gravitational field (Galilean metric), the Newtonian weak field metric, Schwarzschild’s exterior and interior solutions, black holes, and cosmological spacetimes. A brief discussion on distances, areas and volumes in curved spaces is also given.

Afterthoughts

If you arrived at this page by reading the entire book (and didn’t skip), then you most certainly deserve a very hearty: Congratulations! Oh well, let’s admit it, even if you cheated a bit and skipped some chapters, you still deserve a (perhaps not so hearty) cheer!...

Coordinate Systems and Vectors

This chapter introduces the various types of coordinate systems that exist in three dimensions and develops the basic concept of ‘metric’ to describe their properties. It introduces vectors in these coordinate systems and develops the notions of the ‘index language,’ dependence on the metric, and the covariance of vectors. Early familiarity with the metric tensor, index or component notation, symmetric and anti-symmetric manipulation is intended.