An introduction to general relativity using interactive symbolic computation

Abstract

Just over a century after Einstein’s writing of 1914-1917, The Berlin years, General relativity (GR) is still the most robust theory of physics. This theory became the first to approximately predict the perihelion behaviour of Mercury[1], as well as offer mathematical justification for photon deflection. During the year 1916 of Einstein’s Berlin years, he predicted the existence of Gravitational waves, which was only confirmed a century later by the LIGO experiment[2]. His theory was later used by a Russian and soviet physicist and mathematician Alexander Friedmann in 1922, to explain that a homogeneous and isotropic model of the universe is an expanding one. Friedmann derived the equations of physical cosmology governing this expansion phenomenon. These Friedmann equations formed the foundation of the FLRWmodel[3, 4, 5], the first empirically plausible cosmological model of our universe. This interactive text is focussed on building a foundation of GR tooling, grounded in computation, intended to give students a well-rounded approach to GR as well as insight to build and/or solve problems all the way up to complex scenarios. This tooling foundation comprises of three key areas: tensor calculus, physics, and the computation and visualisation of results. Asmost of the computation in GR is often cumbersome to do by hand, the approach of this text places emphasis on the use of modern symbolic computation inMathematica to provide a direct framework for exploring the introductory implications of GR through computation interaction. AsMathematica is praised for its design consistency, aesthetics and automation, it became the logical choice of software for this endeavour. We begin by introducing curvilinear coordinates, tensor calculus, investigate the Eulerian and Lagrangian formalisms for the equations of fluid mechanics in tensor notation. Now that the reader has been exposed to familiar material in this notation, we define and discuss the geodesic equations, making a link to the Euler- Lagrange equations. The reader is then introduced to the principles of GR and supplied with a discussion of how Einstein unified inertia and gravity. This sets the foundation on which different representations of GR are introduced and discussed. A chapter dedicated to exact and analytical solutions is supplied, addressing theMinkowski, Schwarzschild and Kerr space-times as well as the FLRWuniverse model. The final chapter is dedicated to the introduction of perturbation theory in GR and its role in the detection of gravitational waves. Lastly, an introduction to the foundation of numerical relativity with an application to an FLRW universe is supplied. Throughout the text the reader is asked to do several written and Mathematica exercises, of which the solutions are supplied in the two appendices. In no way is this work to be interpreted as novel, as it is an amalgamation of several texts, but rather as a modern approach to the teaching of GR with symbolic computation at the postgraduate level. It is aimed at retaining the historic development of GR as a key approach to unifying Inertia and Gravity; rather than perspectives aimed at simplifying Gravity to bemerely considered as an artefact of geometric representation. The project was conducted under the supervision ofDr. R.S.Herbst andDr. T. Gebbie, and was inspired by the detailed handwritten notes of Professor David P. Mason on GR and tensors. These notes, that were written in the 1990’s, have been widely used by lecturers and graduate students from the School over the decades. Prof. Mason aimed at presenting a unified theory for fluid mechanics and elasticity, developing the theory using curvilinear coordinates, making his notes an invaluable resource. There is considerable pedagogical value in returning to the roots of fluid modelling and its influence on the development of ideas that led to GR, particularly through hands-on computation and manipulation of solutions to the Einstein Field Equations using a symbolic computation framework, such as Mathematica