Advanced Mechanics and General Relativity

Abstract

Joel Franklin's textbook `Advanced Mechanics and General Relativity'comprises two partially overlapping, partially complementaryintroductory paths into general relativity at advanced undergraduatelevel.Path I starts with the Lagrangian and Hamiltonian formulations ofNewtonian point particle motion, emphasising the action principle andthe connection between symmetries and conservation laws. The conceptsare then adapted to point particle motion in Minkowski space,introducing Lorentz transformations as symmetries of the action.There follows a focused development of tensor calculus, paralleltransport and curvature, using examples from Newtonian mechanics andspecial relativity, culminating in the field equations of generalrelativity. The Schwarzschild solution is analysed, including adetailed discussion of the tidal forces on a radially infallingobserver. Basics of gravitational radiation are examined, highlightingthe similarities to and differences from electromagneticradiation. The final topics in Path I are equatorial geodesics in Kerrand the motion of a relativistic string in Minkowski space.Path II starts by introducing scalar field theory on Minkowski spaceas a limit of point masses connected by springs, emphasising theaction principle, conservation laws and the energy–momentumtensor. The action principle for electromagnetism is introduced, andthe coupling of electromagnetism to a complex scalar field isdeveloped in a detailed and pedagogical fashion. A free symmetricsecond-rank tensor field on Minkowski space is introduced, and theaction principle of general relativity is recovered from coupling thesecond-rank tensor to its own energy-momentum tensor.Path II then merges with Path I and, supplanted with judicious earlyselections from Path I, can proceed to the Schwarzschild solution.The choice of material in each path is logical and focused. A notableexample in Path I is that Lorentz transformations in Minkowki spaceare introduced efficiently and with a minimum of fuss, as symmetriesof a geodesic action principle. Another example is a similarlyefficient and hands-on introduction of Killing vectors. A consequenceof this focus is that some perhaps traditional material is omitted. For example, Lorentz contraction appears briefly in the incompatibility discussion of special relativity and Newtoniangravity but is not introduced in a more systematic manner. The style is informal and very readable, with detailed explanations,frequent summaries of what has been achieved and pointers to what isabout to follow. There are plenty of examples and some 150 well-chosenexercises, and the author's website hosts relevant Maple samplescripts for tensor manipulations and variational problems. The textconveys an enthusiasm for explaining the subject, frequentlyreminiscent of the Feynman lectures.The presentation emphasises explicit calculations and examples,largely avoiding technical definitions of abstract mathematicalconcepts. The author negotiates the challenge between readability andtechnical accuracy with admirable skill, striking a balance that willbe much appreciated by the target audience. For example, the notion ofspherical symmetry in curved spacetime is introduced informally as ageneralisation of a spherically symmetric vector field in Minkowskispace, and spherically symmetric vacuum and electrovacuum solutionsare then carefully discussed so that a formal definition of sphericalsymmetry is not required. A rare instance that may border onoversimplification is the brief discussion of curvature scalars versusspacetime singularities.Towards the end of the book, the text mentions with increasing explicitness that inserting a gauge condition or an ansatzin an action before varying may not always give the correctequations of motion. It would be useful to be more explicit about thispoint already earlier in the book. In particular, the text refers tothe reparametrisation-invariant square root action of a relativisticpoint particle as being `in proper time parametrisation', while theactual calculations of course impose the proper time condition only inthe equation of motion after the action has been varied.Two presentational conventions surprised me. First, the speed of lightis throughout kept explicitly as c: might advanced undergraduatesappreciate being trusted with geometric units, reinstating c bydimensional analysis when desired? Second, in Minkowski space fieldtheory, the overall coefficient in the action is chosen so that thetime derivative term is negative, with the consequence that theHamiltonian is negative (as explicitly noted in an exercise) and thedefinition of the energy-momentum tensor must include a minus sign toachieve the usual choice T00 > 0. This convention eliminates someminus signs in the computations with the spin two field: does thiscomputational saving outweigh the adjustment awaiting those whocontinue with the topic at graduate level?Overall, Franklin's book is an excellent addition to the literature,and its readability and explicitness will be appreciated by the targetaudience. Should I be teaching an introductory undergraduate class ingeneral relativity in the near future, I would seriously consider this book for the main class text.