General Relativity: An Introduction for Physicists

When Euclidean coordinate lengths are replaced by the metric lengths of a curved geometry within Newton’s second law of motion, the metric form of the second law can be shown to be identical to the geodesic equation of motion of general relativity. The metric coefficients are contained in the metric lengths and satisfy the field equations of general relativity. Because metric lengths are the physically measured lengths, their use makes it possible to understand general relativity directly in terms of physical quantities such as energy and momentum within a curved space–time. The metric form of the second law contains gravitational effects in exactly the same manner as occurs in relativity. Its mathematical derivation uses vectors rather than tensors, and nongravitational forces can occur in this modified second law without a tensor form. Because quantum mechanics is based on Newtonian concepts of energy and momentum, it is shown that when metric lengths replace coordinate lengths in Dirac’s wave equation, it has a covariant form under a metric transformation of the physically measured distances themselves, rather than a coordinate transformation. Metric transformations are also used to describe the Dirac equation for the gravitational central field in a Schwarzschild metric.

The aim of this chapter is a detailed study of applicability and applications of distributional geometry as developed in the previous chapters—with its special focus, of course, on the generalized pseudo-Riemannian geometry of Section 3.2.5—to the theory of general relativity. We shall begin with a general introduction into our theme in Section 5.1, where we also introduce our notational conventions concerning general relativity. Section 5.2 is divided into two parts. The first one is concerned with a brief overview of applications of linear distributional geometry to relativity with a strong focus on a “no-go”-theorem by Geroch and Traschen (cf. [Ger87]). In the second part of Section 5.2 we use the nonlinear distributional geometry of Chapter 3 to define the generalized curvature tensor as well its contractions appearing in the field equations of general relativity. Moreover, we present a guideline for applying this setting to the study of singular spacetime metrics and discuss consistency with respect to the classical theory. Finally in Section 5.3 we give a complete distributional description of impulsive gravitational pp-waves based upon a variety of concepts introduced so far.

In this work, we examine the geometric character of the field equations of general relativity and propose to formulate relativistic field equations in terms of the Riemann curvature tensor. The resulted relativistic field equations are also integrated into the general framework that we have presented in our previous works that all known classical fields can be expressed in the same dynamical form. We also discuss a possibility to reformulate the field equations of general relativity so that the Ricci curvature tensor and the energy-momentum tensor can appear symmetrically in the field equations without violating the conservation law stated by the covariant derivative.

The metric tensor of a Riemannian space-time is composed quadratically of timelike and spacelike tetrad fields. A system of first-order field equations for the tetrad fields is postulated, which determines the geometric structure of space-time as well as the second-order dynamics of the fields. The energy-momentum content of the fields automatically corresponds to the geometric structure of the space-time according to the Einstein field equations. The spin contribution of the fields is mutually compensated and therefore spin does not influence the space-time geometry. As a consequence, it is not necessary to include torsion into the general theory of relativity and nevertheless the equivalence principle can be used to consistently transfer energy-momentum tensors from flat to curved space.

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.

November 2015 marks the 100th anniversary of Einstein's theory of general relativity (GR). He published four papers on the subject in November 1915. Remarkable! The first two papers led to the field equations of GR, and the other two laid out vital consequences of the theory, including the anomalous precession of the perihelion of Mercury (unexplained since 1859) and the gravitational bending of light. The latter phenomenon was confirmed famously by Eddington's 1919 observation of a total solar eclipse and the shift of apparent position for stars, the light of which passed very near the sun. Einstein, already famous, became a household name — an early 20th-century superstar.

Einstein's special theory of relativity revolutionized physics by teaching us that space and time are not separate entities, but join as ‘spacetime’. His general theory of relativity further taught us that spacetime is not just a stage on which dynamics takes place, but is a participant: the field equation of general relativity connects matter dynamics to the curvature of spacetime. Curvature is responsible for gravity, carrying us beyond the Newtonian conception of gravity that had been in place for the previous two and a half centuries. Much research in gravitation since then has explored and clarified the consequences of this revolution; the notion of dynamical spacetime is now firmly established in the toolkit of modern physics. Indeed, this notion is so well established that we may now contemplate using spacetime as a tool for other sciences. One aspect of dynamical spacetime—its radiative character, ‘gravitational radiation’—will inaugurate entirely new techniques for observing violent astrophysical processes. Over the next 100 years, much of this subject's excitement will come from learning how to exploit spacetime as a tool for astronomy. This paper is intended as a tutorial in the basics of gravitational radiation physics.

It is proved that the combined neutrino gravitational field equations in general relativity admit no nontrivial solutions in a static space-time provided that the energy flow vector of the neutrino field is timelike or null for all observers.

The theory of special and general relativity causes a “schizophrenic” dilemma in physics. It undeniably provides mathematically correct values, but it is undeniably epistemologically wrong in many respects. Including the relativistic explanation of the gravitational “time dilation” and the curvature of light beams at the surfaces of large masses, the author demonstrates the illogical character of relativistic physics. When one thinks the relativistic explanations of gravitational time dilatation and of the curvature of light rays by masses through to the end, they lead to absurd and contradictory logical conclusions.

In the framework of classical field theories, the notions of conserved quantities and entropy for stationary solutions of covariant theories of gravitation, e. g. of Einstein field equations of General Relativity, are discussed. Nother theorem is used to provide the correct definition of (covariantly) conserved quantities such as mass and angular momentum. The variation of entropy is then defined as a macroscopical quantity which satisfies a Clausiuslike first principle of thermodynamics. Finally, a proposal for the entropy of non-stationary solutions is discussed.

The happiest thought of Einstein's life

The covariant wedge products of tetrads in General Relativity are introduced and related field equations are derived. The electromagnetic dual field is treated in detail and it is shown that this field is closely related with one component field, which is eigenfunction of d'Alembertian operator. The formalism is developed to incorporate internal symmetry.

This book deals with Colombeau solutions to Einstein field equations in general relativity: Gravitational singularities, distributional SAdS BH spacetime-induced vacuum dominance. This book covers key areas of Colombeau nonlinear generalized functions, distributional Riemannian, geometry, distributional schwarzschild geometry, Schwarzschild singularity, Schwarzschild horizon, smooth regularization, nonsmooth regularization, quantum fields, curved spacetime, vacuum fluctuations, vacuum dominance etc. This book contains various materials suitable for students, researchers and academicians of this area.

While the physical properties of the gravitational field in the radiation regime are reasonably well understood, several mathematical questions remain unanswered. The question here is that of existence and properties of gravitational fields with asymptotic behavior compatible with existence of gravitational radiation. A framework to study those questions has been proposed by R. Penrose (R. Penrose, “Zero rest-mass fields including gravitation”, Proc. Roy. Soc. London A284 (1965), 159-203), and developed by H. Friedrich (H. Friedrich, “Cauchy problem for the conformal vacuum field equations in general relativity”, Commun. Math. Phys. 91 (1983), 445-472.), (H. Friedrich, “On the existence of n-geodesically complete or future complete solutions of Einstein's field equations with smooth asymptotic structure”, Commun. Math. Phys. 107 (1986), 587-609.), (-,“On the global existence and the asymptotic behavior of solutions to the Einstein-Maxwell-Yang-Mills equations”, Jour. Diff. Geom. 34 (1991), 275-345) using conformal completions techniques. In this conformal approach one has to 1) construct initial data, which satisfy the general relativistic constraint equations, with appropriate behavior near the conformal boundary, and 2) show a local (and perhaps also a global) existence theorem for the associated evolution problem. In this context solutions of the constraint equations can be found by solving a nonlinear elliptic system of equations, one of which resembles the Yamabe equation (and coincides with this equation in some cases), with the system degenerating near the conformal boundary. In the first part of the talk I (PTC) will describe the existence and boundary regularity results about this system obtained some years ago in collaboration with Helmut Friedrich and Lars Andersson. Some new applications of those techniques are also presented. In the second part of the talk I will describe some new results, obtained in collaboration with Olivier Lengard, concerning the evolution problem.

Inthe present work we show that the existence of non-vanishing torsion field may solve, at least, one of the problems FRW-cosmology, the particle horizons problem. The field equations of general relativity (GR) are written in a space having non-vanishing torsion, the absolute parallelism (AP) space. An AP-Structure, satisfying the cosmological principle, is used to construct a world model. Energy density and pressure, purely induced by torsion, are defined from the building blocks of the AP-geometry using GR. When these quantities are used in the FRW-dynamical equations, we get a world model free from particle horizons.