Correspondence between modified gravity and general relativity with scalar fields

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.

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.

The field equations of general relativity with electromagnetic stress tensor and zeromass scalar meson field are investigated. The metric coefficients are assumed to be functions of three variables only. It is then shown that, if one assumes a functional relation between some one of the metric coefficients and the electromagnetic potentials, that one can find a solution of the coupled Einstein-Maxwell equations in terms of a solution of the Einstein equations with zeromass scalar meson field as source.

General formula for symmetry factors (S-factor) of Feynman diagrams containing fields with high spins is derived. We prove that symmetry factors of Feynman diagrams of well-known theories do not depend on spins of fields. In contributions to S-factors, self-conjugate fields and non self-conjugate fields play the same roles as real scalar fields and complex scalar fields, respectively. Thus, the formula of S-factors for scalar theories --- theories include only real and complex scalar fields --- works on all well-known theories of fields with high spins.Two interesting consequences deduced from our result are : (i) S-factors of all external connected diagrams consisting of only vertices with three different fields, e.g., spinor QED, are equal to unity; (ii) some diagrams with different topologies can contribute the same factor, leading to the result that the inverse S-factor for the total contribution is the sum of inverse S-factors, i.e., 1/S = \sum_i (1/S_i).

In this paper we construct asymptotically locally AdS and flat black holes in\nthe presence of a scalar field whose kinetic term is constructed out from a\nlinear combination of the metric and the Einstein tensor. The field equations\nas well as the energy-momentum tensor are second order in the metric and the\nfield, therefore the theory belongs to the ones defined by Horndeski. We show\nthat in the presence of a cosmological term in the action, it is possible to\nhave a real scalar field in the region outside the event horizon. The solutions\nare characterized by a single integration constant, the scalar field vanishes\nat the horizon and it contributes to the effective cosmological constant at\ninfinity. We extend these results to the topological case. The solution is\ndisconnected from the maximally symmetric AdS background, however, within this\nfamily there exits a gravitational soliton which is everywhere regular. This\nsoliton is therefore used as a background to define a finite Euclidean action\nand to obtain the thermodynamics of the black holes. For a certain region in\nthe space of parameters, the thermodynamic analysis reveals a critical\ntemperature at which a Hawking-Page phase transition between the black hole and\nthe soliton occurs. We extend the solution to arbitrary dimensions grater than\nfour and show that the presence of a cosmological term in the action allows to\nconsider the case in which the standard kinetic term for the scalar it's not\npresent. In such scenario, the solution reduces to an asymptotically flat black\nhole.\n

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 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.

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.

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.

Rotating cosmological cylindrical wormholes in GR and TEGR sourced by anisotropic fluids

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.

General Relativity: An Introduction for Physicists provides a clear mathematical introduction to Einstein's theory of general relativity. It presents a wide range of applications of the theory, concentrating on its physical consequences. After reviewing the basic concepts, the authors present a clear and intuitive discussion of the mathematical background, including the necessary tools of tensor calculus and differential geometry. These tools are then used to develop the topic of special relativity and to discuss electromagnetism in Minkowski spacetime. Gravitation as spacetime curvature is then introduced and the field equations of general relativity derived. After applying the theory to a wide range of physical situations, the book concludes with a brief discussion of classical field theory and the derivation of general relativity from a variational principle. Written for advanced undergraduate and graduate students, this approachable textbook contains over 300 exercises to illuminate and extend the discussion in the text.

In this paper the quantum vacuum energies induced by massive fluctuations of\none real scalar field on a configuration of two partially transparent plates\nare analysed. The physical properties of the infinitely thin plates are\ncharacterized by two Dirac-$\\delta$ potentials. We find that an\nattractive/repulsive Casimir force arises between the plates when the weights\nof the $\\delta$'s have equal/different sign. If some of the plates absorbs\nfluctuations below some threshold of energy (the corresponding weight is\nnegative) there is the need to set a minimum mass to the scalar field\nfluctuations to preserve unitarity in the corresponding quantum field theory.\nTwo repulsive $\\delta$-interactions are compatible with massless fluctuations.\nThe effect of Dirichlet boundary conditions at the endpoints of the interval\n$(-a,a)$ on a massless scalar quantum field theory defined on this interval is\ntantamount letting the weights of the repulsive $\\delta$-interactions to\n$+\\infty$.\n

Domain walls naturally arise in any model that has a spontaneously broken discrete symmetry. In the simplest example, symmetry breaking is accomplished by a real scalar field. We investigate infinite domain walls in general relativity using the full Einstein scalar-field equations. We study the gravitational effects on test particles both inside and outside the walls and find that matter is repelled by the walls. Also, we find that there is pressure in the direction perpendicular to the plane of the wall.

We present the first symmetry inheritance analysis of fields non-minimally coupled to gravity. In this work we are focused on the real scalar field ϕ with nonminimal coupling of the form . Possible cases of symmetry noninheriting fields are constrained by the properties of the Ricci tensor and the scalar potential. Examples of such spacetimes can be found among those which are ‘dressed’ with the stealth scalar field, a nontrivial scalar field configuration with the vanishing energy–momentum tensor. We classify the scalar field potentials which allow symmetry noninheriting stealth field configurations on top of the exact solutions of the Einstein’s gravitational field equation with the cosmological constant.