Which action for brane worlds?

We classify all fundamental electrically charged thin shells in general relativity, i.e., static spherically symmetric perfect fluid thin shells with a Minkowski spacetime interior and a Reissner-Nordstr\"om spacetime exterior, characterized by the spacetime mass and electric charge. The fundamental shell can exist in three states, nonextremal, extremal, and overcharged. The nonextremal state allows the shell to be located such that its radius can be outside its own gravitational radius, or can be inside its own Cauchy radius. The extremal state allows the shell to be located such that its radius can be outside its own gravitational radius, or can be inside it. The overcharged state allows the shell to be located anywhere. There is a further division, one has to specify the orientation of the shell, i.e., whether the normal out of the shell points toward increasing or decreasing radii. There is still a subdivision in the extremal state when the shell is at the gravitational radius, in that the shell can approach it from above or from below. The shell is assumed to be composed of an electrically charged perfect fluid, and the energy conditions are tested. Carter-Penrose diagrams are drawn for the shell spacetimes. There are fourteen cases in the classification of the fundamental shells, namely, nonextremal star shells, nonextremal tension shell black holes, nonextremal tension shell regular and nonregular black holes, nonextremal compact shell naked singularities, Majumdar-Papapetrou star shells, extremal tension shell singularities, extremal tension shell regular and nonregular black holes, Majumdar-Papapetrou compact shell naked singularities, Majumdar-Papapetrou shell quasiblack holes, extremal null shell quasinonblack holes, extremal null shell singularities, Majumdar-Papapetrou null shell singularities, overcharged star shells, and overcharged compact shell naked singularities.

On the electromagnetic field of a charged collapsing spherical shell in general relativity

This paper shows how the structure and dynamics of a thin shell traveling at the speed of light can be obtained from a simple and convenient prescription that is a straightforward extension and continuous limit of the familiar extrinsic-curvature algorithm for subluminal shells. It allows the space-time coordinates to be chosen freely and independently on the two sides of the shell. The prescription is applied to several examples of interest in general relativity and cosmology.

Section V discusses quasinormal modes. It is shown how the transition conditions at the shell which are derived in Sec. III can be used to calculate numerically the quasinormal mode frequencies of the shell. Section VI deals with the Newtonian limit. The ordinary differential equations describing linearized self-gravitating Newtonian shells are derived as limits from the relativistic equations. This limit process is rather subtle, because the Gaussian gauge, which is natural in the relativistic picture, is not so natural for the usual formulation of Newton’s theory in an inertial frame. Nevertheless, the correct Newtonian equations can be found as a limit from Einstein’s equations and the transition conditions at the shell.

The Einstein-Hilbert action for general relativity is not well posed in terms of the metric $g_{ab}$ as a dynamical variable. There have been many proposals to obtain an well posed action principle for general relativity, e.g., addition of the Gibbons-Hawking-York boundary term to the Einstein-Hilbert action. These boundary terms are dependent on what one fixes on the boundary and in particular on spacetime dimensions as well. Following recent works of Padmanabhan we will introduce two new variables to describe general relativity and the action principle with these new dynamical variables will turn out to be well posed. Then we will connect these dynamical variables and boundary term obtained thereof to existing literature and shall comment on a few properties of Einstein-Hilbert action which might have been unnoticed earlier in the literature. Before concluding with future prospects and discussions, we will perform a general analysis of the boundary term of Einstein-Hilbert action for null surfaces as well.

Seven new solutions to the interior static and spherically symmetric Einstein's field equations (EFE) are found and investigated. These new solutions are a generalisation of the quadratic density fall-off profile of the Tolman VII solution. The generalisation involves the addition of anisotropic pressures and electric charge to the density profile. Of these new solutions three are found to obey all the necessary conditions of physical acceptability, including linear stability under radial perturbations, and causality of the speed of pressure waves inside the object. Additionally an equation of state can be found for all the physically viable solutions. The generalised pulsation equation for interior solutions to the EFE that include both electric charge and pressure anisotropy is derived and used to determine the stability of the solutions. However the pulsation equation found is general and can be used for all new solutions that contain these ingredients.

I discuss the relation between arbitrarily high-order theories of gravity and scalar--tensor gravity at the level of the field equations and the action. I show that (2n+4)-order gravity is dynamically equivalent to Brans--Dicke gravity with an interaction potential for the Brans--Dicke field and n further scalar fields. This scalar--tensor action is then conformally equivalent to the Einstein--Hilbert action with n+1 scalar fields. This clarifies the nature and extent of the conformal equivalence between extended gravity theories and general relativity with many scalar fields.

We investigate the time-dependent nature of the cosmological constant, Λ, of the Einstein Field Equation (EFE). Beginning with the Einstein–Hilbert action as our fundamental principle we develop a modified version of the EFE allowing the value of Λ to vary as a function of time, Λ(t), indirectly, for an expanding universe. We follow the evolving Λ presuming four-dimensional space–time and a flat universe geometry and present derivations of Λ(t) as functions of the Hubble constant, matter density, and volume changes which can be traced back to the radiation epoch. The models are more detailed descriptions of the Λ dependence on cosmological factors than previous, allowing calculations of the important parameters, Ω_m and Ω_r, to deep lookback times. Since we derive these without the need for extra dimensions or other special conditions our derivations are useful for model evaluation with astronomical data. This should aid resolution of several difficult problems of astronomy such as the best value for the Hubble constant at present and at recombination.

The post-Minkowskian expansion of Einstein’s general theory of relativity has received much attention in recent years due to the possibility of harnessing the computational power of modern amplitude calculations in such a classical context. In this brief review, we focus on the post-Minkowskian expansion as applied to the two-body problem in general relativity without spin, and we describe how relativistic quantum field theory can be used to greatly simplify analytical calculations based on the Einstein–Hilbert action. Subtleties related to the extraction of classical physics from such quantum mechanical calculations highlight the care which must be taken when both positive and negative powers of Planck’s constant are at play. In the process of obtaining classical results in both Einstein gravity and supergravity, one learns new aspects of quantum field theory that are obscured when using units in which Planck’s constant is set to unity. The scattering amplitude approach provides a self-contained framework for deriving the two-body scattering valid in all regimes of energy. There is hope that the full impact of amplitude computations in this field may significantly alter the way in which gravitational wave predictions will advance in the coming years.

We develop a strictly geometric reformulation of General Relativity (GR) in which a scalar clock field ϕ defines the spacetime foliation through its timelike gradient nµ = -∇µϕ/ -(∇ϕ) 2 . The field ϕ is not a new dynamical degree of freedom and does not enter the action; its sole role is to generate a covariant, hypersurface-orthogonal slicing that renders the ADM lapse, shift, and extrinsic curvature algebraic functionals of (gµν , ϕ).Starting from the pure Einstein-Hilbert action, we carry out a complete 3 + 1 decomposition adapted to the clock-field foliation and derive the Hamiltonian constraint, momentum constraint, and full constraint algebra. All coincide exactly with their canonical GR counterparts, confirming that the theory propagates only the two tensorial graviton modes and contains no additional scalar or vector excitations. The Einstein equations are recovered identically, the ADM algebra remains first class, and both cosmological dynamics and linear perturbations match those of standard GR.The novelty of the construction is conceptual rather than dynamical: ϕ provides a covariant, relational time variable that fixes the foliation non-perturbatively, offering a clean geometric notion of time for canonical quantization and Wheeler-DeWitt evolution. Because the framework introduces no new fields, couplings, or potentials, it avoids the instabilities and extra degrees of freedom typical of scalar-tensor or modified-gravity theories.This yields a mathematically consistent and fully covariant formulation of GR with an internally defined geometric clock, opening new directions for relational observables, deparametrized quantum gravity, and emergent-time frameworks.

General relativity as a hybrid theory: The genesis of Einstein's work on the problem of motion

Plane walls (including plane domain walls) without reflection symmetry are studied in the framework of Einstein's general relativity. Using the distribution theory, all the Einstein field equations and Bianchi identities are split into two groups: one holding in the regions outside of the wall and the other holding at the wall. The Einstein field equations at the wall are found to take a very simple form, and given explicitly in terms of the discontinuities of the metric coefficients and their derivatives. The Bianchi identities at the wall are also given explicitly. Using the latter, the interaction of a plane wall with gravitational waves and some specific matter fields is studied. In particular, it is found that, when a gravitational plane wave passes through a wall, if the wall has no reflection symmetry, the phenomena, such as reflection, stimulation, or absorption, in general, occur. It is also found that, unlike for gravitational waves, a massless scalar wave or an electromagnetic wave continuously passes through a wall without any reflection. The repulsion and attraction of a plane wall are also studied. It is found that the acceleration of an observer who is at rest relative to the wall usually consists of threemore » parts: one is due to the force produced by the wall, the second is due to the force produced by the space-time curvature, which is zero if the wall has reflection symmetry, and the last is due to the accelerated motion of the wall. As a result, a repulsive (attractive) plane wall may not be repulsive (attractive) at all. Finally, the collision and interaction among the walls are studied.« less

Einstein's field equations are based on Riemannian geometry. One of the great important in Riemannian geometry is the curves that minimize the distance between two given points. The Ricci curvature tensors are also broadly applicable to modern Riemannian geometry and general theory of relativity (GTR). The solution of Einstein's field equations, need Ricci tensors. The line elements based on the ellipsoidal coordinates systems usually are difficult to work but are more perfect than spherical coordinate systems. Here we are trying to get all non-zero Ricci tensors coefficients of a line element in an oblate spheroidal coordinate system for an elliptical shaped object. These objects can be as galaxies, stars or planets and other celestial objects that we know today.

In the framework of general relativity, systematically deal with the problems of galaxy formation and evolution. Einstein’s equation of gravitational field is first improved, space-time is proved to be infinite and its expansion and contraction are in circles, the singular point of big bang didn’t exist, Besides, it is disclosed that galaxies or celestial bodies form from unceasing growth but not the assemblage of existent matter after big bang, new matter continuously creates in the interior of celestial bodies depending on the work done by the negative pressure, and in the same time celestial bodies, galaxies and space synchronously enlarge in accordance with Hubble expansion, cosmic density and pressure keep unchanged.

A student-friendly style, over 100 illustrations, and numerous exercises are brought together in this textbook for advanced undergraduate and beginning graduate students in physics and mathematics. Lewis Ryder develops the theory of general relativity in detail. Covering the core topics of black holes, gravitational radiation, and cosmology, he provides an overview of general relativity and its modern ramifications. The book contains chapters on gravitational radiation, cosmology, and connections between general relativity and the fundamental physics of the microworld. It explains the geometry of curved spaces and contains key solutions of Einstein's equations - the Schwarzschild and Kerr solutions. Mathematical calculations are worked out in detail, so students can develop an intuitive understanding of the subject, as well as learn how to perform calculations. The book also includes topics concerned with the relation between general relativity and other areas of fundamental physics. Password protected solutions for instructors are available at www.cambridge.org/9780521845632.