Initial Conditions in General Relativity: Lapse and Shift Formulation

1969 ◽  
Vol 10 (8) ◽  
pp. 1503-1507 ◽  
Author(s):  
Elliot P. Belasco ◽  
Hans C. Ohanian
Keyword(s):  
General Relativity ◽  
Initial Conditions

scholarly journals Secular evolution of close-in planets: the effects of general relativity

2020 ◽  
Vol 493 (1) ◽  
pp. 427-436
Author(s):  
F Marzari ◽  
M Nagasawa
Keyword(s):  
General Relativity ◽  
Numerical Integration ◽  
Time Scales ◽  
Initial Conditions ◽  
Outer Planet ◽  
Secular Perturbations ◽  
Secular Evolution ◽  
Eccentric Orbits ◽  
Wide Range ◽  
Comparable Time

ABSTRACT Pairs of planets in a system may end up close to their host star on eccentric orbits as a consequence of planet–planet scattering, Kozai, or secular migration. In this scenario, general relativity and secular perturbations have comparable time-scales and may interfere with each other with relevant effects on the eccentricity and pericenter evolution of the two planets. We explore, both analytically and via numerical integration, how the secular evolution is changed by general relativity for a wide range of different initial conditions. We find that when the faster secular frequency approaches the general relativity precession rate, which typically occurs when the outer planet moves away from the inner one, it relaxes to it and a significant damping of the proper eccentricity of the inner planet occurs. The proper eccentricity of the outer planet is reduced as well due to the changes in the secular interaction of the bodies. The lowering of the peak eccentricities of the two planets during their secular evolution has important implications on their stability. A significant number of two-planet systems, otherwise chaotic because of the mutual secular perturbations, are found stable when general relativity is included.

scholarly journals The dynamics of brane-world cosmological models

2005 ◽  
Vol 83 (5) ◽  
pp. 475-525 ◽  
Author(s):  
A A Coley
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Initial Conditions ◽  
High Energy ◽  
Cosmological Models ◽  
Brane World ◽  
Initial Singularity ◽  
Field Equations ◽  
Cosmological Singularity ◽  
Spatially Homogeneous

Brane-world cosmology is motivated by recent developments in string/M-theory and offers a new perspective on the hierarchy problem. In the brane-world scenario, our Universe is a four-dimensional subspace or brane embedded in a higher-dimensional bulk spacetime. Ordinary matter fields are confined to the brane while the gravitational field can also propagate in the bulk, and it is not necessary for the extra dimensions to be small, or even compact, leading to modifications of Einstein's theory of general relativity at high energies. In particular, the Randall–Sundrum-type models are relatively simple phenomenological models that capture some of the essential features of the dimensional reduction of eleven-dimensional supergravity introduced by Hořava and Witten. These curved (or warped) models are self-consistent and simple and allow for an investigation of the essential nonlinear gravitational dynamics. The governing field equations induced on the brane differ from the general relativistic equations in that there are nonlocal effects from the free gravitational field in the bulk, transmitted via the projection of the bulk Weyl tensor, and the local quadratic energy-momentum corrections, which are significant in the high-energy regime close to the initial singularity. In this review, we investigate the dynamics of the five-dimensional warped Randall–Sundrum brane worlds and their generalizations, with particular emphasis on whether the currently observed high degree of homogeneity and isotropy can be explained. In particular, we discuss the asymptotic dynamical evolution of spatially homogeneous brane-world cosmological models containing both a perfect fluid and a scalar field close to the initial singularity. Using dynamical systems techniques, it is found that, for models with a physically relevant equation of state, an isotropic singularity is a past-attractor in all orthogonal spatially homogeneous models (including Bianchi type IX models). In addition, we describe the dynamics in a class of inhomogeneous brane-world models, and show that these models also have an isotropic initial singularity. These results provide support for the conjecture that typically the initial cosmological singularity is isotropic in brane-world cosmology. Consequently, we argue that, unlike the situation in general relativity, brane-world cosmological models may offer a plausible solution to the initial conditions problem in cosmology. PACS Nos.: 98.89.Cq/Jk, 04.20–q

scholarly journals Isochronous cosmologies

2014 ◽  
Vol 11 (06) ◽  
pp. 1450054 ◽  
Author(s):  
Fabio Briscese ◽  
Francesco Calogero
Keyword(s):  
General Relativity ◽  
Physical Interpretation ◽  
Body Problem ◽  
Initial Conditions ◽  
Many Body ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Long Time ◽  
Geodesically Complete ◽  
Relevant Class

The possibility has been recently demonstrated to manufacture (nonrelativistic, Hamiltonian) many-body problems which feature an isochronous time evolution with an arbitrarily assigned period T yet mimic with good approximation, or even exactly, any given many-body problem (within a large, physically relevant, class) over times [Formula: see text] which may also be arbitrarily large (but of course such that [Formula: see text]). Purpose and scope of this paper is to explore the possibility to extend this finding to a general relativity context. For simplicity we restrict our consideration to the case of homogeneous and isotropic metrics and show that, via an approach analogous to that used for the nonrelativistic many-body problem, a class of homogeneous and isotropic cyclic solutions of Einstein's equations may be obtained. For these solutions the duration of the cycles does not depend on the initial conditions, so we call these models isochronous cosmologies. We give a physical interpretation of such metrics and in particular we show that they may behave arbitrarily closely, or even identically, to the Friedman–Robertson–Walker solutions of Einstein's equations for an arbitrarily long time (of course shorter than their period, which can also be assigned arbitrarily), so that they may reproduce all the satisfactory phenomenological features of the standard cosmological Λ-CDM model in a portion of their cycle; while these isochronous cosmologies may be geodesically complete and therefore singularity-free.

scholarly journals REPARAMETRIZATION-INVARIANT PATH INTEGRAL IN GR AND "BIG BANG" OF QUANTUM UNIVERSE

2001 ◽  
Vol 16 (10) ◽  
pp. 1715-1742 ◽  
Author(s):  
M. PAWLOWSKI ◽  
V. N. PERVUSHIN
Keyword(s):  
General Relativity ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Big Bang ◽  
Constrained System ◽  
Generating Functional ◽  
Finite Space ◽  
Creation Of Matter ◽  
The Big Bang ◽  
Friedmann Robertson Walker

The reparametrization-invariant generating functional for the unitary and causal perturbation theory in general relativity in a finite space–time is obtained. The region of validity of the Faddeev–Popov–DeWitt functional is studied. It is shown that the invariant content of general relativity as a constrained system can be covered by two "equivalent" unconstrained systems: the "dynamic" (with "dynamic" evolution parameter as the metric scale factor) and "geometric" (given by the Levi–Civita type canonical transformation to the action-angle variables where the energy constraint converts into a new momentum). "Big Bang," the Hubble evolution, and creation of matter fields by the "geometric" vacuum are described by the inverted Levi–Civita transformation of the geomeric system into the dynamic one. The particular case of the Levi–Civita transformations are the Bogoliubov ones of the particle variables (diagonalizing the dynamic Hamiltonian) to the quasiparticles (diagonalizing the equations of motion). The choice of initial conditions for the "Big Bang" in the form of the Bogoliubov (squeezed) vacuum reproduces all stages of the evolution of the Friedmann–Robertson–Walker universe in their conformal (Hoyle–Narlikar) versions.

Primordial quantum perturbations

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
General Relativity ◽  
Scalar Field ◽  
Energy Momentum Tensor ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Momentum Tensor ◽  
Slow Roll ◽  
Inflationary Models

This chapter shows that fluctuations of quantum origin are generated during inflation and that this process supplies initial conditions compatible with the observations. These fluctuations are therefore an important prediction of inflationary models. The chapter thus begins with a study of perturbations during inflation, proceeding in a similar manner to the previous chapter by finding the perturbation of the energy–momentum tensor of the scalar field. Another method of deriving the equations of motion of the perturbations is to start from the action of general relativity coupled to a scalar field, and expand to second order in the metric and scalar field perturbations. The chapter then proceeds with the determination of the initial conditions and the slow-roll inflation.

Beyond the Dynamical Universe

2018 ◽  
Author(s):  
Michael Silberstein ◽  
W.M. Stuckey ◽  
Timothy McDevitt
Keyword(s):  
General Relativity ◽  
Initial Conditions ◽  
Big Bang ◽  
Theoretical Physics ◽  
Global Constraints ◽  
New Paradigm ◽  
Least Action Principle ◽  
Initial State ◽  
The Past ◽  
Foundations Of Physics

Theoretical physics and foundations of physics have not made much progress in the last few decades. There is no consensus among researchers on how to approach unifying general relativity and quantum field theory (quantum gravity), explaining so-called dark energy and dark matter (cosmology), or the interpretation and implications of quantum mechanics and relativity. In addition, both fields are deeply puzzled about various facets of time including, above all, time as experienced. This book argues that this impasse is the result of the “dynamical universe paradigm,” the idea that reality fundamentally comprises physical entities that evolve in time from some initial state according to dynamical laws. Thus, in the dynamical universe, the initial conditions plus the dynamical laws explain everything else going exclusively forward in time. In cosmology, for example, the initial conditions reside in the Big Bang and the dynamical law is supplied by general relativity. Accordingly, the present state of the universe is explained exclusively by its past. A completely new paradigm (called Relational Blockworld) is offered here whereby the past, present, and future co-determine each other via “adynamical global constraints,” such as the least action principle. Accordingly, the future is just as important for explaining the present as the past is. Most of the book is devoted to showing how Relational Blockworld resolves many of the current conundrums of both theoretical physics and foundations of physics, including the mystery of time as experienced and how that experience relates to the block universe.

ON STABILITY OF SIMPLEST NONSINGULAR INFLATIONARY COSMOLOGICAL MODELS WITHIN GENERAL RELATIVITY AND GAUGE THEORIES OF GRAVITY

2004 ◽  
Vol 13 (04) ◽  
pp. 695-707 ◽  
Author(s):  
G. V. VERESHCHAGIN
Keyword(s):  
General Relativity ◽  
Scalar Field ◽  
Gauge Theories ◽  
Initial Conditions ◽  
Solvable Model ◽  
Cosmological Models ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Theories Of Gravity ◽  
Analysis Of Stability

In this paper we provide approximate analytical analysis of stability of nonsingular inflationary chaotic-type cosmological models. Initial conditions for nonsingular solutions at the bounce correspond to dominance of potential part of the energy density of the scalar field over its kinetic part both within general relativity and gauge theories of gravity. Moreover, scalar field at the bounce exceeds the Planckian value and on expansion stage these models correspond to chaotic inflation. Such solutions can be well approximated by explicitly solvable model with constant effective potential (cosmological term) and massless scalar field during the bounce and on stages of quasi-exponential contraction and expansion. Perturbative analysis shows that nonsingular inflationary solutions are exponentially unstable during contraction stage. This result is compared with numerical calculations.

scholarly journals Toward Interactions through Information in a Multifractal Paradigm

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 987
Author(s):  
Maricel Agop ◽  
Alina Gavriluț ◽  
Claudia Grigoraș-Ichim ◽  
Ștefan Toma ◽  
Tudor-Cristian Petrescu ◽  
...  
Keyword(s):  
General Relativity ◽  
Initial Conditions ◽  
Harmonic Mappings ◽  
Poincaré Metric ◽  
Minimization Principle ◽  
Ernst Potential ◽  
Lobachevsky Plane ◽  
Local Manifestation ◽  
Newtonian Type ◽  
Multifractal Dynamics

In a multifractal paradigm of motion, Shannon’s information functionality of a minimization principle induces multifractal–type Newtonian behaviors. The analysis of these behaviors through motion geodesics shows the fact that the center of the Newtonian-type multifractal force is different from the center of the multifractal trajectory. The measure of this difference is given by the eccentricity, which depends on the initial conditions. In such a context, the eccentricities’ geometry becomes, through the Cayley–Klein metric principle, the Lobachevsky plane geometry. Then, harmonic mappings between the usual space and the Lobachevsky plane in a Poincaré metric can become operational, a situation in which the Ernst potential of general relativity acquires a classical nature. Moreover, the Newtonian-type multifractal dynamics, perceived and described in a multifractal paradigm of motion, becomes a local manifestation of the gravitational field of general relativity.

scholarly journals INITIAL CONDITIONS FOR NUMERICAL RELATIVITY: INTRODUCTION TO NUMERICAL METHODS FOR SOLVING ELLIPTIC PDEs

2013 ◽  
Vol 28 (22n23) ◽  
pp. 1340016 ◽  
Author(s):  
HIROTADA OKAWA
Keyword(s):  
Black Holes ◽  
General Relativity ◽  
Numerical Methods ◽  
Elliptic Equations ◽  
Initial Conditions ◽  
Numerical Relativity ◽  
Elliptic Partial Differential Equations ◽  
Elliptic Pdes ◽  
Proper Understanding ◽  
Binary Problems

Numerical relativity became a powerful tool to investigate the dynamics of binary problems with black holes or neutron stars as well as the very structure of General Relativity. Although public numerical relativity codes are available to evolve such systems, a proper understanding of the methods involved is quite important. Here, we focus on the numerical solution of elliptic partial differential equations. Such equations arise when preparing initial data for numerical relativity, but also for monitoring the evolution of black holes. Because such elliptic equations play an important role in many branches of physics, we give an overview of the topic, and show how to numerically solve them with simple examples and sample codes written in C ++ and Fortran90 for beginners in numerical relativity or other fields requiring numerical expertise.

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