General relativistic dynamics of compact binary systems

The equations of motion of compact binary systems have been derived in the post-Newtonian (PN) approximation of general relativity. The current level of accuracy is 3.5PN order. The conservative part of the equations of motion (neglecting the radiation reaction damping terms) is deducible from a generalized Lagrangian in harmonic coordinates, or equivalently from an ordinary Hamiltonian in ADM coordinates. As an application, we investigate the problem of the dynamical stability of circular binary orbits against gravitational perturbations up to the 3PN order. We find that there is no innermost stable circular orbit or ISCO at the 3PN order for equal masses. To cite this article: L. Blanchet, C. R. Physique 8 (2007).     

Post-Newtonian parameter <Emphasis Type="Italic">γ</Emphasis> in generalized non-local gravity

We investigate the post-Newtonian parameter γ and derive its formalism in generalized non-local (GNL) gravity, which is the modified theory of general relativity (GR) obtained by adding a term m ^2n?2 R?^?n R to the Einstein-Hilbert action. Concretely, based on parametrizing the generalized non-local action in which gravity is described by a series of dynamical scalar fields ? ⁱ in addition to the metric tensor g _μν, the post-Newtonian limit is computed, and the effective gravitational constant as well as the post-Newtonian parameters are directly obtained from the generalized non-local gravity. Moreover, by discussing the values of the parametrized post-Newtonian parameters γ, we can compare our expressions and results with those in Hohmann and Järv et al. (2016), as well as current observational constraints on the values of γ in Will (2006). Hence, we draw restrictions on the nonminimal coupling terms F? around their background values.     

The importance of gravitational self-field effects in binary systems with compact objects. I. The “static two-body problem” and the attraction law in a post-Newtonian approximation of general relativity

As a first step toward a proper treatment of compact objects in binary systems the attraction force of two massive bodies connected by a rod is calculated in a post-Newtonian expansion. Contrary to a calculation by Weyl und Bach we start without specializing the internal structure of the bodies. We consider general anisotropic pressures and do not require axial symmetry for the bodies. We calculate the attraction force first in a post-Newtonian approximation and then (in paper II) we shall be concerned with the post-post-Newtonian approximation. In both approximations we obtain Newton's attraction forceM _S1 M _S2/R ² plus terms of order 1/R ³ and higher, whereM _S1,M _S2 are the Schwarzschild masses of the bodies.     

Vortex Equations in Abelian Gauged σ-Models

We consider nonlinear gauged σ-models with Kähler domain and target. For a special choice of potential these models admit Bogomolny (or self-duality) equations — the so-called vortex equations. Here we describe the space of solutions and energy spectrum of the vortex equations when the gauge group is a torus T ⁿ , the domain is compact, and the target is We also obtain a large family of solutions when the target is a compact Kähler toric manifold.     

Compactification and supersymmetry in d = 11 supergravity

We examine the conditions under which the ground state of d = 11 supergravity can be supersymmetric and be of the form M⁴ ? B⁷ with M⁴ Minkowski spacetime and B⁷ a compact seven-dimensional manifold. Since we have in mind a background that renders the effective action stationary we make no use of the classical field equations. We find that the requirement that the four-space be flat is very restrictive. It requires all components of the background four-index field to vanish and the compact manifold to be Ricci-flat and hence to have at most the abelian symmetries associated with tori.     

Non-Abelian Vortices on Riemann Surfaces: an Integrable Case

We consider U(n + 1) Yang–Mills instantons on the space Σ × S ², where Σ is a compact Riemann surface of genus g. Using an SU(2)-equivariant dimensional reduction, we show that the U(n + 1) instanton equations on Σ × S ² are equivalent to non-Abelian vortex equations on Σ. Solutions to these equations are given by pairs (A,?), where A is a gauge potential of the group U(n) and ? is a Higgs field in the fundamental representation of the group U(n). We briefly compare this model with other non-Abelian Higgs models considered recently. Afterwards we show that for g > 1, when Σ × S ² becomes a gravitational instanton, the non-Abelian vortex equations are the compatibility conditions of two linear equations (Lax pair) and therefore the standard methods of integrable systems can be applied for constructing their solutions.     

Second order electromagnetic radiative corrections toe + e −→γ*Z 0→μ+μ−

We derive a compact expression for the cross sectione ⁺ e ^?→μ⁺μ^?, containing a full treatment of the electromagnetic radiative corrections from both initial and final states. By using a formulation in terms of evolution equations, we obtain an expression for the cross-section to theO(α²) which includes soft and hard photons and resums to all orders dominant and non-dominant logarithmic contributions. Comparisons with previous results are made.     

Concentration of Laplace Eigenfunctions and Stabilization of Weakly Damped Wave Equation

In this article, we prove some universal bounds on the speed of concentration on small (frequency-dependent) neighbourhoods of sub-manifolds of L²-norms of quasi modes for Laplace operators on compact manifolds. We deduce new results on the rate of decay of weakly damped wave equations.     

The black hole and FRW geometries of non-relativistic gravity

We consider the recently proposed non-relativistic Ho?ava–Lifshitz four-dimensional theory of gravity. We study a particular limit of the theory which admits flat Minkowski vacuum and we discuss thoroughly the quadratic fluctuations around it. We find that there are two propagating polarizations of the metric. We then explicitly construct a spherically symmetric, asymptotically flat, black hole solution that represents the analog of the Schwarzschild solution of GR. We show that this theory has the same Newtonian and post-Newtonian limits as GR and thus, it passes the classical tests. We also consider homogeneous and isotropic cosmological solutions and we show that although the equations are identical with GR cosmology, the couplings are constrained by the observed primordial abundance of ⁴He.     

Post-Newtonian Expansions for Perfect Fluids

We prove the existence of a large class of dynamical solutions to the Einstein-Euler equations that have a first post-Newtonian expansion. The results here are based on the elliptic-hyperbolic formulation of the Einstein-Euler equations used in [15], which contains a singular parameter , where v _T is a characteristic velocity associated with the fluid and c is the speed of light. As in [15], energy estimates on weighted Sobolev spaces are used to analyze the behavior of solutions to the Einstein-Euler equations in the limit , and to demonstrate the validity of the first post-Newtonian expansion as an approximation.     

Slow-motion approximations in general relativity I. Decomposition of the field equations

A new (3+1)-dimensional decomposition of the Einstein gravitational field equations is obtained for a general spacetime. The metric is taken in the form $$ds^2 = e^{ - 2u} k_{ab} (dx^a + \xi ^a dt)(dx^b + \xi ^b dt) - c^2 e^{2u} dt^2 $$ and the resulting equations treatk _ab as the metric in the space-like hypersurfacest=constant. It is shown that this decompostion forms a more convenient starting point for slow motion approximations than does their usual 4-dimensional formulation. This is illustrated by a derivation of the first post-Newtonian approximation to the field equations, the simplicity there resulting fromk _ab being still flat to this order.     

Dynamics of a system of compact spin bodies with charges and magnetic moments

Universal equations of motion of the second kind previously proposed by the author are used to investigate the dynamics of a system of compact spin bodies with charges and magnetic moments. It is shown that the resulting post-Newtonian equations of motion do not coincide with existing results.Astrophysics Institute, Kazakhstan Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 56–62, July, 1994.     

Equations of hydrodynamics in general relativity in the slow motion approximation

By means of a formal solution to the Einstein gravitational field equations a slow motion expansion in inverse powers of the speed of light is developed for the metric tensor. The formal solution, which satisfies the deDonder coordinate conditions and the Trautman outgoing radiation condition, is in the form of an integral equation which is solved iteratively. A stress-energy tensor appropriate to a perfect fluid is assumed and all orders of the metric needed to obtain the equations of motion and conserved quantities to the 21/2post-Newtonian approximation are found. The results are compared to those obtained in another gauge by S. Chandrasekhar. In addition, the relation of the fast motion approximation to the slow motion approximation is examined.     

Post-post-Newtonian deflection of light ray in multiple systems with PPN parameters

The post-Newtonian scheme in multiple systems with post-Newtonian parameters presented by Klioner and Soffel is extended to the post-post-Newtonian (PPN) order for light propagation problem in the solar system.Under considering the solar system experiment requirement,a new parameter ε is introduced.This extension does not change the virtue of the scheme on the linear partial differential equations of the potential and vector potential mentioned in previous work.Furthermore,this extension is based on the former work done by Richter and Matzner in one global system theory.As an application,we also consider the deflection of light ray in the global coordinates.And the deflection angle of light ray is obtained with post-Newtonian parameters.     

Relativistic Charged Star Solutions in Higher Dimensions

We present a class of relativistic solutions of the Einstein-Maxwell equations for a spherically symmetric charged static fluid sphere in higher dimensions. The interior space at t=constant considered here possess (D?1) dimensional spheroidal geometry described by a higher dimensional Vaidya-Tikekar metric. A class of new static solutions of coupled Einstein-Maxwell equations is obtained in a D-dimensional space-time by prescribing the geometry of a (D?1) dimensional hyper spheroid in hydrostatic equilibrium. The solutions of the Einstein-Maxwell field equations are employed to obtain relativistic models for charged compact stars with a suitable law for variation of electric field in terms of the charged fluid content in the interior of the sphere. The central density is found to depend on the space-time dimensions and a physically realistic model is permitted for (D≥4). The validity of both Strong Energy Condition (SEC), Weak Energy Condition (WEC) are studied for a given configuration and compactness of compact objects. We found new class of solutions with interesting stellar models where it permits a star with a core having different property than the rest which however disappears in higher dimensions. The effect of dimensions on the Electric charge of the compact object is studied. We note that the upper limit of the electric field is determined by the space-time dimensions which are determined.     

A variational principle for symplectic connections

We introduce a variational principle for symplectic connections and study the corresponding field equations. For two-dimensional compact symplectic manifolds we determine all solutions of the field equations. For two-dimensional non-compact simply connected symplectic manifolds we give an essentially exhaustive list of solutions of the field equations. Finally we indicate how to construct from solutions of the field equations on (M, ω) solutions of the field equations on the cotangent bundle to M with its standard symplectic structure.     

Space-time geometry and symmetry breaking

We look for solutions of the Einstein-Yang-Mills equations in a 4 + D dimensional space-time. We find solutions where the first 4 dimensions are a flat Minkowskian space-time, while the D others are a compact, space-like manifold of small size. Such solutions can be obtained for an arbitrary compact gauge group K and are invariant under a sub-group G of K related to the space-time geometry. This shows that 4 + D dimensional gravity can give a mechanism for the super-strong symmetry breaking needed in grand unified field theories without introducing Higgs scalars.     

Unified compact ADI methods for solving nonlinear viscous and nonviscous wave equations

In this paper, two unified alternating direction implicit (ADI) methods, based on the combination of fourth-order compact difference for the approximations of the second spatial derivatives with approximation factorization of difference operators, are presented for solving a two-dimensional (2D) and three-dimensional (3D) nonlinear viscous and nonviscous wave equations, respectively. By the discrete energy method, it is shown that their solutions converge to exact solutions with an order of two in time and four in space in L²- and H¹-norms. Finally, numerical findings testify the computational efficiency of the algorithms.     

Symmetries of cosmological Cauchy horizons

We consider analytic vacuum and electrovacuum spacetimes which contain a compact null hypersurface ruled byclosed null generators. We prove that each such spacetime has a non-trivial Killing symmetry. We distinguish two classes of null surfaces, degenerate and non-degenerate ones, characterized by the zero or non-zero value of a constant analogous to the “surface gravity” of stationary black holes. We show that the non-degenerate null surfaces are always Cauchy horizons across which the Killing fields change from spacelike (in the globally hyperbolic regions) to timelike (in the acausal, analytic extensions). For the special case of a null surface diffeomorphic toT ³ we characterize the degenerate vacuum solutions completely. These consist of an infinite dimensional family of “plane wave” spacetimes which are entirely foliated by compact null surfaces. Previous work by one of us has shown that, when one dimensional Killing symmetries are allowed, then infinite dimensional families of non-degenerate, vacuum solutions exist. We recall these results for the case of Cauchy horizons diffeomorphic toT ³ and prove the generality of the previously constructed non-degenerate solutions. We briefly discuss the possibility of removing the assumptions of closed generators and analyticity and proving an appropriate generalization of our main results. Such a generalization would provide strong support for the cosmic censorship conjecture by showing that causality violating, cosmological solutions of Einstein's equations are essentially an artefact of symmetry.