Frontiers in Relativistic Celestial Mechanics: Volume 1 Theory

• Publisher: De Gruyter
• Year: 2014
• Tongue: English
• Leaves: 420
• Series: De Gruyter Studies in Mathematical Physics; 21
• Category: Library

✦ Synopsis

Relativistic celestial mechanics – investigating the motion celestial bodies under the influence of general relativity – is a major tool of modern experimental gravitational physics. With a wide range of prominent authors from the field, this two-volume series consists of reviews on a multitude of advanced topics in the area of relativistic celestial mechanics – starting from more classical topics such as the regime of asymptotically-flat spacetime, light propagation and celestial ephemerides, but also including its role in cosmology and alternative theories of gravity as well as modern experiments in this area.

This first volume of a two-volume series is concerned with theoretical foundations such as post-Newtonian solutions to the two-body problem, light propagation through time-dependent gravitational fields, as well as cosmological effects on the movement of bodies in the solar systems.

On the occasion of his 80-th birthday, these two volumes honor V. A. Brumberg – one of the pioneers in modern relativistic celestial mechanics.

Contributions include:

• M. Soffel: On the DSX-framework
• T. Damour: The general relativistic two body problem
• G. Schaefer: Hamiltonian dynamics of spinning compact binaries through high post-Newtonian approximations
• A. Petrov and S. Kopeikin: Post-Newtonian approximations in cosmology
• T. Futamase: On the backreaction problem in cosmology
• Y. Xie and S. Kopeikin: Covariant theory of the post-Newtonian equations of motion of extended bodies
• S. Kopeikin and P. Korobkov: General relativistic theory of light propagation in multipolar gravitational fields

A very renowned team of contributors in the field of relativistic celestial mechanics provides reviews on a breadth of advanced topics.

✦ Table of Contents

Contents
List of figures
Preface
The general relativistic two-body problem
1 Introduction
2 Multichart approach to the N-body problem
3 EOB description of the conservative dynamics of two-body systems
4 EOB description of radiation reaction and of the emitted waveform during inspiral
5 EOB description of the merger of binary black holes and of the ringdown of the final black hole
6 EOB vs NR
6.1 EOB[NR] waveforms vs NR ones
6.2 EOB[3PN] dynamics vs NR one
7 Other developments
7.1 EOB with spinning bodies
7.2 EOB with tidally deformed bodies
7.3 EOB and GSF
8 Conclusions
References
Hamiltonian dynamics of spinning compact binaries through high post-Newtonian approximations
1 Introduction
2 Hamiltonian formulation of general relativity
2.1 Point particles
2.2 Spinning particles
2.3 Introducing the Routhian
3 The Poincaré algebra
4 Post-Newtonian binary Hamiltonians
4.1 Spinless binaries
4.2 Spinning binaries
5 Binary motion
5.1 Spinless two-body systems
5.2 Particle motion in Kerr geometry
5.3 Two-body systems with spinning components
References
Covariant theory of the post-Newtonian equations of motion of extended bodies
1 Introduction
2 A theory of gravity for post-Newtonian celestial mechanics
2.1 The field equations
2.2 The energy–momentum tensor
3 Parameterized post-Newtonian celestial mechanics
3.1 External and internal problems of motion
3.2 Solving the field equations by post-Newtonian approximations
3.3 The post-Newtonian field equations
3.4 Conformal harmonic gauge
4 Parameterized post-Newtonian coordinates
4.1 The global post-Newtonian coordinates
4.2 The local post-Newtonian coordinates
5 Post-Newtonian coordinate transformations by asymptotic matching
5.1 General structure of the transformation
5.2 Matching solution
6 Post-Newtonian equations of motion of extended bodies in local coordinates
6.1 Microscopic post-Newtonian equations of motion
6.2 Post-Newtonian mass of an extended body
6.3 Post-Newtonian center of mass and linear momentum of an extended body
6.4 Translational equation of motion in the local coordinates
7 Post-Newtonian equations of motion of extended bodies in global coordinates
7.1 STF expansions of the external gravitational potentials in terms of the internal multipoles
7.2 Translational equations of motion
8 Covariant equations of translational motion of extended bodies
8.1 Effective background manifold
8.2 Geodesic motion and 4-force
8.3 Four-dimensional form of multipole moments
8.4 Covariant translational equations of motion
8.5 Comparison with Dixon’s translational equations of motion
References
On the DSX-framework
1 Introduction
2 The post-Newtonian formalism
2.1 The general form of the metric
3 Field equations and the gauge problem
4 The gravitational field of a body
4.1 Post-Newtonian multipole moments
5 Geodesic motion in the PN-Schwarzschild field
6 Astronomical reference frames
6.1 Transformation between global and local systems: first results
6.2 Split of local potentials, multipole moments
6.3 Tetrad induced local coordinates
6.4 The standard transformation between global and local coordinates
6.5 The description of tidal forces
7 The gravitational N-body problem
7.1 Local evolution equations
7.2 The translational motion
8 Further developments
References
General relativistic theory of light propagation in multipolar gravitational fields
1 Introduction
1.1 Statement of the problem
1.2 Historical background
1.3 Notations and conventions
2 The metric tensor, gauges and coordinates
2.1 The canonical form of the metric tensor perturbation
2.2 The harmonic coordinates
2.3 The ADM coordinates
3 Equations of propagation of electromagnetic signals
3.1 Maxwell equations in curved spacetime
3.2 Maxwell equations in the geometric optics approximation
3.3 Electromagnetic eikonal and light-ray geodesics
3.4 Polarization of light and the Stokes parameters
4 Mathematical technique for analytic integration of light-ray equations
4.1 Monopole and dipole light-ray integrals
4.2 Light-ray integrals from quadrupole and higher order multipoles
5 Gravitational perturbations of the light ray
5.1 Relativistic perturbation of the electromagnetic eikonal
5.2 Relativistic perturbation of the coordinate velocity of light
5.3 Perturbation of the light-ray trajectory
6 Observable relativistic effects
6.1 Gravitational time delay of light
6.2 Gravitational deflection of light
6.3 Gravitational shift of frequency
6.4 Gravity-induced rotation of the plane of polarization of light
7 Light propagation through the field of gravitational lens
7.1 Small parameters and asymptotic expansions
7.2 Asymptotic expressions for observable effects
8 Light propagation through the field of plane gravitational waves
8.1 Plane-wave asymptotic expansions
8.2 Asymptotic expressions for observable effects
References
On the backreaction problem in cosmology
1 Introduction
2 Formulation and averaging
3 Calculation in the Newtonian gauge
4 Definition of the background
5 Conclusions
References
Post-Newtonian approximations in cosmology
1 Introduction
2 Derivatives on the geometric manifold
2.1 Variational derivative
2.2 Lie derivative
3 Lagrangian and field variables
3.1 Action functional
3.2 Lagrangian of the ideal fluid
3.3 Lagrangian of scalar field
3.4 Lagrangian of a localized astronomical system
4 Background manifold
4.1 Hubble flow
4.2 Friedmann–Lemître–Robertson–Walker metric
4.3 Christoffel symbols and covariant derivatives
4.4 Riemann tensor
4.5 The Friedmann equations
4.6 Hydrodynamic equations of the ideal fluid
4.7 Scalar field equations
4.8 Equations of motion of matter of the localized astronomical system
5 Lagrangian perturbations of FLRW manifold
5.1 The concept of perturbations
5.2 The perturbative expansion of the Lagrangian
5.3 The background field equations
5.4 The Lagrangian equations for gravitational field perturbations
5.5 The Lagrangian equations for dark matter perturbations
5.6 The Lagrangian equations for dark energy perturbations
5.7 Linearized post-Newtonian equations for field variables
6 Gauge-invariant scalars and field equations in 1+3 threading formalism
6.1 Threading decomposition of the metric perturbations
6.2 Gauge transformation of the field variables
6.3 Gauge-invariant scalars
6.4 Field equations for the scalar perturbations
6.5 Field equations for vector perturbations
6.6 Field equations for tensor perturbations
6.7 Residual gauge freedom
7 Post-Newtonian field equations in a spatially flat universe
7.1 Cosmological parameters and scalar field potential
7.2 Conformal cosmological perturbations
7.3 Post-Newtonian field equations in conformal spacetime
7.4 Residual gauge freedom in the conformal spacetime
8 Decoupled system of the post-Newtonian field equations
8.1 The universe governed by dark matter and cosmological constant
8.2 The universe governed by dark energy
8.3 Post-Newtonian potentials in the linearized Hubble approximation
8.4 Lorentz invariance of retarded potentials
8.5 Retarded solution of the sound-wave equation
References
Index