Chapter 2. Post-Newtonian Approximation

Asada, Hideki; Futamase, Toshifumi

doi:10.1143/ptps.128.123

Cited by 36 publications

(62 citation statements)

References 43 publications

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“…We thereby obtain M with sufficient accuracy for controlling the energy at the 2PN order 13 . The result…”

Section: B the Adm Massmentioning

confidence: 95%

“…The function g introduced here represents an elementary "kernel" playing an important role in post-Newtonian calculations [8,9,10,11,12,13]. It depends on the "field" point x on the one hand, and on the pair of "source" points y 1 , y 2 on the other hand; it is defined by g(x; y 1 , y 2 ) = ln (r 1 + r 2 + r 12 ) , δ ij T kk ; so the trace of the metric (2.1) is normalized to be γ ii = 3.…”

Section: Momentarily At Rest (Vmentioning

confidence: 99%

“…The integral (4.12) can be computed analytically 12 , and yields the following contribution to the ADM mass: 13) where the remainder O (α) indicates that the terms proportional to α 1 or α 2 (in this secondorder perturbation ∝ β 2 ), are not to be considered for the present calculation. We thereby obtain M with sufficient accuracy for controlling the energy at the 2PN order 13 .…”

Section: B the Adm Massmentioning

confidence: 99%

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Time-symmetric initial data for binary black holes in numerical relativity

Blanchet¹

2003

Phys. Rev. D

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We look for physically realistic initial data in numerical relativity which are in agreement with post-Newtonian approximations. We propose a particular solution of the time-symmetric constraint equation, appropriate to two momentarily static black holes, in the form of a conformal decomposition of the spatial metric. This solution is isometric to the post-Newtonian metric up to the 2PN order. It represents a non-linear deformation of the solution of Brill and Lindquist, i.e. an asymptotically flat region is connected to two asymptotically flat (in a certain weak sense)sheets, that are the images of the two singularities through appropriate inversion transformations.The total ADM mass M as well as the individual masses m 1 and m 2 (when they exist) are computed by surface integrals performed at infinity. Using second order perturbation theory on the Brill-Lindquist background, we prove that the binary's interacting mass-energy M − m 1 − m 2 is well-defined at the 2PN order and in agreement with the known post-Newtonian result. 04.80.Nn, 97.60.Jd, 97.60.Lf * Electronic address: blanchet@iap.fr 1 I. MOTIVATION AND RELATION TO OTHER WORKSThe numerical computation of the collision of two black holes is of paramount importance for the observation of gravitational waves by the network of laser-interferometric detectors.When investigating this problem the ten Einstein field equations are separated into: (i) four constraint equations that are to be satisfied by some initial data given on an initial threedimensional Cauchy hypersurface; (ii) six hyperbolic-like equations describing the dynamical evolution of the gravitational field on neighbouring hypersurfaces. The Bianchi identities guarantee that the constraint equations are satisfied on neighbouring hypersurfaces if they are on the initial hypersurface. There are infinitely many ways that the initial data can be chosen to represent the starting state of the evolution of black holes. It is widely admitted that the problem of choosing physically realistic initial conditions for the collision of two black holes has not yet been solved. There has been a lot of concern in the litterature [1] for knowing what would really motivate physically a particular choice of initial data.Let us consider the problem of time-symmetric initial data, which are physically appropriate to two momentarily static black holes, i.e. with zero initial velocities. The dynamical evolution of time-symmetric data describes the subsequent head-on collision of the two black holes. In this situation the constraint equations reduce to the Hamiltonian or scalar contraint equation R = 0 (in vacuum -the case appropriate to black holes), with R being the three-dimensional scalar curvature. Considering as usual [2,3,4] a conformal decomposition of the spatial metric (spatial indices i, j, · · · = 1, 2, 3),where γ ij is the physical metric and γ ij denotes the conformal (unphysical) metric, we obtain the Lichnerowicz [2] equation, which is an elliptic-type equation to be satisfied by the conformal fa...

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“…We thereby obtain M with sufficient accuracy for controlling the energy at the 2PN order 13 . The result…”

Section: B the Adm Massmentioning

confidence: 95%

Section: Momentarily At Rest (Vmentioning

confidence: 99%

Section: B the Adm Massmentioning

confidence: 99%

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Time-symmetric initial data for binary black holes in numerical relativity

Blanchet¹

2003

Phys. Rev. D

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“…Well known applications include the precession of Mercury's perihelion, and various solar system tests of Einstein's gravity theory like the light deflection [12]. Recent successful applications include the generation of gravitational waves from compact binary objects, and the weakly relativistic evolution stages of isolated systems of celestial bodies [13]. Another important application is relativistic celestial mechanics which is required by recent technology driven precise measurements of the solar system bodies [14,15].…”

Section: Introductionmentioning

confidence: 99%

Cosmological non-linear hydrodynamics with post-Newtonian corrections

Hwang¹,

Noh²,

Puetzfeld³

2008

J. Cosmol. Astropart. Phys.

106

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The purpose of this paper is to present general relativistic cosmological hydrodynamic equations in Newtonian-like forms using the post-Newtonian (PN) method. The PN approximation, based on the assumptions of weak gravitational fields and slow motions, provides a way to estimate general relativistic effects in the fully nonlinear evolution stage of the large-scale cosmic structures. We extend Chandrasekhar's first order PN (1PN) hydrodynamics based on the Minkowski background to the Robertson-Walker background. We assume the presence of Friedmann's cosmological spacetime as a background. In the background we include the three-space curvature, the cosmological constant and general pressure; we show that our 1PN approach is successful only for spatially flat cosmological background. In the Newtonian order and 1PN order we include general pressure, stress, and flux. The Newtonian hydrodynamic equations appear naturally in the 0PN order. The spatial gauge degree of freedom is fixed in a unique manner and the basic equations are arranged without taking the temporal gauge condition. In this way we can conveniently try alternative temporal gauge conditions. We investigate a number of temporal gauge conditions under which all the remaining variables are equivalently gauge-invariant. Our aim is to present the fully nonlinear 1PN equations in a form suitable for implementation in conventional Newtonian hydrodynamic simulations with minimal extensions. The 1PN terms can be considered as relativistic corrections added to the well known Newtonian equations. The proper arrangement of the variables and equations in combination with suitable gauge conditions would allow the possible future 1PN cosmological simulations to become more tractable. Our equations and gauges are arranged for that purpose. We suggest ways of controlling the numerical accuracy. The typical 1PN order terms are about 10 −6 ∼ 10 −4 times smaller than the Newtonian terms. However, we cannot rule out possible presence of cumulative effects due to the time-delayed propagation of the relativistic gravitational field with finite speed, in contrast to the Newtonian case where changes in the gravitational field are felt instantaneously. The quantitative estimation of such effects is left for future numerical simulations.

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“…One is the post-Newtonian (PN) approximation [9] with previous studies [10,11,12,13,14,15,16,17,18]. The other one is the fully nonlinear and exact perturbation theory in Einstein's gravity [19].…”

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confidence: 95%

Cosmological post-Newtonian equations from nonlinear perturbation theory

Noh¹,

Hwang²

2013

J. Cosmol. Astropart. Phys.

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Abstract.We derive the basic equations of the cosmological first-order post-Newtonian approximation from the recently formulated fully nonlinear and exact cosmological perturbation theory in Einstein's gravity. Apparently the latter, being exact, should include the former, and here we use this fact as a new derivation of the former. The complete sets of equations in both approaches are presented without fixing the temporal gauge conditions so that we can use the gauge choice as an advantage. Comparisons between the two approaches are made. Both are potentially important in handling relativistic aspects of nonlinear processes occurring in cosmological structure formation. We consider an ideal fluid and include the cosmological constant.

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Chapter 2. Post-Newtonian Approximation

Cited by 36 publications

References 43 publications

Time-symmetric initial data for binary black holes in numerical relativity

Time-symmetric initial data for binary black holes in numerical relativity

Cosmological non-linear hydrodynamics with post-Newtonian corrections

Cosmological post-Newtonian equations from nonlinear perturbation theory

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