Examples of Newtonian limits of relativistic spacetimes

Ehlers, Jürgen

doi:10.1088/0264-9381/14/1a/010

Cited by 56 publications

(79 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the case of the post-Newtonian approximation (limit c → ∞), Jürgen Ehlers has provided with his frame theory [9][10][11] a conceptual framework in which the post-Newtonian approximation can be clearly formulated (among other purposes). This theory unifies the theories of Newton and Einstein into a single generally covariant theory, with a parameter 1/c taking the value zero in the case of Newton and being the inverse of the speed of light in the case of Einstein.…”

Section: Introduction a On Approximation Methods In General Relatmentioning

confidence: 99%

“…Now if h µν satisfies for instance (9), so does the pseudo-tensor τ µν built on it, and then it is clear that the retarded integral of τ µν satisfies itself the same condition. Therefore one infers that the unique solution of the Einstein equation (3) satisfying the condition (9) is 10) where the retarded integral takes the standard form…”

Section: B Field Equations and The No-incoming Radiation Conditionmentioning

confidence: 99%

“…The field h (skipping space-time indices), solution in IR 4 of the relaxed field equations and the no-incoming radiation condition, is given as the retarded integral (10). We now assume that outside the isolated system, say, in the region r > R where R is a constant radius strictly larger than d, we have h = M(h) where M(h) denotes the multipole expansion of h, a solution of the vacuum field equations in IR 4 deprived from the spatial origin r = 0, and admitting a spherical-harmonics expansion of a certain structure (see below).…”

Section: Multipole Decompositionmentioning

confidence: 99%

“…Up to now we have solved the relaxed field equation (10) in the exterior zone, with result the multipole decomposition (24)- (25). In this section we further impose the harmonic gauge condition (13a), and from this we find a solution of the linearized vacuum equation, appearing as the first approximation in a post-Minkowskian expansion of the multipole expansion M(h).…”

Section: B Linearized Approximation To the Exterior Fieldmentioning

confidence: 99%

See 3 more Smart Citations

Post-Newtonian Gravitational Radiation

Blanchet

1999

Einstein’s Field Equations and Their Physical Implications

View full text Add to dashboard Cite

Section: Introduction a On Approximation Methods In General Relatmentioning

confidence: 99%

Section: B Field Equations and The No-incoming Radiation Conditionmentioning

confidence: 99%

Section: Multipole Decompositionmentioning

confidence: 99%

Section: B Linearized Approximation To the Exterior Fieldmentioning

confidence: 99%

See 2 more Smart Citations

Post-Newtonian Gravitational Radiation

Blanchet

1999

Einstein’s Field Equations and Their Physical Implications

View full text Add to dashboard Cite

“…We shall follow the formulation by Futamase and Schutz 47) . We will not mention other formulation of Newtonian limit by Ehlers 38), 39) , because it has not yet used to construct the post-Newtonian approximation.…”

Section: §1 Introductionmentioning

confidence: 99%

Chapter 2. Post-Newtonian Approximation

Asada¹,

Futamase

1997

Prog. Theor. Phys. Suppl.

View full text Add to dashboard Cite

We discuss various aspects of the post-Newtonian approximation in general relativity. After presenting the foundation based on the Newtonian limit, we use the (3+1) formalism to formulate the post-Newtonian approximation for the perfect fluid. As an application we show the method for constructing the equilibrium configuration of nonaxisymmetric uniformly rotating fluid. We also discuss the gravitational waves including tail from post-Newtonian systems. §1. IntroductionThe motion and associated emission of gravitational waves (GW) of self gravitating systems have been a main research interest in general relativity. The problem is complicated conceptually as well as mathematically because of the nonlinearity of Einstein's equations. There is no hope in any foreseeable future to have exact solutions describing motions of arbitrary shaped, massive bodies so that we have to adopt some sort of approximation scheme for solving Einstein's equations to study such problems. In the past years many types of approximation schemes have been developed depending on the nature of the system under consideration. Here we shall focus on a particular scheme called the post-Newtonian (PN) approximation. There are many systems in astrophysics where Newtonian gravity is dominant, but general relativistic gravity plays also important roles in their evolution. For such systems it would be nice to have an approximation scheme which gives Newtonian description in the lowest order and general relativistic effects as higher order perturbations. The post-Newtonian approximation is perfectly suited for this purpose. Historically Einstein computed first the post-Newtonian effects, the precession of the perihelion 36) , but a systematic study of the post-Newtonian approximation was not made until the series of papers by Chandrasekhar and associates 21), 22), 23), 24), 25) . Now it is widely known that the post-Newtonian approximation is important in analyzing a number of relativistic problems, such as the equation of motion of binary pulsar 19), 40), 53), 31) , solar-system tests of general relativity 97), 98) , and gravitational radiation reaction 25), 20) .Any approximation scheme nesseciates a small parameter(s) characterizing the nature of the system under consideration. Typical parameter which most of schemes adopt is the magnitude of the metric deviation away from a certain background metric. In particular if the background is Minkowski spacetime and there is no other parameter, the scheme is sometimes called as the post-Minkowskian approximation typeset using PTPT E X.sty

show abstract

Manifestly invariant Lagrangians for geophysical fluids

Zadra

Charron

2015

Quart J Royal Meteoro Soc

View full text Add to dashboard Cite

Three manifestly invariant Lagrangians are presented from which the covariant equations of motion for inviscid classical fluids are derived using the least action principle. Invariance and covariance are here defined with respect to synchronous, but otherwise arbitrary, coordinate transformations, i.e. supposing that time intervals are absolute as required by Newtonian mechanics. In the first Lagrangian, the flow is formulated in terms of fluid particles, but conservation of mass and entropy is assumed a priori. In the second Lagrangian, the flow is described by fields, and conservation of mass and entropy is obtained with Lagrange multipliers. The third Lagrangian is also based on a field formulation, but has no Lagrange multipliers and produces all the desired equations of motion, including conservation of mass and entropy. The differences and similarities between these formulations are discussed. Hydrostatic equations are rederived from an asymptotic expansion of the action using the covariant field formulation.

show abstract

Examples of Newtonian limits of relativistic spacetimes

Abstract: Abstract.A frame theory encompassing general relativity and Newton-Cartan theory is reviewed. With its help, a definition is given for a one-parameter family of general relativistic spacetimes to have a Newton-Cartan or a Newtonian limit. Several examples of such limits are presented.

Cited by 56 publications

References 14 publications

Post-Newtonian Gravitational Radiation

Post-Newtonian Gravitational Radiation

Chapter 2. Post-Newtonian Approximation

Manifestly invariant Lagrangians for geophysical fluids

Contact Info

Product

Resources

About