Einstein's equations near spatial infinity

Beig, Robert; Schmidt, Bernd

doi:10.1007/bf01211056

Cited by 135 publications

(228 citation statements)

References 14 publications

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“…In the literature several mathematically inequivalent model for the spatial infinity have been suggested (see e.g. [13][14][15][16]). However, the notion of asymptotic flatness at the spatial infinity based on a spacelike hypersurface is expected to be the weakest possible in the sense that in every reasonable model of spatial infinity the existence of such a hypersurface is expected.…”

Section: Introductionmentioning

confidence: 99%

On the roots of the Poincar structure of asymptotically flat spacetimes

Szabados

2003

Class. Quantum Grav.

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The analysis of canonical vacuum general relativity by R. Beig and N.Ó Murchadha (Ann. Phys. 174 463-498 (1987)) is extended in numerous ways. The weakest possible power-type fall-off conditions for the energy-momentum tensor of the matter fields, the metric, the extrinsic curvature, the lapse and the shift are determined which, together with the parity conditions, are preserved by the energy-momentum conservation law T ab ;b = 0 and the evolution equations for the geometry. The algebra of the asymptotic Killing vectors, defined with respect to a foliation of the spacetime, is shown to be the Lorentz Lie algebra for slow fall-off of the metric, but it is the Poincare algebra for 1/r or faster fall-off.It is shown that the applicability of the symplectic formalism already requires the 1/r (or faster) fall-off of the metric. The connection between the Poisson algebra of the Beig-Ó Murchadha Hamiltonians (and, in particular, the constraint algebra) and the asymptotic Killing vectors is clarified. Their Hamiltonian H[K a ] is shown to be constant in time modulo constraints for those asymptotic Killing vectors K a that are defined with respect to the foliation by the constant time slices.The energy-momentum and angular momentum are defined by the boundary term Q[K a ] in H[K a ] even in the presence of matter. Although the energy-momentum is well defined even for slightly faster than the r −1/2 fall-off, we show that the angular momentum and centre-of-mass are finite only if the metric falls off as 1/r or faster. Q[K a ] is constant in time for those K a 's that are asymptotic Killing vectors with respect to the foliation by the constant time slices. If the foliation corresponds to proper time evolution (i.e. its lapse tends to 1 at infinity), then Q[K a ] reproduces the ADM energy, the spatial momentum and spatial angular momentum, but the centre-of-mass deviates from that of Beig andÓ Murchadha by the spatial momentum times the coordinate time. The spatial angular momentum and the new centre-of-mass form an anti-symmetric Lorentz tensor, which transforms in the expected way under Poincare transformations. * Dedicated to Jim Nester on the occasion of his 60th birthday.1

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Section: Introductionmentioning

confidence: 99%

On the roots of the Poincar structure of asymptotically flat spacetimes

Szabados

2003

Class. Quantum Grav.

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“…any such expression is pseudotensorial or, in the tetrad formalism of gravity, depends on the tetrad field too. For asymptotically flat spacetimes, however, one can define the total energy-momentum [1][2][3]. One of the most important results of the last decade in the classical relativity theory is the better understanding of the energy-momentum of localized gravitating systems, especially the proof of the positivity of the ADM and Bondi-Sachs masses [4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Two-dimensional Sen connections in general relativity

Szabados

1994

Class. Quantum Grav.

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The two dimensional version of the Sen connection for spinors and tensors on spacelike 2-surfaces is constructed. A complex metric γ AB on the spin spaces is found which characterizes both the algebraic and extrinsic geometrical properties of the 2-surface $. The curvature of the two dimensional Sen operator ∆ e is the pull back to $ of the anti-self-dual part of the spacetime curvature while its 'torsion' is a boost gauge invariant expression of the extrinsic curvatures of $. The difference of the 2 dimensional Sen and the induced spin connections is the anti-self-dual part of the 'torsion'. The irreducible parts of ∆ e are shown to be the familiar 2-surface twistor and the Weyl-SenWitten operators. Two Sen-Witten type identities are derived, the first is an identity between the 2 dimensional twistor and the Weyl-Sen-Witten operators and the integrand of Penrose's charge integral, while the second contains the 'torsion' as well. For spinor fields satisfying the 2-surface twistor equation the first reduces to Tod's formula for the kinematical twistor.

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“…Linearized gravitational fluctuations about asymptotically flat space are similar to the linear scalar solutions reviewed above (see e.g. [29,30,31]. We may take, e.g., the boundary conditions of [32] (for d = 4) or [17] (for d ≥ 4) to define a notion of "fast fall-off."…”

Section: Interacting and Non-scalar Fieldsmentioning

confidence: 99%

“…While a reasonable theory of such deformations may exist, it is clear from e.g. [30] in d = 4 (or [31] in higher dimensions) that such deformations destroy the entire asymptotic structure near spatial infinity. One expects that such deformations correspond to non-renormalizable deformations of the dual theory.…”

Section: Interacting and Non-scalar Fieldsmentioning

confidence: 99%

Asymptotic flatness, little string theory, and holography

Marolf

2007

J. High Energy Phys.

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We argue that any non-gravitational holographic dual to asymptotically flat string theory in d-dimensions naturally resides at spacelike infinity. Since spacelike infinity can be resovled as a (d − 1)-dimensional timelike hyperboloid (i.e., as a copy of de Sitter space in (d − 1) dimensions), the dual theory is defined on a Lorentz signature spacetime. Conceptual issues regarding such a duality are clarified by comparison with linear dilaton boundary conditions, such as those dual to little string theory. We compute both timeordered and Wightman boundary 2-point functions of operators dual to massive scalar fields in the asymptotically flat bulk.

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Einstein's equations near spatial infinity

Cited by 135 publications

References 14 publications

On the roots of the Poincar structure of asymptotically flat spacetimes

On the roots of the Poincar structure of asymptotically flat spacetimes

Two-dimensional Sen connections in general relativity

Asymptotic flatness, little string theory, and holography

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