Solutions to the Einstein-scalar field constraint equations with a small TT-tensor

Gicquaud, Romain; Nguyen, Cang

doi:10.1007/s00526-016-0963-1

Cited by 9 publications

(4 citation statements)

References 41 publications

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“…For further informations on the initial value problem for the Einstein equations we refer to the recent surveys [5,7,11], and the references therein. The scalar equation in (1.2) is called the Lichnerowicz equation; since it is the main source of nonlinearity in the system (1.2), a good understanding of its solutions is a crucial step toward the resolution of the Einstein equations by the conformal method, see for instance [7,8,10,18,25,21,22,26,27,29,30,33,35,35]. Here we generalize many of the results obtained in the above papers, especially relaxing the assumptions on M .…”

Section: Introductionsupporting

confidence: 53%

Lichnerowicz-type equations with sign-changing nonlinearities on complete manifolds with boundary

Albanese

Rigoli

2017

Journal of Differential Equations

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Section: Introductionsupporting

confidence: 53%

Lichnerowicz-type equations with sign-changing nonlinearities on complete manifolds with boundary

Albanese

Rigoli

2017

Journal of Differential Equations

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“…However, after it was shown in 2011 [126] that multiple solutions are possible in the non-CMC case, a number of additional techniques were developed that have led to a more refined understanding of the conformal method. In [64], scaling and blowup techniques were developed for the conformal method, giving a new approach to obtain non-CMC existence results [84,85]; this was further refined in [133], giving the best characterization to date for multiplicity of general solutions in the non-CMC case. Analytic bifurcation theory and numerical continuation methods are now also being used where possible [158,98,138,48,70] to characterize fold and bifurcation phenomena in the conformal method.…”

Section: The Discovery Of Gw150914mentioning

confidence: 99%

The emergence of gravitational wave science: 100 years of development of mathematical theory, detectors, numerical algorithms, and data analysis tools

Holst

Sarbach

Tiglio

et al. 2016

Bull. Amer. Math. Soc.

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Abstract. On September 14, 2015, the newly upgraded Laser Interferometer Gravitational-wave Observatory (LIGO) recorded a loud gravitational-wave (GW) signal, emitted a billion light-years away by a coalescing binary of two stellar-mass black holes. The detection was announced in February 2016, in time for the hundredth anniversary of Einstein's prediction of GWs within the theory of general relativity (GR). The signal represents the first direct detection of GWs, the first observation of a black-hole binary, and the first test of GR in its strong-field, high-velocity, nonlinear regime. In the remainder of its first observing run, LIGO observed two more signals from black-hole binaries, one moderately loud, another at the boundary of statistical significance. The detections mark the end of a decades-long quest and the beginning of GW astronomy: finally, we are able to probe the unseen, electromagnetically dark Universe by listening to it. In this article, we present a short historical overview of GW science: this young discipline combines GR, arguably the crowning achievement of classical physics, with record-setting, ultra-low-noise laser interferometry, and with some of the most powerful developments in the theory of differential geometry, partial differential equations, high-performance computation, numerical analysis, signal processing, statistical inference, and data science. Our emphasis is on the synergy between these disciplines and how mathematics, broadly understood, has historically played, and continues to play, a crucial role in the development of GW science. We focus on black holes, which are very pure mathematical solutions of Einstein's gravitationalfield equations that are nevertheless realized in Nature and that provided the first observed signals.

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“…However, after it was shown in 2011 [126] that multiple solutions are possible in the non-CMC case, a number of additional techniques were developed that have led to a more refined understanding of the conformal method. In [64] scaling and blow-up techniques were developed for the conformal method, giving a new approach for obtaining non-CMC existence results [85,86]; this was further refined in [133], giving the best characterization to date for multiplicity of general solutions in the non-CMC case. Analytic bifurcation theory and numerical continuation methods are now also being used where possible [48,70,98,138,159] to characterize fold and bifurcation phenomena in the conformal method.…”

Section: The Discovery Of Gw150914mentioning

confidence: 99%

Untitled

2016

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

On September 14, 2015, the newly upgraded Laser Interferometer Gravitational-wave Observatory (LIGO) recorded a loud gravitational-wave (GW) signal, emitted a billion light-years away by a coalescing binary of two stellar-mass black holes. The detection was announced in February 2016, in time for the hundredth anniversary of Einstein's prediction of GWs within the theory of general relativity (GR). The signal represents the first direct detection of GWs, the first observation of a black-hole binary, and the first test of GR in its strong-field, high-velocity, nonlinear regime. In the remainder of its first observing run, LIGO observed two more signals from black-hole binaries, one moderately loud, another at the boundary of statistical significance. The detections mark the end of a decades-long quest and the beginning of GW astronomy: finally, we are able to probe the unseen, electromagnetically dark Universe by listening to it. In this article, we present a short historical overview of GW science: this young discipline combines GR, arguably the crowning achievement of classical physics, with record-setting, ultra-low-noise laser interferometry, and with some of the most powerful developments in the theory of differential geometry, partial differential equations, high-performance computation, numerical analysis, signal processing, statistical inference, and data science. Our emphasis is on the synergy between these disciplines and how mathematics, broadly understood, has historically played, and continues to play, a crucial role in the development of GW science. We focus on black holes, which are very pure mathematical solutions of Einstein's gravitationalfield equations that are nevertheless realized in Nature and that provided the first observed signals.

show abstract

Solutions to the Einstein-scalar field constraint equations with a small TT-tensor

Abstract: ABSTRACT. In this paper, we prove a far-from-CMC result similar to [15,20,21,29] for the conformal Einstein-scalar field constraint equations on compact Riemannian manifolds with positive (modified) Yamabe invariant.

Cited by 9 publications

References 41 publications

Lichnerowicz-type equations with sign-changing nonlinearities on complete manifolds with boundary

Lichnerowicz-type equations with sign-changing nonlinearities on complete manifolds with boundary

The emergence of gravitational wave science: 100 years of development of mathematical theory, detectors, numerical algorithms, and data analysis tools

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