Bifurcating Solutions of the Lichnerowicz Equation

Chruściel, Piotr T.; Gicquaud, Romain

doi:10.1007/s00023-016-0501-x

Cited by 15 publications

(21 citation statements)

References 31 publications

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“…In particular, the method of [10] cannot work any longer and the Lichnerowicz equation may admits multiple solutions, see e.g. [8], [31] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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

Gicquaud

Nguyen

2016

Calc. Var.

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“…In particular, the method of [10] cannot work any longer and the Lichnerowicz equation may admits multiple solutions, see e.g. [8], [31] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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

Gicquaud

Nguyen

2016

Calc. Var.

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“…Namely, these can be taken to be constant linear combinations of the Euclidean coordinates (z 1 , z 2 , z 3 , z 4 ) (see (9)), restricted to S 3 . As a corollary they satisfy…”

Section: Smentioning

confidence: 99%

Vacuum initial data on $\mathbb{S}^3$ from Killing vectors

Beig¹,

Bizoń²,

Simon³

2019

Class. Quantum Grav.

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We construct compact initial data of constant mean curvature K for Einstein's 4d vacuum equations with Λ = Λ − ( K 2 /3) positive, where Λ is the cosmological constant, via the conformal method. To construct a transverse, trace-free (TT) momentum tensor explicitly we first observe that, if the seed manifold has two orthogonal Killing vectors, their symmetrized tensor product is a natural TT candidate. Without the orthogonality requirement, but on locally conformally flat seed manifolds there is a generalized construction for the momentum which also involves the derivatives of the Killing fields found in work by Beig and Krammer [2]. We consider in particular the round three sphere and classify the TT tensors resulting from all possible pairs of its six Killing vectors, focusing on the commuting case where the seed data are U(1) × U(1) -symmetric. As to solving the Lichnerowicz equation, we discuss in particular potential "symmetry breaking" by which we mean that solutions have less symmetries than the equation itself; we compare with the case of the "round donut" of topology S 2 × S. In the absence of symmetry breaking, the Lichnerowicz equation for a U(1) × U(1) symmetric momentum on S 3 reduces to an ODE. We analyze distinguished families of solutions and the resulting data via a combination of analytical and numerical techniques. Finally we investigate marginally trapped surfaces of toroidal topology in our data.

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“…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. These studies could point the way to generalizations of the conformal method, such as the drift system [122,127,128], that may provide better parametrizations of the initial data for GR in the truly non-CMC setting.…”

Section: The Discovery Of Gw150914mentioning

confidence: 99%

Untitled

2016

Bull. Amer. Math. Soc.

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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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Bifurcating Solutions of the Lichnerowicz Equation

Abstract: We give an exhaustive description of bifurcations and of the number of solutions of the vacuum Lichnerowicz equation with positive cosmological constant on S 1 × S 2 with U (1) × SO(3)-invariant seed data. The resulting CMC slicings of Schwarzschild-de Sitter and Nariai are described.

Cited by 15 publications

References 31 publications

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

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

Vacuum initial data on $\mathbb{S}^3$ from Killing vectors

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