Solutions globales des équations d'Einstein–Maxwell avec jauge harmonique et jauge de Lorenz

Loizelet, Julien

doi:10.1016/j.crma.2006.01.018

Cited by 9 publications

(13 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Now, one can ask the question of geodesic completeness or at least Lorentzian maximality of the maximal Cauchy development around a solution with vanishing spinor fields, e.g. around the Minkowski solution applying the same machinery as for the Klein-Gordon equation or the Maxwell equation as in [LR10] and [Loi06].…”

Section: Existence Of a Maximal Cauchy Developmentmentioning

confidence: 99%

A universal spinor bundle and the Einstein–Dirac–Maxwell equation as a variational theory

Müller

Nowaczyk

2016

Lett Math Phys

View full text Add to dashboard Cite

Not only the Dirac operator, but also the spinor bundle of a pseudo-Riemannian manifold depends on the underlying metric. This leads to technical difficulties in the study of problems where many metrics are involved, for instance in variational theory. We construct a natural finite dimensional bundle, from which all the metric spinor bundles can be recovered including their extra structure. In the Lorentzian case, we also give some applications to Einstein-Dirac-Maxwell theory as a variational theory and show how to coherently define a maximal Cauchy development for this theory.where ρ := ρ r,s : Spin r,s → GL(Σ r,s ) is a complex spinor representation, Θ g : Spin g M → SO g M is a metric spin structure and Spin r,s ⊂ GL + m is the spin group. Main Theorem 1 (universal spinor bundle, cf. Theorem 2.25). There exists a finite dimensional vector bundleπ Σ SM :ΣM → J 1 π r,s such that for each metric g ∈ S r,s (M ), there existsῑ g such thatcommutes. Here, J 1 π r,s denotes the first jet bundle of π r,s . Moreover,π Σ SM carries a connection, a metric and a Clifford multiplication such thatῑ g is a morphism of (generalized) Dirac bundles (see Definition 2.27). In addition,π Σ SM is natural with respect to spin diffeomorphisms. ♦The claims of the last assertions mean that not only the vector bundle structure of any metric spinor bundle π g M can be recovered fromπ Σ SM , but also its spinorial connection, its metric and its Clifford multiplication, see Theorem 2.25 for the precise meaning and a proof. We would like to emphasize that the morphism j 1 (g) in (1.2) is the 1-jet of the metric g and that the naturality assertion is formulated with respect to diffeomorphisms and not just isometries, see Section 2.6 for details. arXiv:1504.01034v3 [math.DG] 6 Oct 2015 ∼ = 9 9 We setκ V := κ V • θ. ♦ Diagram (2.1) easily extends to spin manifolds as follows: Let {ρ r,s : Spin r,s → GL(Σ r,s )} r,s∈N be a fixed choice of Spin r,s -representations. The construction of κ V andκ V induces bundle maps κ M : GL + M → S r,s M,κ M : GL + M → S r,s M, (2.2) by setting κ M | GL + x M := κ TxM andκ M | GL + x M :=κ TxM for any x ∈ M . Definition 2.3 (universal spinor bundle). The map π Σ SM : ΣM → S r,s M [b, σ] →κ M (b), where ΣM := GL + M × ρr,s Σ r,s is called universal spinor (vector) bundle. We obtain the following diagram ΣM π Σ SM / / π Σ M 9 9 S r,s M π r,s / / M, (2.3)where π Σ M := π r,s • π Σ SM . The map π Σ M is called universal spinor (fiber) bundle of M . Its sections are called universal spinor fields.♦

show abstract

Section: Existence Of a Maximal Cauchy Developmentmentioning

confidence: 99%

A universal spinor bundle and the Einstein–Dirac–Maxwell equation as a variational theory

Müller

Nowaczyk

2016

Lett Math Phys

View full text Add to dashboard Cite

show abstract

“…We refer the reader to [23,24] for details and for some information on the asymptotic behaviour of the fields. …”

Section: Nonlinear Stability In Higher Dimensionsmentioning

confidence: 99%

Global solutions of the Einstein–Maxwell equations in higher dimensions

Choquet–Bruhat

Chruściel

Loizelet

2006

Class. Quantum Grav.

Self Cite

View full text Add to dashboard Cite

We consider the Einstein-Maxwell equations in space-dimension n. We point out that the Lindblad-Rodnianski stability proof applies to those equations whatever the space-dimension n 3. In even spacetime dimension n + 1 6, we use the standard conformal method on a Minkowski background to give a simple proof that the maximal globally hyperbolic development of initial data sets which are sufficiently close to the data for Minkowski spacetime and which are Schwarzschildian outside of a compact set lead to geodesically complete spacetimes, with a complete Scri, with a smooth conformal structure, and with the gravitational field approaching the Minkowski metric along null directions at least as fast as r −(n−1)/2 .

show abstract

“…The Einstein-Maxwell equations, in harmonic and Lorenz gauge, take the form (6.13) (see [4,16,18]) with the following replacements there:…”

Section: Existence Of a Solutionmentioning

confidence: 99%

“…In particular, in dimensions n + 1 ≥ 9 the small data solutions of [18,19] evolving out from data stationary outside of a compact set are polyhomogeneous.…”

Section: Polyhomogeneous Solutions Of the Einstein-maxwell Equationsmentioning

confidence: 99%

Solutions of Quasi-Linear Wave Equations Polyhomogeneous at Null Infinity in High Dimensions

Chruściel

Wafo

2011

J. Hyper. Differential Equations

View full text Add to dashboard Cite

We prove propagation of weighted Sobolev regularity for solutions of the hyperboloidal Cauchy problem for a class of quasi-linear symmetric hyperbolic systems, under structure conditions compatible with the Einstein-Maxwell equations in space-time dimensions n + 1 ≥ 7. Similarly we prove propagation of polyhomogeneity in dimensions n+1 ≥ 9. As a byproduct we obtain, in those last dimensions, polyhomogeneity at null infinity of small data solutions of vacuum Einstein, or Einstein-Maxwell equations evolving out of initial data which are stationary outside of a ball.Proof: This theorem is a generalization of the semi-linear case, Theorem 3.7 of [12], and can be proved by following step by step the proof given there. A detailed exposition can be found in [25].

show abstract

Solutions globales des équations d'Einstein–Maxwell avec jauge harmonique et jauge de Lorenz

Cited by 9 publications

References 1 publication

A universal spinor bundle and the Einstein–Dirac–Maxwell equation as a variational theory

A universal spinor bundle and the Einstein–Dirac–Maxwell equation as a variational theory

Global solutions of the Einstein–Maxwell equations in higher dimensions

Solutions of Quasi-Linear Wave Equations Polyhomogeneous at Null Infinity in High Dimensions

Contact Info

Product

Resources

About