D. Bao, Y. Choquet-Bruhat, J. Isenberg, and P. B. Yasskin :
“The well-posedness of (\( N=1 \) ) classical supergravity J. Math. Phys.
26 : 2
(1985 ),
pp. 329–333 .
MR
776502 Zbl
0563.53060 article
Abstract
People
BibTeX
In this paper the authors investigate whether classical (\( N=1 \) ) supergravity has a well-posed locally causal Cauchy problem. One defines well-posedness to mean that any choice of initial data (from an appropriate function space) which satisfies the supergravity constraint equations and a set of gauge conditions can be continuously developed into a space-time solution of the supergravity field equations around the initial surface. Local causally means that the domains of dependence of the evolution equations coincide with those determined by the light cones. They show that when the fields of classical supergravity are treated as formal objects, the field equations are (under certain gauge conditions) equivalent to a coupled system of quasilinear nondiagonal second-order partial differential equations which is formally nonstrictly hyperbolic (in the sense of Leray-Ohy). Hence, if the fields were numerical valued, there would be an applicable existence theorem leading to well-posedness.
@article {key776502m,
AUTHOR = {Bao, David and Choquet-Bruhat, Yvonne
and Isenberg, James and Yasskin, Philip
B.},
TITLE = {The well-posedness of (\$N=1\$) classical
supergravity},
JOURNAL = {J. Math. Phys.},
FJOURNAL = {Journal of Mathematical Physics},
VOLUME = {26},
NUMBER = {2},
YEAR = {1985},
PAGES = {329--333},
DOI = {10.1063/1.526663},
NOTE = {MR:776502. Zbl:0563.53060.},
ISSN = {0022-2488},
}
Y. Choquet-Bruhat, J. Isenberg, and V. Moncrief :
“Solutions of constraints for Einstein equations ,”
C. R. Acad. Sci. Paris Sér. I Math.
315 : 3
(1992 ),
pp. 349–355 .
MR
1179734 Zbl
0796.35161 article
Abstract
People
BibTeX
@article {key1179734m,
AUTHOR = {Choquet-Bruhat, Yvonne and Isenberg,
James and Moncrief, Vincent},
TITLE = {Solutions of constraints for {E}instein
equations},
JOURNAL = {C. R. Acad. Sci. Paris S\'{e}r. I Math.},
FJOURNAL = {Comptes Rendus de l'Acad\'{e}mie des
Sciences. S\'{e}rie I. Math\'{e}matique},
VOLUME = {315},
NUMBER = {3},
YEAR = {1992},
PAGES = {349--355},
NOTE = {MR:1179734. Zbl:0796.35161.},
ISSN = {0764-4442},
}
Y. Choquet-Bruhat, J. Isenberg, and J. W. York, Jr. :
“Einstein constraints on asymptotically Euclidean manifolds Phys. Rev. D (3)
61 : 8
(2000 ),
pp. 084034, 20 .
MR
1791413 article
Abstract
People
BibTeX
@article {key1791413m,
AUTHOR = {Choquet-Bruhat, Yvonne and Isenberg,
James and York, Jr., James W.},
TITLE = {Einstein constraints on asymptotically
{E}uclidean manifolds},
JOURNAL = {Phys. Rev. D (3)},
FJOURNAL = {Physical Review. D. Third Series},
VOLUME = {61},
NUMBER = {8},
YEAR = {2000},
PAGES = {084034, 20},
DOI = {10.1103/PhysRevD.61.084034},
NOTE = {MR:1791413.},
ISSN = {0556-2821},
}
Y. Choquet-Bruhat, J. Isenberg, and V. Moncrief :
“Topologically general U(1) symmetric vacuum space-times with AVTD behavior Nuovo Cimento Soc. Ital. Fis. B
119 : 7–9
(2004 ),
pp. 625–638 .
MR
2136898 article
Abstract
People
BibTeX
We use Fuchsian methods to show that, for any two-dimensional manifold \( \Sigma^2 \) , there is a large family of \( U (1) \) symmetric solutions of the vacuum Einstein equations on the manifold
\[ \Sigma \times S^1 \times \mathbb{R} ,\]
each of which has AVTD behavior in theneighborhood of its singularity.
@article {key2136898m,
AUTHOR = {Choquet-Bruhat, Y. and Isenberg, J.
and Moncrief, V.},
TITLE = {Topologically general {U}(1) symmetric
vacuum space-times with {AVTD} behavior},
JOURNAL = {Nuovo Cimento Soc. Ital. Fis. B},
FJOURNAL = {Il Nuovo Cimento della Societ\`a Italiana
di Fisica B},
VOLUME = {119},
NUMBER = {7-9},
YEAR = {2004},
PAGES = {625--638},
DOI = {10.1393/ncb/i2004-10174-x},
NOTE = {MR:2136898.},
ISSN = {1594-9982},
}
Y. Choquet-Bruhat, J. Isenberg, and D. Pollack :
“The Einstein-scalar field constraints on asymptotically Euclidean manifolds Chinese Ann. Math. Ser. B
27 : 1
(2006 ),
pp. 31–52 .
MR
2209950 Zbl
1112.83008 article
Abstract
People
BibTeX
By using the conformal method, solutions of the Einstein-scalar field gravitational constraint equations are obtained. Handling scalar fields is a bit more challenging than handling matter fields such as fluids, Maxwell fields or Yang-Mills fields, because the scalar field introduces three extra terms into the Lichnerowicz equation, rather than just one. The proofs are constructive and allow for arbitrary dimension (\( > 2 \) ) as well as low regularity initial data.
@article {key2209950m,
AUTHOR = {Choquet-Bruhat, Yvonne and Isenberg,
James and Pollack, Daniel},
TITLE = {The {E}instein-scalar field constraints
on asymptotically {E}uclidean manifolds},
JOURNAL = {Chinese Ann. Math. Ser. B},
FJOURNAL = {Chinese Annals of Mathematics. Series
B},
VOLUME = {27},
NUMBER = {1},
YEAR = {2006},
PAGES = {31--52},
DOI = {10.1007/s11401-005-0280-z},
NOTE = {MR:2209950. Zbl:1112.83008.},
ISSN = {0252-9599},
}
Y. Choquet-Bruhat and J. Isenberg :
“Half polarized \( \mathrm{ U}(1) \) symmetric vacuum spacetimes with AVTD behavior J. Geom. Phys.
56 : 8
(2006 ),
pp. 1199–1214 .
MR
2234438 Zbl
1113.83006 article
Abstract
People
BibTeX
In a previous work, we used a polarization condition to show that there is a family of \( U(1) \) symmetric solutions of the vacuum Einstein equations on
\[ \Sigma \times S^1 \times R \]
(\( \Sigma \) any two-dimensional manifold) such that each exhibits AVTD1 behavior in the neighbourhood of its singularity. Here we consider the general case of \( S^1 \) bundles over the base \( \Sigma \times R \) and determine a condition, called the half polarization condition, necessary and sufficient in our context, for AVTD behavior near the singularity.
@article {key2234438m,
AUTHOR = {Choquet-Bruhat, Yvonne and Isenberg,
James},
TITLE = {Half polarized \${\rm U}(1)\$ symmetric
vacuum spacetimes with {AVTD} behavior},
JOURNAL = {J. Geom. Phys.},
FJOURNAL = {Journal of Geometry and Physics},
VOLUME = {56},
NUMBER = {8},
YEAR = {2006},
PAGES = {1199--1214},
DOI = {10.1016/j.geomphys.2005.06.011},
NOTE = {MR:2234438. Zbl:1113.83006.},
ISSN = {0393-0440},
}
Y. Choquet-Bruhat, J. Isenberg, and D. Pollack :
“Applications of theorems of Jean Leray to the Einstein-scalar field equations J. Fixed Point Theory Appl.
1 : 1
(2007 ),
pp. 31–46 .
MR
2282342 Zbl
1169.83007 article
Abstract
People
BibTeX
The Einstein-scalar field theory can be used to model gravitational physics with scalar field sources. We discuss the initial value formulation of this field theory, and show that the ideas of Leray can be used to show that the Einstein-scalar field system of partial differential equations is well-posed as an evolutionary system. We also show that one can generate solutions of the Einstein-scalar field constraint equations using conformal methods.
@article {key2282342m,
AUTHOR = {Choquet-Bruhat, Yvonne and Isenberg,
James and Pollack, Daniel},
TITLE = {Applications of theorems of {J}ean {L}eray
to the {E}instein-scalar field equations},
JOURNAL = {J. Fixed Point Theory Appl.},
FJOURNAL = {Journal of Fixed Point Theory and Applications},
VOLUME = {1},
NUMBER = {1},
YEAR = {2007},
PAGES = {31--46},
DOI = {10.1007/s11784-006-0006-1},
NOTE = {MR:2282342. Zbl:1169.83007.},
ISSN = {1661-7738},
}
Y. Choquet-Bruhat, J. Isenberg, and D. Pollack :
“The constraint equations for the Einstein-scalar field system on compact manifolds Classical Quantum Gravity
24 : 4
(2007 ),
pp. 809–828 .
MR
2297268 Zbl
1111.83002 article
Abstract
People
BibTeX
We study the constraint equations for the Einstein-scalar field system on compact manifolds. Using the conformal method we reformulate these equations as a determined system of nonlinear partial differential equations. By introducing a new conformal invariant, which is sensitive to the presence of the initial data for the scalar field, we are able to divide the set of free conformal data into subclasses depending on the possible signs for the coefficients of terms in the resulting Einstein-scalar field Lichnerowicz equation. For many of these subclasses we determine whether or not a solution exists. In contrast to other well studied field theories, there are certain cases, depending on the mean curvature and the potential of the scalar field, for which we are unable to resolve the question of existence of a solution. We consider this system in such generality so as to include the vacuum constraint equations with an arbitrary cosmological constant, the Yamabe equation and even (all cases of) the prescribed scalar curvature problem as special cases.
@article {key2297268m,
AUTHOR = {Choquet-Bruhat, Yvonne and Isenberg,
James and Pollack, Daniel},
TITLE = {The constraint equations for the {E}instein-scalar
field system on compact manifolds},
JOURNAL = {Classical Quantum Gravity},
FJOURNAL = {Classical and Quantum Gravity},
VOLUME = {24},
NUMBER = {4},
YEAR = {2007},
PAGES = {809--828},
DOI = {10.1088/0264-9381/24/4/004},
NOTE = {MR:2297268. Zbl:1111.83002.},
ISSN = {0264-9381},
}
