Y. Choquet-Bruhat, P. T. Chruściel, and J. M. Martín-García :
“The light-cone theorem Classical Quantum Gravity
26 : 13
(2009 ),
pp. 135011, 22 .
MR
2515694 Zbl
1171.83001 article
Abstract
People
BibTeX
We prove that the area of cross-sections of light cones, in spacetimes satisfying suitable energy conditions, is smaller than or equal to that of the corresponding cross-sections in Minkowski, or de Sitter, or anti-de Sitter spacetime. The equality holds if and only if the metric coincides with the corresponding model in the domain of dependence of the light cone.
@article {key2515694m,
AUTHOR = {Choquet-Bruhat, Yvonne and Chru\'{s}ciel,
Piotr T. and Mart\'{\i}n-Garc\'{\i}a,
Jos\'{e} M.},
TITLE = {The light-cone theorem},
JOURNAL = {Classical Quantum Gravity},
FJOURNAL = {Classical and Quantum Gravity},
VOLUME = {26},
NUMBER = {13},
YEAR = {2009},
PAGES = {135011, 22},
DOI = {10.1088/0264-9381/26/13/135011},
NOTE = {MR:2515694. Zbl:1171.83001.},
ISSN = {0264-9381},
}
Y. Choquet-Bruhat, P. T. Chruściel, and J. M. Martín-García :
“A property of light-cones in Einstein’s gravity C. R. Math. Acad. Sci. Paris
347 : 15–16
(2009 ),
pp. 971–977 .
MR
2542904 Zbl
1170.83005 article
Abstract
People
BibTeX
We prove that the area of cross-sections of future light-cones in a space-time of arbitrary dimension, solution of the Einstein equations with sources satisfying suitable energy conditions, is smaller than the area of corresponding cross-sections for Minkowskian cones, equal only in space-times which are Minkowskian to the future of the light-cone.
@article {key2542904m,
AUTHOR = {Choquet-Bruhat, Yvonne and Chru\'{s}ciel,
Piotr T. and Mart\'{\i}n-Garc\'{\i}a,
Jos\'{e} M.},
TITLE = {A property of light-cones in {E}instein's
gravity},
JOURNAL = {C. R. Math. Acad. Sci. Paris},
FJOURNAL = {Comptes Rendus Math\'{e}matique. Acad\'{e}mie
des Sciences. Paris},
VOLUME = {347},
NUMBER = {15-16},
YEAR = {2009},
PAGES = {971--977},
DOI = {10.1016/j.crma.2009.06.001},
NOTE = {MR:2542904. Zbl:1170.83005.},
ISSN = {1631-073X},
}
Y. Choquet-Bruhat, P. T. Chruściel, and J. M. Martín-García :
“Einstein constraints on a characteristic cone 93–102
in
Proceedings “WASCOM 2009” 15th Conference on Waves and Stability in Continuous Media .
World Sci. Publ. (Hackensack, NJ ),
2010 .
MR
2762005 Zbl
1243.83015 inproceedings
Abstract
People
BibTeX
We analyse the Cauchy problem on a characteristic cone, including its vertex, for the Einstein equations in arbitrary dimensions. We use a wave map gauge, solve the obtained constraints and show gauge conservation.
@inproceedings {key2762005m,
AUTHOR = {Choquet-Bruhat, Y. and Chru\'{s}ciel,
P. T. and Mart\'{\i}n-Garc\'{\i}a, J.
M.},
TITLE = {Einstein constraints on a characteristic
cone},
BOOKTITLE = {Proceedings ``{WASCOM} 2009'' 15th {C}onference
on {W}aves and {S}tability in {C}ontinuous
{M}edia},
PUBLISHER = {World Sci. Publ.},
ADDRESS = {Hackensack, NJ},
YEAR = {2010},
PAGES = {93--102},
DOI = {10.1142/9789814317429_0016},
NOTE = {MR:2762005. Zbl:1243.83015.},
}
Y. Choquet-Bruhat, P. T. Chruściel, and J. M. Martín-García :
“The Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions Ann. Henri Poincaré
12 : 3
(2011 ),
pp. 419–482 .
MR
2785136 Zbl
1215.83016 article
Abstract
People
BibTeX
We derive explicit formulae for a set of constraints for the Einstein equations on a null hypersurface, in arbitrary space-time dimensions \( n + 1 \geq 3 \) . We solve these constraints and show that they provide necessary and sufficient conditions so that a spacetime solution of the Cauchy problem on a characteristic cone for the hyperbolic system of the reduced Einstein equations in wave-map gauge also satisfies the full Einstein equations. We prove a geometric uniqueness theorem for this Cauchy problem in the vacuum case.
@article {key2785136m,
AUTHOR = {Choquet-Bruhat, Yvonne and Chru\'{s}ciel,
Piotr T. and Mart\'{\i}n-Garc\'{\i}a,
Jos\'{e} M.},
TITLE = {The {C}auchy problem on a characteristic
cone for the {E}instein equations in
arbitrary dimensions},
JOURNAL = {Ann. Henri Poincar\'{e}},
FJOURNAL = {Annales Henri Poincar\'{e}. A Journal
of Theoretical and Mathematical Physics},
VOLUME = {12},
NUMBER = {3},
YEAR = {2011},
PAGES = {419--482},
DOI = {10.1007/s00023-011-0076-5},
NOTE = {MR:2785136. Zbl:1215.83016.},
ISSN = {1424-0637},
}
Y. Choquet-Bruhat and J. M. Martín-García :
“A geometric energy estimate for data on a characteristic cone 17–25
in
Advances in Lorentzian geometry .
Edited by M. Plaue, A. Rendall, and M. Scherfner .
AMS/IP Stud. Adv. Math. 49 .
Amer. Math. Soc. (Providence, RI ),
2011 .
MR
2867849 Zbl
1243.83016 incollection
Abstract
People
BibTeX
The Einstein equations in wave map gauge are a geometric second order system for a Lorentzian metric. To study existence of solutions of this hyperbolic quasi diagonal system with initial data on a characteristic cone which are not zero in a neighbourhood of the vertex one can appeal to theorems due to Cagnac and Dossa, proved for a scalar wave equation, for initial data in functional spaces relevant for their proofs. It is difficult to check that the initial data that we have constructed as solutions of the Einstein wave-map gauge constraints satisfy the more general of the Cagnac-Dossa hypotheses which uses weighted energy estimates. In this paper we start a new study of energy estimates using on the cone coordinates adapted to its null structure which are precisely the coordinates used to solve the constraints, following work of Rendall who considered the Cauchy problem for Einstein equations with data on two intersecting characteristic surfaces.
@incollection {key2867849m,
AUTHOR = {Choquet-Bruhat, Yvonne and Mart\'{\i}n-Garc\'{\i}a,
Jos\'{e} M.},
TITLE = {A geometric energy estimate for data
on a characteristic cone},
BOOKTITLE = {Advances in Lorentzian geometry},
EDITOR = {Matthias Plaue and Alan Rendall and
Mike Scherfner},
SERIES = {AMS/IP Stud. Adv. Math.},
NUMBER = {49},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2011},
PAGES = {17--25},
DOI = {10.1090/amsip/049/03},
NOTE = {MR:2867849. Zbl:1243.83016.},
}
Y. Choquet-Bruhat, P. T. Chruściel, and J. M. Martín-García :
“An existence theorem for the Cauchy problem on a characteristic cone for the Einstein equations 73–81
in
Complex analysis and dynamical systems, IV: Part 2 .
Edited by M. Agranovsky, M. Ben-Artzi, G. Galloway, L. Karp, S. Reich, D. S. G. Weinstein, and L. Zalcman .
Contemp. Math. 554 .
American Mathematical Society; Bar-Ilan University (Providence, RI; Ramat Gan ),
2011 .
MR
2884395 Zbl
1235.83011 incollection
Abstract
People
BibTeX
@incollection {key2884395m,
AUTHOR = {Choquet-Bruhat, Y. and Chru\'{s}ciel,
Piotr T. and Mart\'{\i}n-Garc\'{\i}a,
Jos\'{e} M.},
TITLE = {An existence theorem for the {C}auchy
problem on a characteristic cone for
the {E}instein equations},
BOOKTITLE = {Complex analysis and dynamical systems,
{IV}: {P}art 2},
EDITOR = {Mark Agranovsky and Matania Ben-Artzi
and Greg Galloway and Lavi Karp and
Simeon Reich and David Shoikhet Gilbert
Weinstein and Lawrence Zalcman},
SERIES = {Contemp. Math.},
NUMBER = {554},
PUBLISHER = {American Mathematical Society; Bar-Ilan
University},
ADDRESS = {Providence, RI; Ramat Gan},
YEAR = {2011},
PAGES = {73--81},
DOI = {10.1090/conm/554/10961},
NOTE = {MR:2884395. Zbl:1235.83011.},
}
