Comptes Rendus
Volume articles
Partial differential equations
Une interprétation variationnelle de la relativité générale dans le vide en termes de transport optimal
Yann Brenier1 
1 CNRS, Département de Mathématiques et Applications, École Normale Supérieure, Université PSL, 45 rue d’Ulm 75005 Paris, France
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 25-33.

On revoit les équations d’Einstein de la relativité générale dans le vide comme équations d’optimalité d’une sorte de problème de transport optimal quadratique généralisé.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.275

Yann Brenier 1

1 CNRS, Département de Mathématiques et Applications, École Normale Supérieure, Université PSL, 45 rue d’Ulm 75005 Paris, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{CRMATH_2022__360_G1_25_0,
     author = {Yann Brenier},
     title = {Une interpr\'etation variationnelle de la relativit\'e g\'en\'erale dans le vide en termes de transport optimal},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {25--33},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.275},
     language = {fr},
}
TY  - JOUR
AU  - Yann Brenier
TI  - Une interprétation variationnelle de la relativité générale dans le vide en termes de transport optimal
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 25
EP  - 33
VL  - 360
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.275
LA  - fr
ID  - CRMATH_2022__360_G1_25_0
ER  - 
%0 Journal Article
%A Yann Brenier
%T Une interprétation variationnelle de la relativité générale dans le vide en termes de transport optimal
%J Comptes Rendus. Mathématique
%D 2022
%P 25-33
%V 360
%I Académie des sciences, Paris
%R 10.5802/crmath.275
%G fr
%F CRMATH_2022__360_G1_25_0
Yann Brenier. Une interprétation variationnelle de la relativité générale dans le vide en termes de transport optimal. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 25-33. doi : 10.5802/crmath.275. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.275/

[1] Luigi Ambrosio; Nicola Gigli; Giuseppe Savaré Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2008 | Zbl

[2] Jean-David Benamou; Yann Brenier A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem, Numer. Math., Volume 84 (2000) no. 3, pp. 375-393 | DOI | MR | Zbl

[3] Robert J. Berman; Sébastien Boucksom; Vincent Guedj; Ahmed Zeriahi A variational approach to complex Monge–Ampère equations, Publ. Math., Inst. Hautes Étud. Sci., Volume 117 (2013), pp. 179-245 | DOI | Numdam | Zbl

[4] Bo Berndtsson; Dario Cordero-Erausquin; Bo’az Klartag; Yanir A. Rubinstein Complex Legendre duality, Am. J. Math., Volume 142 (2020) no. 1, pp. 323-339 | DOI | MR | Zbl

[5] Yann Brenier Extended Monge-Kantorovich theory, Optimal transportation and applications. Lectures given at the C.I.M.E. summer school, Martina Franca, Italy, September 2–8, 2001 (Luis A. Caffarelli, ed.) (Lecture Notes in Mathematics), Volume 1813, Springer, 2003, pp. 91-121 | MR | Zbl

[6] Yann Brenier Hydrodynamic structure of the augmented Born–Infeld equations, Arch. Ration. Mech. Anal., Volume 172 (2004) no. 1, pp. 65-91 | DOI | MR | Zbl

[7] Yann Brenier; Dmitry Vorotnikov On optimal transport of matrix-valued measures, SIAM J. Math. Anal., Volume 52 (2020) no. 3, pp. 2849-2873 | DOI | MR | Zbl

[8] Haïm Brézis Analyse fonctionnelle appliquée, Masson, 1982

[9] Haïm Brézis; Jean-Michel Coron; Elliott H. Lieb Harmonic maps with defects, Commun. Math. Phys., Volume 107 (1986), pp. 649-705 | DOI | MR | Zbl

[10] Eric A. Carlen; Jan Maas Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance, J. Funct. Anal., Volume 273 (2017) no. 5, pp. 1810-1869 | DOI | MR | Zbl

[11] Yongxin Chen; Wilfrid Gangbo; Tryphon T. Georgiou; Allen Tannenbaum On the matrix Monge–Kantorovich problem, Eur. J. Appl. Math., Volume 31 (2020) no. 4, pp. 574-600 | DOI | MR

[12] Demetrios Christodoulou; Sergiu Klainerman The global nonlinear stability of the Minkowski space, Princeton Mathematical Series, 41, Princeton University Press, 1993 | Zbl

[13] Uriel Frisch; Sabino Matarrese; Roya Mohayaee; Andrei Sobolevski A reconstruction of the initial conditions of the Universe by optimal mass transportation, Nature, Volume 417 (2002), pp. 260-262 | DOI

[14] Alice Guionnet First order asymptotics of matrix integrals ; a rigorous approach towards the understanding of matrix models, Commun. Math. Phys., Volume 244 (2004) no. 3, pp. 527-569 | DOI | MR | Zbl

[15] Cécile Huneau; Jonathan Luk Trilinear compensated compactness and Burnett’s conjecture in general relativity (2019) (https://arxiv.org/abs/1907.10743)

[16] Lott John; Cédric Villani Ricci curvature for metric-measure spaces via optimal transport, Ann. Math., Volume 169 (2009) no. 3, pp. 903-991 | DOI | MR | Zbl

[17] Hugo Lavenant Harmonic mappings valued in the Wasserstein space, J. Funct. Anal., Volume 277 (2019) no. 3, pp. 688-785 | DOI | MR | Zbl

[18] Robert J. McCann Displacement convexity of Boltzmann’s entropy characterizes the strong energy condition from general relativity, Camb. J. Math., Volume 8 (2020) no. 3, pp. 609-681 | DOI | MR | Zbl

[19] Andrea Mondino; Stefan Suhr An optimal transport formulation of the Einstein equations of general relativity (2018) (https://arxiv.org/abs/1810.13309v1)

[20] Felix Otto; Michael Westdickenberg Eulerian calculus for the contraction in the Wasserstein distance, SIAM J. Math. Anal., Volume 37 (2005) no. 4, pp. 1227-1255 | DOI | MR | Zbl

[21] Gabriel Peyré; Lénaïc Chizat; François-Xavier Vialard; Justin Solomon Quantum entropic regularization of matrix-valued optimal transport, Eur. J. Appl. Math., Volume 30 (2019) no. 6, pp. 1079-1102 | DOI | MR | Zbl

[22] S. Rachev; L. Rûschendorf Mass Transportation Problems. Vol. 1 : Theory. Vol. 2 : Applications, Springer Series in Statistics, Springer, 1998 | Zbl

[23] Filippo Santambrogio Optimal transport for applied mathematicians, Progress in Nonlinear Differential Equations and their Applications, 87, Birkhäuser/Springer, 2015 | DOI | Zbl

[24] Karl-Théodor Sturm On the geometry of metric measure spaces, Acta Math., Volume 196 (2006) no. 1, pp. 65-177 | DOI | MR | Zbl

[25] Cédric Villani Topics in optimal transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, 2003 | Zbl

Cited by Sources:

Comments - Policy