×

zbMATH — the first resource for mathematics

Einstein equations and conformal structure: Existence of anti-de Sitter-type space-times. (English) Zbl 0840.53055
The author develops and discusses in detail the conformal structure of space-time and, in particular, of the Einstein equations. By conformal extension, a space-time can aquire a boundary, so that asymptotic conditions become boundary conditions. Cauchy’s problem to Einstein’s equations is treated as an initial and boundary value problem. The author proves a theorem of existence and uniqueness of a global solution which he calls of “anti-de Sitter type” because of some likeness to the anti-de Sitter space-time. This type is asymptotically simple and has a positive cosmological constant. The analytical tools come from a paper of O. Guès of 1990.

MSC:
53Z05 Applications of differential geometry to physics
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
53A30 Conformal differential geometry (MSC2010)
83C15 Exact solutions to problems in general relativity and gravitational theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Andersson, L.; Chruściel, P.T., On the asymptotic behaviour of solutions to the constraint equations in general relativity with “hyperboloidal” boundary conditions, (1994), preprint
[2] Andersson, L.; Chruściel, P.T.; Friedrich, H., On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein’s field equations, Commun. math. phys., 149, 587-612, (1992) · Zbl 0764.53027
[3] Cartan, E., LES espaces à connexion conforme, Ann. soc. po. math., 2, 171-221, (1923) · JFM 50.0493.01
[4] Choquet-Bruhat, Y.; York, J.W., The Cauchy problem, ()
[5] Friedrich, H., On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure, Commun. math. phys., 107, 587-609, (1986) · Zbl 0659.53056
[6] Friedrich, H., On static and radiative spacetimes, Commun. math. phys., 119, 51-73, (1988) · Zbl 0658.53074
[7] Friedrich, H., On the global existence and the asymptotic behaviour of solutions to the Einstein-Maxwell-Yang-Mills equations, J. diff. geom., 34, 275-345, (1991) · Zbl 0737.53070
[8] Friedrich, H.; Schmidt, B.G., Conformal geodesics in general relativity, (), 171-195 · Zbl 0629.53063
[9] Guès, O., Problème mixte hyperbolique quasi-linéaire charactéristique, Commun. part. diff. equ., 15, 595-645, (1990) · Zbl 0712.35061
[10] Hawking, S.W., The boundary conditions for gauged supergravity, Phys. lett., 126 B, 175-177, (1983)
[11] Hawking, S.W.; Ellis, F.G.R., The large scale structure of space-time, (1973), Cambridge University Press Cambridge
[12] Henneaux, M.; Teitelboim, C., Asymptotically anti-de Sitter spaces, Commun. math. phys., 98, 391-424, (1985) · Zbl 1032.83502
[13] Kobayashi, S., Transformation groups in differential geometry, (1972), Springer Berlin · Zbl 0246.53031
[14] Ogiue, K., Theory of conformal connections, Kodai math. sem. rep., 19, 193-224, (1967) · Zbl 0163.16501
[15] Penrose, R., Conformal treatment of infinity, () · Zbl 0148.46403
[16] Penrose, R., Zero rest-mass fields including gravitation: asymptotic behaviour, (), 159-203 · Zbl 0129.41202
[17] Penrose, R., Structure of space-time, () · Zbl 1001.83040
[18] Penrose, R.; Rindler, W., ()
[19] Rauch, J., Symmetric positive systems with boundary characteristics of constant multiplicity, Trans. amer. math. soc., 291, 167-187, (1985) · Zbl 0549.35099
[20] Taylor, M.E., Pseudodifferential operators and nonlinear PDE, (1991), Birkhäuser Boston · Zbl 0746.35062
[21] Weyl, H., Reine infinitesimalgeometrie, Mathem. zeitschrift, 2, 384-411, (1918) · JFM 46.1301.01
[22] Yano, K.; Yano, K., Sur la théorie des espaces à connexion conforme, (), J. fac. sci. univ. Tokyo sect. 1, 4, 1-59, (1939) · JFM 65.1421.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.