Abstract
In this paper we begin the study of the global initial value problem for Einstein's equations in the spherically symmetric case with a massless scalar field as the material model. We reduce the problem to a single nonlinear evolution equation. Taking as initial hypersurface a future light cone with vertex at the center of symmetry, we prove, the local, in retarded time, existence and global uniqueness of classical solutions. We also prove that if the initial data is sufficiently small there exists a global classical solution which disperses in the infinite future.
Similar content being viewed by others
References
Penrose, R.: Gravitational collapse and space-time singularities. Phys. Rev. Lett.14, 57–59 (1965)
Hawking, S. W., Penrose, R.: The singularities of gravitational collapse and cosmology. Proc. Roy. Soc. Lond.A314, 529–548 (1970)
Hawking, S. W., Ellis G. F. R.: The large scale structure of space-time. Cambridge: Cambridge University Press 1973
Bondi, H., van Burg, M. G. J., Metzner, A. W. K.: Gravitational waves in general relativity VII. Waves from axi-symmetric isolated systems. Proc. Roy. Soc. Lond.A269, 21–48 (1962)
Author information
Authors and Affiliations
Additional information
Communicated by S.-T. Yau
Research supported in part by National Science Foundation grants MCS-8201599 to the Courant Institute and PHY-8318350 to Syracuse University
Rights and permissions
About this article
Cite this article
Christodoulou, D. The problem of a self-gravitating scalar field. Commun.Math. Phys. 105, 337–361 (1986). https://doi.org/10.1007/BF01205930
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01205930