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- Title
On static solutions of the Einstein-Scalar Field equations.
- Authors
Reiris, Martín
- Abstract
In this article we study self-gravitating static solutions of the Einstein-Scalar Field system in arbitrary dimensions. We discuss the existence of geodesically complete solutions depending on the form of the scalar field potential $$V(\phi )$$ , and provide full global geometric estimates when the solutions exist. The most complete results are obtained for the physically important Klein-Gordon field and are summarised as follows. When $$V(\phi )=m^{2}|\phi |^{2}$$ , it is proved that geodesically complete solutions have Ricci-flat spatial metric, have constant lapse and are vacuum, (that is $$\phi $$ is constant and equal to zero if $$m\ne 0$$ ). In particular, when the spatial dimension is three, the only such solutions are either Minkowski or a quotient thereof (no nontrivial solutions exist). When $$V(\phi )=m^{2}|\phi |^{2}+2\Lambda $$ , that is, when a vacuum energy or a cosmological constant is included, it is proved that no geodesically complete solution exists when $$\Lambda >0$$ , whereas when $$\Lambda <0$$ it is proved that no non-vacuum geodesically complete solution exists unless $$m^{2}<-2\Lambda /(n-1)$$ , ( n is the spatial dimension) and the spatial manifold is non-compact. The proofs are based on novel techniques in comparison geometry á la Bakry-Émery that have their own interest.
- Subjects
EINSTEIN field equations; SCALAR field theory; GEODESICS; KLEIN-Gordon equation; RICCI flow; MANIFOLDS (Mathematics)
- Publication
General Relativity & Gravitation, 2017, Vol 49, Issue 3, p1
- ISSN
0001-7701
- Publication type
Article
- DOI
10.1007/s10714-017-2191-1