On static solutions of the Einstein-Scalar Field equations

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 ( ϕ ) , 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 ( ϕ ) = m 2 | ϕ | 2 , it is proved that geodesically complete solutions have Ricci-flat spatial metric, have constant lapse and are vacuum, (that is ϕ is constant and equal to zero if m ≠ 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 ( ϕ ) = m 2 | ϕ | 2 + 2 Λ , that is, when a vacuum energy or a cosmological constant is included, it is proved that no geodesically complete solution exists when Λ > 0 , whereas when Λ < 0 it is proved that no non-vacuum geodesically complete solution exists unless m 2 < - 2 Λ / ( 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.