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General Relativity and Quantum Cosmology, gr-qc,Mathematics, Differential Geometry, math.DG
Abstract:
In this note we study the Einstein-ScalarField static equations 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. As a special case it is
shown that when $V(\phi)$ is the Klein-Gordon potential, i.e.
$V(\phi)=m^{2}|\phi|^{2}$, geodesically complete solutions are necessarily
Ricci-flat, have constant lapse and are vacuum, (that is $\phi=\phi_{0}$ with
$\phi_{0}=0$ if $m\neq 0$). Hence, if the spatial dimension is three, the only
such solutions are either Minkowski or a quotient thereof. For
$V(\phi)=m^{2}|\phi|^{2}+2\Lambda$, that is, including a vacuum energy or a
cosmological constant, 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 manifold is
non-compact.
