5-dimensional space-periodic solutions of the static vacuum Einstein equations

Marcus Khuri,Gilbert Weinstein,Sumio Yamada

doi:10.1007/jhep12(2020)002

Marcus Khuri, Gilbert Weinstein + Show 1 more

Open Access

https://doi.org/10.1007/jhep12(2020)002

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Abstract

An affirmative answer is given to a conjecture of Myers concerning the existence of 5-dimensional regular static vacuum solutions that balance an infinite number of black holes, which have Kasner asymptotics. A variety of examples are constructed, having different combinations of ring S1 × S2 and sphere S3 cross-sectional horizon topologies. Furthermore, we show the existence of 5-dimensional vacuum solitons with Kasner asymptotics. These are regular static space-periodic vacuum spacetimes devoid of black holes. Consequently, we also obtain new examples of complete Riemannian manifolds of nonnegative Ricci curvature in dimension 4, and zero Ricci curvature in dimension 5, having arbitrarily large as well as infinite second Betti number.

Highlights

Let M5 be the domain of outer communication of a stationary bi-axisymmetric 5-dimensional spacetime
An affirmative answer is given to a conjecture of Myers concerning the existence of 5-dimensional regular static vacuum solutions that balance an infinite number of black holes, which have Kasner asymptotics
We show the existence of 5-dimensional vacuum solitons with Kasner asymptotics

Summary

Background

Let M5 be the domain of outer communication of a stationary bi-axisymmetric 5-dimensional spacetime. Weyl-Papapetrou coordinates by g = e2α(dρ2 + dz2) − f −1ρ2dt2 + fij(dφi + vidt)(dφj + vjdt) Note that this exhibits the interpretation of rod structures as vectors (pl, ql)t lying in the (1-dimensional) kernel of the matrix F at an axis rod Γl. The conical singularity at a point (0, z0) on an axis rod Γi, with rod structure v = (v1, v2) = (pl, ql), is measured by the angle defect θ ∈ (−∞, 2π) associated with the 2-dimensional cone formed by the orbits of vj∂φj over the line z = z0. The conical singularity on Γl is said to exhibit an angle deficit if bl > 0, and an angle surplus if bl < 0

Statement of results

Space-periodic solutions

Regularity and asymptotics

Regularity

Asymptotics