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Exact Solution of a Neumann Boundary Value Problem for the Stationary Axisymmetric Einstein Equations
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0001-6191-7769
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2019 (English)In: Journal of nonlinear science, ISSN 0938-8974, E-ISSN 1432-1467, Vol. 29, no 4, p. 1621-1657Article in journal (Refereed) Published
Abstract [en]

For a stationary and axisymmetric spacetime, the vacuum Einstein field equations reduce to a single nonlinear PDE in two dimensions called the Ernst equation. By solving this equation with a Dirichlet boundary condition imposed along the disk, Neugebauer and Meinel in the 1990s famously derived an explicit expression for the spacetime metric corresponding to the Bardeen-Wagoner uniformly rotating disk of dust. In this paper, we consider a similar boundary value problem for a rotating disk in which a Neumann boundary condition is imposed along the disk instead of a Dirichlet condition. Using the integrable structure of the Ernst equation, we are able to reduce the problem to a Riemann-Hilbert problem on a genus one Riemann surface. By solving this Riemann-Hilbert problem in terms of theta functions, we obtain an explicit expression for the Ernst potential. Finally, a Riemann surface degeneration argument leads to an expression for the associated spacetime metric.

Place, publisher, year, edition, pages
Springer, 2019. Vol. 29, no 4, p. 1621-1657
Keywords [en]
Ernst equation, Einstein equations, Boundary value problem, Unified transform method, Fokas method, Riemann-Hilbert problem, Theta function
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-257457DOI: 10.1007/s00332-018-9527-1ISI: 000480743200011Scopus ID: 2-s2.0-85059591310OAI: oai:DiVA.org:kth-257457DiVA, id: diva2:1347117
Note

QC 20190830

Available from: 2019-08-30 Created: 2019-08-30 Last updated: 2019-09-05Bibliographically approved

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Lenells, JonatanPei, Long

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