Exact Solution of a Neumann Boundary Value Problem for the Stationary Axisymmetric Einstein Equations

• 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.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Nyckelord

Ernst equation

Einstein equations

Boundary value problem

Unified transform method

Fokas method

Riemann-Hilbert problem

Theta function

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