Exact Solution of a Neumann Boundary Value Problem for the Stationary Axisymmetric Einstein Equations
Abstract
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.
Keywords
Ernst equation Einstein equations Boundary value problem Unified transform method Fokas method Riemann–Hilbert problem Theta functionMathematics Subject Classification
83C15 37K15 35Q15 35Q761 Introduction
Half a century ago, Bardeen and Wagoner studied the structure and gravitational field of a uniformly rotating, infinitesimally thin disk of dust in Einstein’s theory of relativity (Bardeen and Wagoner 1969, 1971). Although their study was primarily numerical, they pointed out that there may be some hope for finding an exact expression for the solution. Remarkably, such an exact expression was derived in a series of papers by Neugebauer and Meinel in the 1990s (Neugebauer and Meinel 1993, 1994, 1995) (see also Meinel et al. 2008). Rather than analyzing the Einstein equations directly, Neugebauer and Meinel arrived at their exact solution by studying a boundary value problem (BVP) for the socalled Ernst equation.
The Ernst equation is a nonlinear integrable partial differential equation in two dimensions which was first written down by Ernst in the 1960s (Ernst 1968). Ernst made the quite extraordinary discovery that, in the presence of one spacelike and one timelike Killing vector, the full system of the vacuum Einstein field equations reduces to a single equation for a complexvalued function f of two variables (Ernst 1968). This single equation, now known as the (elliptic) Ernst equation, has proved instrumental in the study and construction of stationary axisymmetric spacetimes, cf. Klein and Richter (2005).
In an effort to understand the Neugebauer–Meinel solution from a more general and systematic point of view, Fokas and the first author revisited the solution of the above BVP problem in Lenells and Fokas (2011). It was shown in Lenells and Fokas (2011) that the problem actually is a special case of a socalled linearizable BVP as defined in the general approach to BVPs for integrable equations known as the unified transform or Fokas method (Fokas 1997). In this way, the Neugebauer–Meinel solution could be recovered. Later, an extension of the same approach led to the discovery of a new class of explicit solutions which combine the Kerr and Neugebauer–Meinel solutions (Lenells 2011). The solutions of Lenells (2011) involve a disk rotating uniformly around a central black hole and are given explicitly in terms of theta functions on a Riemann surface of genus four.
In addition to the BVP for the uniformly rotating dust disk, a few other BVPs were also identified as linearizable in Lenells and Fokas (2011). One of these problems has the same form as the BVP for the uniformly rotating disk except that a Neumann condition is imposed along the disk instead of a Dirichlet condition. The purpose of the present paper is to present the solution of this Neumann BVP. Our main result provides an explicit expression for the solution of this problem (both for the solution of the Ernst equation and for the associated spacetime metric) in terms of theta functions on a genus one Riemann surface. In the limit when the rotation axis is approached, the Riemann surface degenerates to a genus zero surface (the Riemann sphere), which means that we can find particularly simple formulas for the spacetime metric in this limit.
Our approach can be briefly described as follows. We first use the integrability of the Ernst equation to reduce the solution of the BVP problem to the solution of a matrix Riemann–Hilbert (RH) problem. The formulation of this RH problem involves both the Dirichlet and Neumann boundary values on the disk. By employing the fact that the boundary conditions are linearizable, we are able to eliminate the unknown Dirichlet values. This yields an effective solution of the problem in terms of the solution of a RH problem. However, as in the case of the Neugebauer–Meinel solution, it is possible to go even further and obtain an explicit solution by reducing the matrix RH problem to a scalar RH problem on a Riemann surface. By solving this scalar problem in terms of theta functions, we find exact formulas for the Ernst potential and two of the metric functions. Finally, a Riemann surface condensation argument is used to find an expression for the third and last metric function.
Although our approach follows steps which are similar to those set forth for the Dirichlet problem in Lenells and Fokas (2011) [which were in turn inspired by Neugebauer and Meinel (1993, 1994, 1995)], the Neumann problem considered here is different in a number of ways. One difference is that the underlying Riemann surface has genus one instead of genus two. This means that we are able to derive simpler formulas for the spectral functions and for the solution on the rotation axis. Another difference is that the jump of the scalar RH problem for the Neumann problem does not vanish at the endpoints of the contour. This means that a new type of condensation argument is needed to determine the last metric function. We expect this new argument to be of interest also for other BVPs and for the construction of exact solutions via solutiongenerating techniques.
We do not explore the possible physical relevance of the solved Neumann BVP here. Instead, our solution of this BVP is motivated by the following two reasons: (a) As already mentioned, very few BVPs for rotating objects in general relativity have been solved constructively by analytic methods. Our solution enlarges the class of constructively solvable BVPs and expands the mathematical toolbox used to solve such problems. (b) An outstanding problem in the context of rotating objects in Einstein’s theory consists of finding solutions which describe disk/black hole systems (Meinel et al. 2008; Klein and Richter 2005). The solutions derived in Lenells (2011) are of this type. However, the disks in these solutions reach all the way to the event horizon. Physically, there should be a gap between the horizon and the inner rim of the disk (so that the disk actually is a ring). Such a ring/black hole problem can be formulated as a BVP for the Ernst equation with a mixed Neumann/Dirichlet condition imposed along the gap. Thus, we expect the solution of a pure Neumann BVP (in addition to the already known solution of the analogous Dirichlet problem) to provide insight which is useful for analyzing ring/black hole BVPs.
1.1 Organization of the Paper
In Sect. 2, we introduce some notation and state the Neumann boundary value problem which is the focus of the paper. The main results are presented in Sect. 3. In Sect. 4, we illustrate our results with a numerical example. In Sect. 5, we begin the proofs by constructing an eigenfunction \(\Phi (z,k)\) of the Lax pair associated with the Ernst equation. We set up a RH problem for \(\Phi (z,k)\) with a jump matrix defined in terms of two spectral functions F(k) and G(k). Using the equatorial symmetry and the Neumann boundary condition, we formulate an auxiliary RH problem which is used to determine F(k) and G(k). This provides an effective solution of the problem in terms of the solution of a RH problem. However, as mentioned above, it is possible to obtain a more explicit solution. Thus, in Sect. 6, we combine the RH problem for \(\Phi \) and the auxiliary RH problem into a scalar RH problem, which can be solved for the Ernst potential f. In Sect. 7, we use tools from algebraic geometry to express f and two of the associated metric functions in terms of theta functions. In Sect. 8, we use a branch cut condensation argument to derive a formula for the last metric function. In Sect. 9, we study the behavior of the solution near the rotation axis and complete the proofs of the main results.
2 Preliminaries
2.1 The Ernst Equation
2.2 The Boundary Value Problem
2.3 The Riemann Surface \(\Sigma _z\)
We will present the solution of the BVP (2.8) in terms of theta functions associated with a family of Riemann surfaces \(\Sigma _z\) parameterized by \(z = \rho + i\zeta \in {\mathcal {D}}\). Before stating the main results, we need to define this family of Riemann surfaces. In view of the equatorial symmetry, it suffices to determine the solution f(z) of (2.8) for \(z=\rho +i \zeta \) with \(\zeta >0\). We therefore assume that \(\zeta > 0\) in the following.
The cut \(C_{k_{1}}\) does not intersect \(\Gamma \) at the endpoints, because the assumption \(2\Omega \rho _0 < 1\) implies that \(\rho _0 < k_1\). Thus, for each \(z \in {\mathcal {D}}\), the branch cuts and the contour \(\Gamma \) are organized as in Fig. 2 with \(C_{k_{1}}\) and \([i z, i{\bar{z}}]\) to the left and right of \(\Gamma \), respectively.
We denote by \(\Sigma _z^+\) and \(\Sigma _z^\) the upper and lower sheets of \(\Sigma _z\), respectively, where the upper (lower) sheet is characterized by \(y \sim k^2\) (\(y \sim k^2\)) as \(k \rightarrow \infty \). Let \({\hat{\mathbb {C}}} = \mathbb {C}\cup \infty \) denote the Riemann sphere. For \(k\in {\hat{\mathbb {C}}} {\setminus } (C_{k_1} \cup [i z, i{\bar{z}}])\), we write \(k^{+}\) and \(k^{}\) for the points in \(\Sigma _{z}^{+}\) and \(\Sigma _{z}^{}\), respectively, which project onto \(k\in \mathbb {C}\). More generally, we let \(A^{+}\) and \(A^{}\) denote the lifts of a subset \(A \subset {\hat{\mathbb {C}}} {\setminus } (C_{k_1} \cup [i z, i{\bar{z}}])\) to \(\Sigma _{z}^+\) and \(\Sigma _{z}^\), respectively.
2.4 The Riemann Surface \(\Sigma '\)
3 Main Results
The next theorem, which is our main result, gives an explicit expression for the solution of the boundary value problem (2.8) and the associated metric functions in terms of the theta function \(\Theta \) associated with the Riemann surface \(\Sigma _z\).
Theorem 3.1
Remark 3.2
(Solution for \(\zeta \le 0\)) Theorem 3.1 provides expressions for the Ernst potential and the metric functions for \(\zeta > 0\). If \(\rho > \rho _0\), these expressions extend continuously to \(\zeta = 0\). For negative \(\zeta \), analogous expressions follow immediately from the equatorial symmetry. In this way, the solution of BVP (2.8) is obtained in all of the exterior disk domain \({\mathcal {D}}\).
Remark 3.3
(The assumption \(2\Omega \rho _0 < 1\)) We have stated Theorem 3.1 under the assumption that \(2\Omega \rho _0 < 1\). If the rotation speed \(\Omega \) and/or the radius \(\rho _0\) are so large that \(2\Omega \rho _0 \ge 1\), then the branch points \(k_1\) and \({\bar{k}}_1\) lie on \(\Gamma \). Nevertheless, the formulas of Theorem 3.1 can easily be adjusted to include these (possibly singular) solutions.
Remark 3.4
Remark 3.5
3.1 Solution Near the Rotation Axis
Theorem 3.6
 The Ernst potential f(z) satisfieswhere$$\begin{aligned} f(\rho +i\zeta )=f(i \zeta )+O(\rho ^{2}), \end{aligned}$$(3.11)$$\begin{aligned} f(i \zeta )=\frac{1 + e^{J^{\prime }}d}{e^{J^{\prime }} + d}. \end{aligned}$$(3.12)
 The metric functions \(e^{2U}\), a and \(e^{2\kappa }\) satisfywhere$$\begin{aligned} e^{2U(\rho +i \zeta )}=e^{2U(i \zeta )}+O(\rho ^{2}),\quad a(\rho +i \zeta )=O(\rho ^{2}),\quad e^{2\kappa (\rho +i \zeta )}=1+O(\rho ^{2}),\nonumber \\ \end{aligned}$$(3.13)$$\begin{aligned} e^{2U(i \zeta )}=\frac{(1d^2)e^{J'}}{e^{2J^{\prime }}  d^2}. \end{aligned}$$(3.14)
Remark 3.7
4 Numerical Example
5 Lax Pair and Spectral Theory
5.1 Lax Pair
5.2 The Main RH Problem
The following lemma can be found in Lenells and Fokas (2011).
Lemma 5.1
 F(k) and G(k) descend to functions on \({\hat{\mathbb {C}}}\); namely, when viewed as functions on \({\mathscr {S}}_{z}\), they satisfy$$\begin{aligned} F(k^+)=F(k^),\quad G(k^+)=G(k^), \quad k\in {\hat{\mathbb {C}}}. \end{aligned}$$(5.10)

F(k) and G(k) are analytic for \(k \in \hat{\mathbb {C}} \setminus \Gamma \).

\(F(k)=\overline{F(\overline{k})}\) and \(G(k)=\overline{G(\overline{k})}\) for \(k\in \mathbb {C}\backslash \Gamma \).
 As \(k\rightarrow \infty \),$$\begin{aligned} F(k)=1+O(k^{1}),\quad G(k)=O(k^{1}). \end{aligned}$$(5.11)
5.3 The Global Relation
Since f is equatorially symmetric, the values of F(k) and G(k) on the left and right sides of \(\Gamma \) satisfy an important relation called the global relation.
Lemma 5.2
Proof
See Lenells and Fokas (2011, Proposition 4.3). \(\square \)
5.4 An Additional Relation
The fact that \(f_\Omega \) obeys a Neumann condition along the disk implies that F(k) and G(k) satisfy an additional relation beyond the global relation.
We use the superscripts L and R on a function of k to indicate that this function should be evaluated with k lying on the left or right side of the branch cut \([i z,i{\bar{z}}]\), respectively. If one of the superscripts L or R is present, we always assume that the evaluation point k lies to the right of \(\Gamma \). The latter specification is needed when \(z=\rho +i 0\) so that the branch cut \([i z,i {\bar{z}}]\) runs infinitesimally close to \(\Gamma \).
Lemma 5.3
Proof
5.5 The Auxiliary RH Problem
Lemma 5.4

\({\mathcal {M}}(k)\) is analytic for \(k \in \hat{\mathbb {C}} \setminus \Gamma \).
 Across \(\Gamma \), \({\mathcal {M}}(k)\) satisfies the jump conditionwhere \({\mathcal {S}}(k)\) is defined by$$\begin{aligned} {\mathcal {S}}(k){\mathcal {M}}_(k)={\mathcal {M}}_+(k){\mathcal {S}}(k), \quad k\in \Gamma , \end{aligned}$$(5.20)$$\begin{aligned} {\mathcal {S}}(k) =\begin{pmatrix} 0&{}\quad 1 \\ 1 &{}\quad 4i k\Omega \\ \end{pmatrix}, \quad k\in \mathbb {C}. \end{aligned}$$(5.21)

\({\mathcal {M}}(k)\) has at most logarithmic singularities at the endpoints of \(\Gamma \).
 \({\mathcal {M}}(k)\) has the asymptotic behavior$$\begin{aligned} {\mathcal {M}}(k)=\sigma _1+O(k^{1}), \quad k\rightarrow \infty . \end{aligned}$$(5.22)
Proof
5.6 Solution of the Auxiliary RH Problem
By solving the RH problem of Lemma 5.4, we can obtain explicit expressions for F(k) and G(k).
Lemma 5.5
Proof
5.7 The Ernst Potential on the Rotation Axis
6 The Scalar RH Problem
Substitution of the expressions for F(k) and G(k) obtained in Lemma 5.5 into (5.13) gives an explicit expression for the jump matrix D(k). Thus, we have an effective solution of BVP (2.8) in terms of the solution of the matrix RH problem (5.12). In what follows, we instead employ the main and auxiliary RH problems to formulate a scalar RH problem on the Riemann surface \(\Sigma _z\). The solution of this scalar RH problem leads to the exact formulas of Theorem 3.1.
6.1 The Functions \({\mathcal {L}}(z,k)\) and \({\mathcal {Q}}(z,k)\)
Lemma 6.1
 \({\mathcal {L}}\) and \({\mathcal {Q}}\) satisfy the trace and determinant relationsand the symmetries$$\begin{aligned} \mathrm {tr}\,{\mathcal {Q}} =0, \quad \mathrm {tr}\,{\mathcal {L}}=0, \quad \det {\mathcal {L}} =1,\quad \det {\mathcal {Q}}=1w^{2}, \end{aligned}$$(6.3)$$\begin{aligned} {\mathcal {Q}}(z,k^)=\sigma _3{\mathcal {Q}}(z,k^+)\sigma _3,\quad {\mathcal {L}}(z,k^)=\sigma _3{\mathcal {L}}(z,k^+)\sigma _3. \end{aligned}$$(6.4)
 \({\mathcal {Q}}\) can be rewritten as$$\begin{aligned} {\mathcal {Q}}(z,k)=\Phi (z,k)\sigma _1 {\mathcal {A}}^{1}(k){\mathcal {S}}(k) {\mathcal {A}}(k)\sigma _1\Phi ^{1}(z,k)+w(k){\mathbb {I}}. \end{aligned}$$(6.5)
 \({\mathcal {Q}}\) has no jump across \(\Gamma ^+ \cup \Gamma ^\), whereas \({\mathcal {L}}\) satisfies the jump conditions$$\begin{aligned} {\left\{ \begin{array}{ll} ({\mathcal {Q}}+w{\mathbb {I}}){\mathcal {L}}_{} ={\mathcal {L}}_{+}({\mathcal {Q}}+w{\mathbb {I}}), &{} k\in \Gamma ^+, \\ ({\mathcal {Q}}w{\mathbb {I}}){\mathcal {L}}_{} ={\mathcal {L}}_{+}({\mathcal {Q}} w{\mathbb {I}}), &{} k\in \Gamma ^. \end{array}\right. } \end{aligned}$$(6.6)

\({\mathcal {L}}\) and \({\mathcal {Q}}\) anticommute, i.e., \({\mathcal {Q}}{\mathcal {L}}={\mathcal {L}}{\mathcal {Q}}\).
 The function \(\hat{{\mathcal {L}}}\) defined bysatisfies$$\begin{aligned} \hat{{\mathcal {L}}} ={\mathcal {L}}\begin{pmatrix} 1 &{}\quad {\mathcal {Q}}_{11} \\ 0 &{}\quad {\mathcal {Q}}_{21} \\ \end{pmatrix}, \end{aligned}$$$$\begin{aligned} \hat{{\mathcal {L}}}_{22}^{2}{\mathcal {L}}_{21}^{2}(w^{2}1)={\mathcal {Q}}^{2}_{21}. \end{aligned}$$(6.7)
 For each z, there exists a point \(m_{1} \equiv m_{1}(z) \in \mathbb {C}\) such that$$\begin{aligned} {\mathcal {Q}}_{21}(z,k)=\frac{4i \Omega f}{f+{\bar{f}}}(km_{1}). \end{aligned}$$(6.8)
Proof
6.2 The Riemann Surface \(\hat{{\mathscr {S}}}_z\) and the Function H(z, k)
6.3 The Scalar RH Problem
Proposition 6.2

\(\psi (z,k)\) is analytic for \(k\in {\hat{\mathbb {C}}}\backslash (\Gamma \cup [k_{1},m_{1}])\).
 Across \(\Gamma \), \(\psi (z,k)\) satisfies the jump relation$$\begin{aligned} \psi _{}(z,k)=\psi _{+}(z,k)+\frac{2}{y(z,k^+)}\log \left( \frac{\sqrt{w^{2}1}w}{\sqrt{w^{2}1}+w}(k^+)\right) , \quad k\in \Gamma . \end{aligned}$$(6.20)
 Across the oriented straightline segment \([k_{1},m_{1}]\), \(\psi (z,k)\) satisfies the jump relationwhere \(k^+\) is the point in the upper sheet of \(\Sigma _z\) which projects onto k, and \(\psi _{}(z,k)\) and \(\psi _{+}(z,k)\) denote the values of \(\psi \) on the left and right sides of \([k_{1},m_{1}]\).$$\begin{aligned} \psi _{}(z,k)=\psi _{+}(z,k)+\frac{4\pi i}{y(z,k^+)}, \quad k\in [k_{1},m_{1}], \end{aligned}$$(6.21)
 As \(k\rightarrow m_{1}\),$$\begin{aligned} \psi (z,k)\rightarrow \frac{2}{y(z,k^+)}\log (km_{1}). \end{aligned}$$(6.22)
 As \(k\rightarrow \infty \),$$\begin{aligned} \psi (z,k) = \frac{2\log f}{k^{2}}+O(k^{3}). \end{aligned}$$(6.23)
 As \(k\rightarrow k_{1}\),where \(y=y(z,k^+)\) for \(k^+\) just to the left of the cut \([k_1,m_1^{+}]\) on \(\Sigma _z^{+}\), and is analytically continued around the endpoint \(k_1\) so that it equals \(y(z,k^)\) when \(k^\) lies just to the left of the cut \([m_1^{},k_1]\) on \(\Sigma _z^{}\).$$\begin{aligned} \psi (z,k) \rightarrow \frac{2\pi i}{y}, \end{aligned}$$(6.24)
Proof
6.4 Solution of the Scalar RH Problem
7 Theta Functions
In this section, we derive the expressions for f, \(e^{2U}\) and a given in (3.3) and (3.4).
7.1 Proof of Expression (3.3) for f
7.2 Proof of Expressions (3.4) for \(e^{2U}\) and a
Since \(u\in i {\mathbb {R}}\) and \(I\in {\mathbb {R}}\), the expression for \(e^{2U}\) in (3.4) can be derived as in Lenells (2011). With the expression for \(e^{2U}\) at hand, the expression for a in (3.4) can be obtained by following the argument in Klein and Richter (1998, Sect. V).
8 The Metric Function \(e^{2\kappa }\)
One useful tool in the study of the Ernst equation (2.5) is the use of branch point condensation arguments (Korotkin and Matveev 2000). In this section, we apply such arguments to derive expression (3.6) for \(e^{2\kappa }\). An analogous derivation was considered in Lenells (2011). However, in contrast to the situation in Lenells (2011), our function h(k) which determines the jump of the scalar RH problem for \(\psi (z,k)\) does not vanish at the endpoints of \(\Gamma \). This means that the condensation argument has to be modified.
8.1 Proof of Expression (3.6) for \(e^{2\kappa }\)
9 Solution Near the Rotation Axis
In this section, we first complete the proof Theorem 3.6 by studying the asymptotic behavior of the Ernst potential and the metric functions near the rotation axis \(\rho = 0\). We then use these results to establish expressions for \(a_0\) and \(K_0\), thus completing also the proof of Theorem 3.1.
Lemma 9.1
We have \(e^{K'} = d\), where \(d(\zeta )\) is the function in (3.10).
Proof
The following lemma gives the behavior of several quantities near the rotation axis.
Lemma 9.2
Proof
Proof of Theorem 3.6
The fact that \(a = O(\rho ^2)\) as \(\rho \rightarrow 0\) is a consequence of (2.4), (3.4) and Lemma 9.2. Similarly, in view of (2.7) and (3.6), it is clear that \(e^{2\kappa }1 = O(\rho ^2)\). This completes the proof of Theorem 3.6. \(\square \)
Proof
Footnotes
 1.
The result is independent of whether \(\Gamma \) is deformed to the left or right of the singularity.
Notes
Acknowledgements
Pei would like to thank Mats Ehrnström and Yuexun Wang for their hospitality at NTNU where part of the research presented in this paper was conducted. The support from the European Research Council, Grant Agreement No. 682537, the Swedish Research Council, Grant No. 201505430, and the Göran Gustafsson Foundation is acknowledged.
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