A continuous Riemann–Hilbert problem for colliding plane gravitational waves

Palenta, Stefan; Meinel, Reinhard

doi:10.1088/1361-6382/aa88dd

Cited by 3 publications

(3 citation statements)

References 17 publications

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“…A post-Newtonian expansion of the disc of charged dust solution will be published elsewhere. 24 It remains a challenge to find an explicit analytic solution as is possible for the uncharged case (ǫ = 0).…”

Section: Rotating Discs Of Charged Dustmentioning

confidence: 99%

Black Holes and Quasiblack Holes in Einstein-Maxwell Theory

Meinel¹,

Breithaupt²,

Liu³

2015

The Thirteenth Marcel Grossmann Meeting

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Continuous sequences of asymptotically flat solutions to the Einstein-Maxwell equations describing regular equilibrium configurations of ordinary matter can reach a black hole limit. For a distant observer, the spacetime becomes more and more indistinguishable from the metric of an extreme Kerr-Newman black hole outside the horizon when approaching the limit. From an internal perspective, a still regular but non-asymptotically flat spacetime with the extreme Kerr-Newman near-horizon geometry at spatial infinity forms at the limit. Interesting special cases are sequences of Papapetrou-Majumdar distributions of electrically counterpoised dust leading to extreme Reissner-Nordström black holes and sequences of rotating uncharged fluid bodies leading to extreme Kerr black holes.

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Section: Rotating Discs Of Charged Dustmentioning

confidence: 99%

Black Holes and Quasiblack Holes in Einstein-Maxwell Theory

Meinel¹,

Breithaupt²,

Liu³

2015

The Thirteenth Marcel Grossmann Meeting

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“…While many exact solutions of the hyperbolic Ernst equation have been found (see [5,8,10,22]), it seems that the problem of determining the solution from given data has only been treated in the series of papers [15][16][17][18] and in [1,11,23]. In [15][16][17][18], the authors relate the Goursat problem to the solution of a homogeneous Hilbert problem and in [1,11,23] the problem is studied by means of inverse scattering. In the case of collinear polarization, both [11] and [15] provide representations for the solution in terms of Abel integrals.…”

Section: Introductionmentioning

confidence: 99%

“…It describes the collision of two (not necessarily collinearly polarized) plane gravitational waves. While many exact solutions of the hyperbolic Ernst equation have been found (see [5,8,10,22]), it seems that the problem of determining the solution from given data has only been treated in the series of papers [15][16][17][18] and in [1,11,23]. In [15][16][17][18], the authors relate the Goursat problem to the solution of a homogeneous Hilbert problem and in [1,11,23] the problem is studied by means of inverse scattering.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics to all orders of the Euler–Darboux equation in a triangle

Mauersberger

2019

Journal of Mathematical Analysis and Applications

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In Einstein's theory of relativity, the interaction of two collinearly polarized plane gravitational waves can be described by a Goursat problem for the Euler-Darboux equation in a triangular domain. In this paper, using a representation of the solution in terms of Abel integrals, we give a full asymptotic expansion of the solution near the diagonal of the triangle. The expansion is related to the formation of a curvature singularity of the spacetime. In particular, our framework allows for boundary data with derivatives which are singular at the corners. This level of generality is crucial for the application to gravitational waves.PDEs in the case of colliding electromagnetic plane waves [3,7]. In the linear limit, both of these coupled equations reduce to the Euler-Darboux equation (1.1).Of particular interest is the behavior of the solution of (1.1) near the triangle's diagonal edge x+y = 1. Indeed, this behavior is related to the formation of a curvature singularity of the spacetime by the mutual focusing of the colliding waves. In [26], it was observed that the solution must behave like ln(1 − x − y) as x + y → 1. In this paper, using an Abel integral representation of the solution, we derive an asymptotic expansion to all orders as x + y → 1.Our first result (Theorem 1) establishes a mathematically precise version of the classical Abel integral representation of the solution of the Goursat problem for (1.1) with given boundary data. This type of representation is well known (cf. e.g. [11,14,15]) and the main purpose of Theorem 1 is to provide a formulation that is suitable for our needs. We discuss regularity, the behavior at the boundary of D, and uniqueness of the solution under reasonable assumptions on the boundary data. In this context, reasonable means that our results can be applied to the common examples of collinear solutions (cf. e.g. [9,19,26]). In particular, we allow for boundary data with singular derivatives at the corners of D. As a consequence, our representation of the solution of (1.1) contains singular integrands, which is the main challenge in the analysis of the Goursat problem.Our second result (Theorem 2) describes the asymptotic behavior of the solution near the diagonal of D. We show that V (x, 1 − x − ) admits an asymptotic expansion to all orders of the formwhere the coefficients f j , g j are given explicitly in terms of the boundary data V 0 (x) = V (x, 0) and V 1 (y) = V (0, y), and where the error term is uniform on compact subsets

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The hyperbolic Ernst equation in a triangular domain

Lenells

Mauersberger

2020

Anal.Math.Phys.

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The collision of two plane gravitational waves in Einstein's theory of relativity can be described mathematically by a Goursat problem for the hyperbolic Ernst equation in a triangular domain. We use the integrable structure of the Ernst equation to present the solution of this problem via the solution of a Riemann-Hilbert problem. The formulation of the Riemann-Hilbert problem involves only the prescribed boundary data, thus the solution is as effective as the solution of a pure initial value problem via the inverse scattering transform. Our results are valid also for boundary data whose derivatives are unbounded at the triangle's corners-this level of generality is crucial for the application to colliding gravitational waves. Remarkably, for data with a singular behavior of the form relevant for gravitational waves, it turns out that the singular integral operator underlying the Riemann-Hilbert formalism can be explicitly inverted at the boundary. In this way, we are able to show exactly how the behavior of the given data at the origin transfers into a singular behavior of the solution near the boundary.

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A continuous Riemann–Hilbert problem for colliding plane gravitational waves

Cited by 3 publications

References 17 publications

Black Holes and Quasiblack Holes in Einstein-Maxwell Theory

Black Holes and Quasiblack Holes in Einstein-Maxwell Theory

Asymptotics to all orders of the Euler–Darboux equation in a triangle

The hyperbolic Ernst equation in a triangular domain

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