Abstract
We consider the Riemann-Hilbert factorization approach to the construction of Weyl metrics in four space-time dimensions. We present, for the first time, a rigorous proof of the remarkable fact that the canonical Wiener-Hopf factorization of a matrix obtained from a general (possibly unbounded) monodromy matrix, with respect to an appropriately chosen contour, yields a solution to the non-linear gravitational field equations. This holds regardless of whether the dimensionally reduced metric in two dimensions has Minkowski or Euclidean signature. We show moreover that, by taking advantage of a certain degree of freedom in the choice of the contour, the same monodromy matrix generally yields various distinct solutions to the field equations. Our proof, which fills various gaps in the existing literature, is based on the solution of a second Riemann-Hilbert problem and highlights the deep role of the spectral curve, the normalization condition in the factorization and the choice of the contour. This approach allows us to construct explicit solutions, including new ones, to the non-linear gravitational field equations, using simple complex analytic results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G.A. Alekseev, Thirty years of studies of integrable reductions of Einstein’s field equations, in On recent developments in theoretical and experimental general relativity, astrophysics and relativistic field theories. Proceedings, 12th Marcel Grossmann meeting on general relativity, Paris, France, 12–18 July 2009, World Scientific, Singapore (2010), pg. 645 [arXiv:1011.3846] [INSPIRE].
P. Breitenlohner and D. Maison, On the Geroch group, Ann. Inst. H. Poincaŕe Phys. Theor. 46 (1987) 215.
H. Nicolai, Two-dimensional gravities and supergravities as integrable system, Lect. Notes Phys. 396 (1991) 231 [INSPIRE].
D. Katsimpouri, A. Kleinschmidt and A. Virmani, Inverse scattering and the Geroch group, JHEP 02 (2013) 011 [arXiv:1211.3044] [INSPIRE].
M.C. Camara, G.L. Cardoso, T. Mohaupt and S. Nampuri, A Riemann-Hilbert approach to rotating attractors, JHEP 06 (2017) 123 [arXiv:1703.10366] [INSPIRE].
G.L. Cardoso and J.C. Serra, New gravitational solutions via a Riemann-Hilbert approach, JHEP 03 (2018) 080 [arXiv:1711.01113] [INSPIRE].
J. Ehlers and W. Kundt, Exact solutions of the gravitational field equations, in Gravitation: an introduction to current research, L. Witten ed., Wiley, U.S.A. (1962), pg. 49.
J.B. Griffiths and J. Podolsky, Exact space-times in Einstein’s general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, U.K. (2009)
N. Wiener and E. Hopf, Über eine Klasse singulärer Integralgleichungen (in German), S.-B. Preuss. Akad. Wiss. Berlin Phys. Math. Kl. 30/32 (1931) 696.
K. Clancey and I. Gohberg, Factorization of matrix functions and singular integral operators, in Operator theory: advances and applications, volume 3, Birkh¨auser Verlag, Basel, Switzerland (1981).
F.-O. Speck, General Wiener-Hopf factorization methods, Res. Notes Math. 119 (1985) 1.
C. Barrabes and W. Israel, Thin shells in general relativity and cosmology: the lightlike limit, Phys. Rev. D 43 (1991) 1129 [INSPIRE].
C. Barrabes and P. Hogan, Singular null hypersurfaces in general relativity, World Scientific, Singapore (2003).
G. Jones and J.E. Wang, Weyl card diagrams and new S-brane solutions of gravity, hep-th/0409070 [INSPIRE].
G.C. Jones and J.E. Wang, Weyl card diagrams, Phys. Rev. D 71 (2005) 124019 [hep-th/0506023] [INSPIRE].
J.H. Schwarz, Classical symmetries of some two-dimensional models coupled to gravity, Nucl. Phys. B 454 (1995) 427 [hep-th/9506076] [INSPIRE].
H. Lü, M.J. Perry and C.N. Pope, Infinite-dimensional symmetries of two-dimensional coset models coupled to gravity, Nucl. Phys. B 806 (2009) 656 [arXiv:0712.0615] [INSPIRE].
A.R. Its, The Riemann-Hilbert problem and integrable systems, Not. Amer. Math. Soc. 50 (2003) 1389.
G. Litvinchuk and I. Spitkovsky, Factorization of measurable matrix functions, in Oper. Theory Adv. Appl. 25, Birkhäuser Verlag, Basel, Switzerland (1987).
S. Mikhlin and S. Prössdorf, Singular integral operators, Springer-Verlag, Berlin, Germany (1986).
M.C. Câmara, A.B. Lebre and F.-O. Speck, Meromorphic factorization, partial index estimates and elastodynamic diffraction problems, Math. Nachr. 157 (1992) 291.
M.C. Câmara, Toeplitz operators and Wiener-Hopf factorisation: an introduction, Concrete Oper. 4 (2017) 130.
D.R. Brill, Electromagnetic fields in a homogeneous, nonisotropic universe, Phys. Rev. 133 (1964) B845.
H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt, Exact solutions of Einstein’s field equations, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, U.K. (2003).
E. Poisson, A relativist’s toolkit: the mathematics of black-hole mechanics, Cambridge University Press, Cambridge, U.K. (2009).
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 1910.10632
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Aniceto, P., Câmara, M., Cardoso, G. et al. Weyl metrics and Wiener-Hopf factorization. J. High Energ. Phys. 2020, 124 (2020). https://doi.org/10.1007/JHEP05(2020)124
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2020)124