Weyl metrics and Wiener-Hopf factorization
- PDF / 2,612,540 Bytes
- 55 Pages / 595.276 x 841.89 pts (A4) Page_size
- 67 Downloads / 153 Views
Springer
Received: March 28, 2020 Accepted: May 3, 2020 Published: May 26, 2020
Weyl metrics and Wiener-Hopf factorization
Center for Mathematical Analysis, Geometry and Dynamical Systems, Department of Mathematics, Instituto Superior T´ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
E-mail: [email protected], [email protected], [email protected], [email protected] 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. Keywords: 2D Gravity, Black Holes, Integrable Field Theories, Sigma Models ArXiv ePrint: 1910.10632
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)124
JHEP05(2020)124
P. Aniceto, M.C. Cˆ amara, G.L. Cardoso and M. Rossell´ o
Contents 1 Introduction
2
2 Summary of the main results
4
3 Preliminary results
8
3.2 3.3
Properties of ϕ ∈ T
9
Contour properties
11
Affine transformations
11
4 The Breitenlohner-Maison linear system
12
5 Monodromy matrix
14
6 Canonical factorization gives a solution to the BM linear system
16
7 Meromorphic factorizations: a case study
23
8 The Schwarzschild monodromy matrix
26
8.1
σ=1
28
8.2
σ = −1
31
8.2.1
Region I
33
8.2.2
Extending solutions: the interior region of the Schwarzschild solution
36
8.2.3
Regions A and B
38
9 The monodromy matrix with = 0
41
9.1
σ=1
42
9.2
σ = −1
43
10 Solutions with two Killing horizons
45
˜>0 A Expressions for P˜ > Q
49
B Gluing solutions along the lines ρ = ±v + m
51
C A-metrics
52
–1–
JHEP05(2020)124
3.1
1
Introduction
where m ∈ R+ , r > 0, 0 < θ < π and 0 ≤ φ < 2π. By restricting to the subspace of solutions that only depend on two of the D spacetime coordinates, various approaches to solving the field equations become available (see for instance [1] for a recent survey thereof). In this paper, we focus on the Riemann-Hilbert
Data Loading...
