Abstract
The aim of this work is to construct exact solutions of Einstein–Maxwell(-dilaton) equations possessing a discrete translational symmetry. We present two approaches to the problem. The first one is to solve Einstein–Maxwell equations in 4D, and the second one relies on dimensional reduction from 5D. We examine the geometry of the solutions, their horizons and singularities and compare them.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The term extremal refers to the fact that the black hole horizons are degenerate, which in this case translates to their charges equal to their masses.
- 2.
The number of spacelike dimensions is denoted as n.
- 3.
Latin indices range over 1, …, n and label only spatial components, and Greek indices are 0, …, n.
- 4.
The Laplacian for the spatial metric h is defined as \(\Delta _h f \equiv h^{ij} \nabla _{i} \nabla _{j} f = \frac {1}{\sqrt {\mathfrak {h}}} \left (\sqrt {\mathfrak {h}} h^{ij} f_{i}\right )_{,j}\).
- 5.
Dirac comb is a periodic tempered distribution defined as
- 6.
Modified Bessel function of the second kind Kν is defined as
$$\displaystyle \begin{aligned} K_\nu \left(x\right) \equiv \int_{0}^{\infty} \exp \left(-x \text{cosh} t \right) \text{cosh}\left(\nu t\right) \mathrm{d}{t}, x > 0. \end{aligned}$$ - 7.
From the Newton integral test, we get the following inequality for integrable, non-increasing and non-negative Cl:
$$\displaystyle \begin{aligned} C_l \geq 0, \frac{\partial{C_l}}{\partial{l}} \leq 0 \, \forall l \geq 1, \sum_{l=1}^{\infty} C_l < \infty \Rightarrow \int_{1}^{\infty} C_l \mathrm{d}{l} \leq \sum_{l=1}^{\infty} C_l \leq C_1 + \int_{1}^{\infty} C_l \mathrm{d}{l}. \end{aligned}$$ - 8.
For simple binary MP black holes in 5D, the horizon is not smooth. Thanks to the alignment of all black holes in the crystal, the horizons of individual black holes are smooth.
References
S.D. Majumdar, A class of exact solutions of Einstein’s field equations. Phys. Rev. 72, 390–398 (1947)
A. Papaetrou, A static solution of the equations of the gravitational field for an arbitrary charge distribution. Proc. Roy. Irish Acad. (Sect. A) A51, 191–204 (1947)
J.B. Hartle, S.W. Hawking, Solutions of the Einstein-Maxwell equations with many black holes. Commun. Math. Phys. 26(2), 87–101 (1972)
J. Ryzner, M. Žofka, Electrogeodesics in the di-hole Majumdar-Papapetrou spacetime. Classical Quantum Gravity 32(20), 205010 (2015)
J. Ryzner, M. Žofka, Extremally charged line. Classical Quantum Gravity 33(24), 245005 (2016)
J. Ryzner, M. Žofka, Einstein Equations: Physical and Mathematical Aspects of General Relativity (Springer-Verlag GmbH, Berlin, 2019)
J.P.S. Lemos, V.T. Zanchin, Class of exact solutions of Einstein’s field equations in higher dimensional spacetimes, d ≥ 4: Majumdar-Papapetrou solutions. Phys. Rev. D 71, 124021 (2005)
D.L. Welch, On the smoothness of the horizons of multi-black-hole solutions. Phys. Rev. D 52, 985–991 (1995)
V.P. Frolov, A. Zelnikov, Scalar and electromagnetic fields of static sources in higher dimensional Majumdar-Papapetrou spacetimes. Phys. Rev. D 85, 064032 (2012)
R.C. Myers, Higher-dimensional black holes in compactified space-times. Phys. Rev. D 35, 455–466 (1987)
Acknowledgements
J.R. was supported by grant GAUK 80918. M.Ž. acknowledges support by GACR 17-13525S.
Grants GAUK 80918, GACR 17-13525S.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Ryzner, J., Žofka, M. (2022). Exact Solutions of Einstein–Maxwell(-Dilaton) Equations with Discrete Translational Symmetry. In: Cacciatori, S.L., Kamenshchik, A. (eds) Einstein Equations: Local Energy, Self-Force, and Fields in General Relativity. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-21845-3_10
Download citation
DOI: https://doi.org/10.1007/978-3-031-21845-3_10
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-031-21844-6
Online ISBN: 978-3-031-21845-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)