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, the second one relies on dimensional reduction from 5D. We examine the geometry of the solutions, their horizons and singularities and compare them.
Grants GAUK 80918, GACR 14-37086G.
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- 1.
The term extremal refers to the fact that the charges of the black holes are equal to their masses, rendering their horizons degenerate.
- 2.
The number of space-like dimensions is denoted as n.
- 3.
Latin indices range over 1, …, n and label only spatial components, 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.
Let \(f_k(x): I \rightarrow \mathbb {C}\) be functions and assume that there exist a k such that \(\left |{f_k(x) - f(x)}\right | \leq a_k\) for ∀x ∈ I and a k → 0 when k →∞. Then \(f_k(x) \rightrightarrows f(x)\) in I.
- 6.
Leibniz theorem: Let \(f_k(x): I \rightarrow \mathbb {C}\) such that \(f_k \rightrightarrows 0\) in I and let f k(x) be monotonous for ∀x ∈ I. Then \(\sum (-1)^k f_k(x)\) converges uniformly in I.
- 7.
Weierstrass M-test: Let \(f_k(x): I \rightarrow \mathbb {C}\) be functions and assume that there exist a k such that \(\left |{f_k(x)}\right | \leq a_k\) and \(\left |{\sum a_k}\right | < \infty \). Then \(\sum f_k(x)\) converges absolutely uniformly in I.
- 8.
The Hurwitz–Lerch transcendent is defined as \(\Phi (z,s,a)=\sum _{k=0}^{\infty } z^k(a+k)^{-s}\).
- 9.
The Hurwitz zeta function, \(\zeta _s(a)=\sum _{k=0}^{\infty } (a+k)^{-s}\), has singularities at a = −n for non-negative integers n.
- 10.
Here \(\bar {\zeta }(s)=\sum _{k=1}^{\infty } k^{-s}\) is the Riemann zeta function.
- 11.
Moore–Osgood: Let f, f n be defined in a punctured neighbourhood of x 0. Let \(f_n \rightrightarrows f\) in I and \(c_n = \lim _{x \rightarrow x_0} f_n(x)\) be finite. Then \(\lim _{x \rightarrow x_0} \lim _{n \rightarrow \infty } f_n(x) = \lim _{n \rightarrow \infty } \lim _{x \rightarrow x_0} f_n(x)\).
- 12.
Equations of motion for electrogeodesics are derived from the Lagrangian \(\mathcal {L} = \frac {1}{2}g_{\mu \nu }\dot {x}^{\mu }\dot {x}^{\nu }+qA_{\mu }\dot {x}^{\mu }\), the dot denotes derivative with respect to affine parameter τ.
- 13.
Formally \(\nabla \varphi _{\neg 0} = (\nabla f) \sum f^{-1} \varphi _n - f \sum \varphi _n f^{-2} \nabla f + f \sum f^{-1} \nabla \varphi _n\). Under suitable assumptions the first two terms cancel each other.
- 14.
Harmonic number \(H(z) = \gamma _e + \frac {\mathrm {d}{(\ln \Gamma (z+1))}}{\mathrm {d}{z}}\), where γ e is the Euler gamma constant and Γ is the Euler gamma function.
- 15.
Capital Latin indices range over 0, …, D + 1, Greek indices are 0, …, D.
- 16.
denotes the d’Alembertian of a Lorentzian metric g, which is defined as 
.
denotes the d’Alembertian of a Lorentzian metric g, which is defined as 
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Acknowledgements
J.R. was supported by grant No. 80918 of Charles University Grant Agency. M.Ž. Acknowledges support by GACR 17-13525S.
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Ryzner, J., Žofka, M. (2019). Crystal Spacetimes with Discrete Translational Symmetry. In: Cacciatori, S., Güneysu, B., Pigola, S. (eds) Einstein Equations: Physical and Mathematical Aspects of General Relativity. DOMOSCHOOL 2018. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-18061-4_10
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