arXiv subjects

P. Wiederhold

Publications and source records attributed to P. Wiederhold.

3 recordsLinked to original sources

arXiv2026

Exact lambdavacuum solutions in higher dimensions

In this work, we obtain exact solutions to the $(n+2)$-dimensional Einstein Field Equations with a non-zero cosmological constant for $n > 1$. These solutions depend on a set $\{ A_a, a=1,2,\ldots , m \}$ of pairwise commuting constant matrices in $\mathfrak{sl} ( n, \mathbb{R} )$ and on a constant matrix $g_0$ in $\mathcal{I} (\{ A_a, a=1,\ldots , m \})$, determined in previous work. Different choices of $\{ A_a, a=1,\ldots , m \}$ and $g_0$ correspond to different solutions. As examples, we show how to obtain the de Sitter metric, the Anti-de Sitter metric, the Birmingham metric, the Nariai metric and the Anti-Nariai metric in higher dimensions. The generalized Nariai and Anti-Nariai solutions are direct topological products of $AdS_{\frac{n}{2} + 1} \times H^{\frac{n}{2} + 1}$, $dS_{\frac{n}{2} + 1} \times S^{\frac{n}{2} + 1}$, $AdS_2 \times H^n$, $AdS_n \times H^2$, $dS_2 \times S^n$ and $dS_n \times S^2$. In addition, we study a solution in the context of cosmology.

gr-qc↗

arXiv2026

Flat subspaces of the $SL(n,\mathbb{R})$ chiral equations

In this work, we introduce a method for finding exact solutions to the vacuum Einstein field equations in higher dimensions from a given solution to the chiral equation. When considering a $n + 2$-dimensional spacetime with $n$ commutative Killing vectors, the metric tensor can take the form $\hat g = f ( ρ, ζ) ( d ρ^2 + d ζ^2 ) + g_{μν} ( ρ, ζ) d x^μd x^ν$. Then, the Einstein field equations in vacuum reduce to a chiral equation, $( ρg_{, z} g ^{-1} )_{, \bar z} + ( ρg_{, \bar z} g ^{-1} )_{, z} = 0$, and two differential equations, $( \ln f ρ^{1-1/n} )_{, Z} = \fracρ{2} \operatorname{tr} ( g_{, _Z} g^{-1} )^2$, where $g \in SL( n, \mathbb{R} )$ is the normalized matrix representation of $g_{μν}$, $z = ρ+ i ζ$ and $Z = z, \bar z$. We use the ansatz $g = g ( ξ^a )$, where the parameters $ξ^a$ depend on $z$ and $\bar z$ and satisfy a generalized Laplace equation, $( ρξ^a _{, z} )_{, \bar z} + ( ρξ^a _{, \bar z} )_{, z} = 0$. The chiral equation to the Killing equation, $A_{a , ξ^b} + A_{b , ξ^a} = 0$, where $A_a = g_{, ξ^a} g^{-1}$. Furthermore, we assume that the matrices $A_a$ commute with each other; in this way, they fulfill the Killing equation.

gr-qc↗

arXiv2016

Relative Convex Hull Determination from Convex Hulls in the Plane

A new algorithm for the determination of the relative convex hull in the plane of a simple polygon A with respect to another simple polygon B which contains A, is proposed. The relative convex hull is also known as geodesic convex hull, and the problem of its determination in the plane is equivalent to find the shortest curve among all Jordan curves lying in the difference set of B and A and encircling A. Algorithms solving this problem known from Computational Geometry are based on the triangulation or similar decomposition of that difference set. The algorithm presented here does not use such decomposition, but it supposes that A and B are given as ordered sequences of vertices. The algorithm is based on convex hull calculations of A and B and of smaller polygons and polylines, it produces the output list of vertices of the relative convex hull from the sequence of vertices of the convex hull of A.

cs.CG↗