A new approach to static numerical relativity and its application to Kaluza–Klein black holes

By M. Headrick, S. Kitchen, T. Wiseman. 2009

248 citations recorded by the source corpus.

We propose a framework for solving the Einstein equation for static and Euclidean metrics. First, we address the issue of gauge-fixing by borrowing from the Ricci-flow literature the so-called DeTurck trick, which renders the Einstein equation strictly elliptic and generalizes the usual harmonic-coordinate gauge. We then study two algorithms, Ricci-flow and Newton's method, for solving the resulting Einstein–DeTurck equation. We illustrate the use of these methods by studying localized black holes and non-uniform black strings in five-dimensional Kaluza–Klein theory, improving on previous calculations of their thermodynamic and geometric properties. We study spectra of various operators for these solutions, in particular finding the negative modes of the Lichnerowicz operator. We classify the localized solutions into two branches that meet at a minimum temperature. We find good evidence for a merger between the localized and non-uniform solutions. We also find a narrow window of localized solutions that possess negative modes yet have positive specific heat.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/bfd35e778d64158e165f567c0d3690a4beb75eff

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Citations in the corpus

The thermoelectric properties of inhomogeneous holographic lattices (2014, methodology): “In section 4, following the approach of [33], we describe the numerical methodology that we employ to solve the PDEs which leads to the holographic lattice black holes.”
Holographic lattice in Einstein-Maxwell-dilaton gravity (2013, methodology): “3), are solved using the Einstein-DeTurck method [41].”
Further evidence for lattice-induced scaling (2012, methodology): “5) using the Einstein-DeTurck method described in [20].”
Entanglement islands in higher dimensions (2019, methodology): “1 The DeTurck trick To find solutions, we will use the so-called DeTurck trick, which was first proposed in [22], and reviewed extensively in [23,24].”
General relativity and the cuprates (2013, methodology): “ck hole temperature given by T G(1)=4ˇ= (6 2 1 )=8ˇand H 1;2, S 1;2, F and are six functions of xand z, that we determine using the numerical methods described in [1], which were rst introduced in [19] and studied in great detail in [20]. When A 0 = 0 the problem reduces to the original holographic”
Optical conductivity with holographic lattices (2012, methodology): “To overcome this difficulty we will use the DeTurck method, which was first outlined in [13].”
Ricci solitons, Ricci flow and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua (2011, methodology): “Following [1] we have considered static solutions to the EinsteinDeTurck equation by Euclidean continuation to a Riemannian problem which is elliptic.”