Euclidean Schwarzschild Research Articles

Coupling a point-like mass to quantum gravity with causal dynamical triangulations

We present a possibility of coupling a point-like, non-singular, mass distribution to four-dimensional quantum gravity in the nonperturbative setting of causal dynamical triangulations (CDT). In order to provide a point of comparison for the classical limit of the matter-coupled CDT model, we derive the spatial volume profile of the Euclidean Schwarzschild–de Sitter space glued to an interior matter solution. The volume profile is calculated with respect to a specific proper-time foliation matching the global time slicing present in CDT. It deviates in a characteristic manner from that of the pure-gravity model. The appearance of coordinate caustics and the compactness of the mass distribution in lattice units put an upper bound on the total mass for which these calculations are expected to be valid. We also discuss some of the implementation details for numerically measuring the expectation value of the volume profiles in the framework of CDT when coupled appropriately to the matter source.

A (running) bolt for new reasons

We construct a four-parameter family of smooth, horizonless, stationary solutions of ungauged five-dimensional supergravity by using the four-dimensional Euclidean Schwarzschild metric as a base space and "magnetizing" its bolt. We then generalize this to a five-parameter family based upon the Euclidean Kerr-Taub-Bolt. These "running Bolt" solutions are necessarily non-static. They also have the same charges and mass as a non-extremal black hole with a classically-large horizon area. Moreover, in a certain regime their mass can decrease as their charges increase. The existence of these solutions supports the idea that the singularities of non-extremal black holes are resolved by low-mass modes that correct the singularity of the classical black hole solution on large (horizon-sized) scales.

New self-dual solutions ofSU(2)Yang-Mills theory in Euclidean Schwarzschild space

We present a systematic study of spherically symmetric self-dual solutions of SU(2) Yang-Mills theory on Euclidean Schwarzschild space. All the previously known solutions are recovered and a new one-parameter family of instantons is obtained. The newly found solutions have continuous actions and interpolate between the classic Charap and Duff instantons. We examine the physical properties of this family and show that it consists of dyons of unit (magnetic and electric) charge.

First-order action and Euclidean quantum gravity

We show that the on-shell path integral for asymptotically flat Euclidean spacetimes can be given in the first-order formulation of general relativity, without assuming the boundary to be isometrically embedded in Euclidean space and without adding infinite counter-terms. For illustrative examples of our approach, we evaluate the first-order action for the four-dimensional Euclidean Schwarzschild and NUT-charged spacetimes to derive the corresponding on-shell partition functions, and show that the correct thermodynamic quantities for the solutions are reproduced.

THE MASTER EQUATIONS IN THE EUCLIDEAN SCHWARZSCHILD–TANGHERLINI METRIC OF A SMALL STATIC PERTURBATION

The master equations in the Euclidean Schwarzschild–Tangherlini space–time of a small static perturbation are studied. For each harmonic mode on the sphere there are two solutions that behave differently at infinity. One solution goes like the power 2-l-n of the radial variable, the other solution goes like the power l. These solutions occur in power series. The second main statement of the paper is that any eigentensor of the Lichnerowicz operator in a Euclidean Schwarzschild space–time with an eigenvalue different from zero is essentially singular at infinity. Possible applications of the stability of instantons are discussed. We present the analysis of a small static perturbation of the Euclidean Schwarzschild–Tangherlini metric tensor. The higher order perturbations will appear later. We determine independently the static perturbations of the Schwarzschild quantum black hole in dimension 1+n≥4, where the system of equations is reduced to master equations — ordinary differential equations. The solutions are hypergeometric functions which in some cases can be reduced to polynomials. In the same Schwarzschild background, we analyze static perturbations of the scalar mode and show that there does not exist any static perturbation that is regular everywhere outside the event horizon and is well-behaved at the spatial infinity. This confirms the uniqueness of the spherically symmetric static empty quantum black hole, within the perturbation framework. Our strategy for treating the stability problem is also applicable to other symmetric quantum black holes with a nonzero cosmological constant.

Eigentensors of the Lichnerowicz operator in Euclidean Schwarzschild metrics*

AbstractProperties of the eigentensors of the Lichnerowicz Laplacian for the Euclidean Schwarzschild metric are discussed together with possible applications to the linear stability of higher‐dimensional instantons. The main statement of the article is that any eigentensor of the Lichnerowicz operator in a Euclidean (possibly higher‐dimensional) Schwarzschild metric is essentially singular at infinity.

Concerning the state-change due to the measurement process

Properties of the eigentensors of the Lichnerowicz Laplacian for the Euclidean Schwarzschild metric are discussed together with possible applications to the linear stability of higher-dimensional instantons. The main statement of the article is that any eigentensor of the Lichnerowicz operator in a Euclidean (possibly higher-dimensional) Schwarzschild metric is essentially singular at infinity.

Eigentensors of the Lichnerowicz operator in Euclidean Schwarzschild metrics

Properties of the eigentensors of the Lichnerowicz Laplacian for the Euclidean Schwarzschild metric are discussed together with possible applications to the linear stability of higher-dimensional instantons. The main statement of the article is that any eigentensor of the Lichnerowicz operator in a Euclidean (possibly higher-dimensional) Schwarzschild metric is essentially singular at infinity.

Yang-Mills solutions on Euclidean Schwarzschild space

We show that the apparently periodic Charap-Duff Yang-Mills `instantons' in time-compactified Euclidean Schwarzschild space are actually time independent. For these solutions, the Yang-Mills potential is constant along the time direction (no barrier) and therefore, there is no tunneling. We also demonstrate that the solutions found to date are three dimensional monopoles and dyons. We conjecture that there are no time-dependent solutions in the Euclidean Schwarzschild background.

Geometric interpretation of Schwarzschild instantons

We address the problem of finding Abelian instantons of finite energy on the Euclidean Schwarzschild manifold. This amounts to construct self-dual L 2 harmonic 2-forms on the space. Gibbons found a non-topological L 2 harmonic form in the Taub-NUT metric, leading to Abelian instantons with continuous energy. We imitate his construction in the case of the Euclidean Schwarzschild manifold and find a non-topological self-dual L 2 harmonic 2-form on it. We show how this gives rise to Abelian instantons and identify them with SU(2)-instantons of Pontryagin number 2 n 2 found by Charap and Duff in 1977. Using results of Dodziuk and Hitchin we also calculate the full L 2 harmonic space for the Euclidean Schwarzschild manifold.

Kaluza - Klein electrically charged black branes in M-theory

We present a class of Kaluza - Klein electrically charged black p-brane solutions of 10-dimensional, type IIA superstring theory. Uplifting to 11 dimensions these solutions are studied in the context of M-theory. They can be interpreted either as a p + 1 extended object trapped around the 11th dimension along which momentum is flowing or as a boost of the following backgrounds: the Schwarzschild black (p + 1)-brane or the product of the (10-p)-dimensional Euclidean Schwarzschild manifold with the (p + 1)-dimensional Minkowski spacetime.

Euclidean black-hole vortices.

We argue the existence of solutions of the Euclidean Einstein equations that correspond to a vortex sitting at the horizon of a black hole. We find the asymptotic behaviours, at the horizon and at infinity, of vortex solutions for the gauge and scalar fields in an abelian Higgs model on a Euclidean Schwarzschild background and interpolate between them by integrating the equations numerically. Calculating the backreaction shows that the effect of the vortex is to cut a slice out of the Euclidean Schwarzschild geometry. Consequences of these solutions for black hole thermodynamics are discussed.

Euclidean Schwarzschild negative mode

The Euclidean (imaginary time) Schwarzschild solution of general relativity is known to possess a spin-2 metric perturbation which decreases its Euclidean action. This ''negative mode'' contributes an imaginary part to the effective action, and renders hot flat space unstable against the nucleation of black holes. In this paper, we enclose the black hole in a spherical ''box'' by imposing boundary conditions on the perturbations. Two conditions, which correspond to a fixed temperature (isothermal wall) and fixed energy (reflecting wall) are examined. The isothermal boundary condition eliminates the negative mode if the box is small enough, and stabilizes hot flat space.

Quaternion analyticity and conformally K�hlerian structure in Euclidean gravity

Starting from the fact that the d=4 Euclidean flat spacetime is conformally related to the Kahler manifold H2×S2, we show the Euclidean Schwarzschild metric to be conformally related to another Kahler manifold M2×S2 with M2 being conformal to H2 in two dimensions. Both metrics which are conformally Kahlerian, are form-invariant under the infinite parameter Fueter group, the Euclidean counterpart of Milne's group of clock regraduation. The associated Einstein's equations translate into Fueter's quaternionic analyticity. The latter leads to an infinite number of local continuity equations.

Black-hole uniqueness theorems in Euclidean quantum gravity

The Euclidean section of the classical Lorentzian black-hole solutions has been used in approximating the functional integral in the Euclidean path-integral approach to quantum gravity. In this paper the claim that classical black-hole uniqueness theorems apply to the Euclidean section is disproved. In particular, it is shown that although a Euclidean version of Israel's theorem does provide a type of uniqueness theorem for the Euclidean Schwarzschild solution, a Euclidean version of Robinson's theorem does not allow one to form conclusions about the uniqueness of the Euclidean Kerr solution. Despite the failure of uniqueness theorems, ''no-hair'' theorems are shown to exist. Implications are discussed. A precise mathematical statement of the Euclidean black-hole uniqueness conjecture is made and the proof, left as an unsolved problem in Riemannian geometry.

Background metric in supergravity theories

In supergravity theories, we investigate the conformal anomaly of the path-integral determinant and the problem-of fermion zero modes in the presence of a nontrivial background metric. Except in SO(3)-invariant supergravity, there are nonvanishing conformal anomalies. As a consequence, amplitudes around the nontrivial background metric contain unpredictable arbitrariness. The fermion zero modes which are explicitly constructed for the Euclidean Schwarzschild metric are interpreted as an indication of the supersymmetric multiplet structure of a black hole. The degree of degeneracy of a black hole is ${2}^{4n}$ in $\mathrm{SO}(n)$ supergravity.