[Paper Review] Searching for Localized Black-Hole solutions in Brane-World models

Θεόδωρος Νάκας|arXiv (Cornell University)|Jul 26, 2017

Black Holes and Theoretical Physics36 references3 citations

TL;DR

This thesis investigates the existence of localized 5-dimensional black hole solutions in brane-world models, using a generalized Vaidya metric with time- and extra-dimensional (y)-dependent mass parameters. Despite incorporating two non-minimally coupled scalar fields, the study finds no viable 5D localized black hole solutions, suggesting potential incompatibility between brane-world models and key predictions of General Relativity.

ABSTRACT

In the context of this thesis, the question that is going to occupy us, is the existence of a 5-dimensional braneworld black hole solution that is localized close to the 3-brane and has the properties of a regular 4-dimensional one. For this purpose, the 4-dimensional part of the complete 5-dimensional spacetime is considered to be a generalized Vaidya metric, in the context of which, the mass parameter $m$ is allowed to vary with respect to time, while it is also allowed to have both $y$ and $r$ dependence. The dependence on the $r$-coordinate essentially means that our black hole solution can deviate from the conventional Schwarzschild solution. Additionally, the dependence on the $y$-coordinate leads to a non-trivial profile of the black hole along the extra dimension. In order to justify physically the existence of such general mass parameter, we consider the case of two scalar fields $ϕ(v,r,y)$, $χ(v,r,y)$ which interact with each other and they are also non-minimally coupled to gravity via a general coupling function $f(ϕ,χ)$. In all the cases that were investigated in the context of this particular scenario, the result for the existence of a viable 5-dimensional localized black hole solution was negative, a result that causes concern about the compatibility of brane-world models with basic predictions of General Theory of Relativity.

Motivation & Objective

• To explore whether 5-dimensional braneworld black hole solutions can be localized on a 3-brane while retaining 4D-like properties.
• To analyze the viability of such solutions under a generalized Vaidya metric with time- and y-dependent mass parameters.
• To assess the physical consistency of these solutions by introducing two interacting, non-minimally coupled scalar fields.
• To test whether the inclusion of scalar fields with general coupling functions f(ϕ,χ) enables stable, localized black hole solutions.
• To determine whether brane-world models can accommodate realistic black hole physics consistent with 4D General Relativity.

Proposed method

• The 5D spacetime geometry is modeled using a generalized Vaidya metric, allowing the mass parameter m to depend on both time (v) and the extra dimension (y).
• The mass parameter's dependence on r introduces deviations from the Schwarzschild solution, while y-dependence enables non-trivial localization along the bulk.
• Two scalar fields ϕ(v,r,y) and χ(v,r,y) are introduced, interacting with each other and non-minimally coupled to gravity via a general function f(ϕ,χ).
• The field equations are derived from a modified action with non-minimal coupling, and the energy-momentum tensor is computed for consistency checks.
• The analysis includes explicit computation of Christoffel symbols, Riemann and Ricci tensors, and Einstein tensor in Gaussian normal coordinates.
• Mathematica-based symbolic computations are used to evaluate components of the energy-momentum tensor and verify vanishing off-diagonal terms.

Experimental results

Research questions

• RQ1 Can a 5-dimensional black hole solution be localized on a 3-brane while preserving 4D-like gravitational behavior?
• RQ2 Does a time- and y-dependent mass parameter in a generalized Vaidya metric allow for a physically viable localized black hole?
• RQ3 Can two interacting, non-minimally coupled scalar fields support a stable, localized black hole in a braneworld scenario?
• RQ4 Is there a consistent solution where the 5D Einstein equations admit a localized black hole with finite mass and proper horizon structure?
• RQ5 Does the absence of such solutions imply a fundamental incompatibility between braneworld models and standard General Relativity predictions?

Key findings

• No viable 5-dimensional localized black hole solution was found in any of the investigated cases involving two interacting scalar fields and non-minimal coupling.
• The energy-momentum tensor components Tμν were found to vanish for all off-diagonal components, indicating no anisotropic stress or non-vacuum contributions.
• All components of the energy-momentum tensor were systematically evaluated and shown to vanish due to vanishing partial derivatives and Christoffel symbols involving y-derivatives.
• The analysis of the field equations and metric components consistently led to Tμν = 0 for all μ,ν, indicating no physical matter content to support a localized black hole.
• The negative result holds across all tested coupling functions f(ϕ,χ), including linear forms such as f(ϕ,χ) = aϕ + bχ, which were explicitly rejected due to inconsistency.
• The study concludes that brane-world models may not support localized black holes consistent with 4D General Relativity, raising concerns about their physical viability.