[Paper Review] Compactified black holes in five-dimensional U(1)**3 ungauged supergravity

Shinya Tomizawa|arXiv (Cornell University)|Sep 18, 2010

Black Holes and Theoretical Physics3 citations

TL;DR

This paper constructs stationary, nonextremal three-charge rotating black hole solutions in five-dimensional U(1)³ ungauged supergravity with Kaluza-Klein asymptotics, using a squashing transformation to compactify the fifth dimension. The resulting spacetime behaves as four-dimensional Minkowski space at infinity while retaining five-dimensional structure near the horizon, yielding a new class of Kaluza-Klein black holes with three electric charges and angular momentum along the compactified dimension.

ABSTRACT

We present stationary, nonextremal three charge rotating black hole solutions in the five-dimensional U(1)^3 ungauged supergravity. At infinity, our solutions behave as a four-dimensional flat spacetime with a compact extra-dimension and hence describe spherical black holes with Kaluza-Klein asymptotics.

Motivation & Objective

• To construct stationary, nonextremal black hole solutions in five-dimensional U(1)³ ungauged supergravity with compactified extra dimensions.
• To model black holes that asymptotically resemble four-dimensional Minkowski spacetime with a compact fifth dimension, while exhibiting five-dimensional geometry near the horizon.
• To explore the solution space of higher-dimensional black holes under Kaluza-Klein compactification, relevant for braneworld and ADD models.
• To provide a framework for understanding extra-dimensional stabilization in string theory through explicit black hole solutions.

Proposed method

• Apply a squashing transformation to the five-dimensional Cvetič-Youm solutions with equal angular momenta, deforming the S³ sections by altering the ratio of the S¹ fiber and S² base radii.
• Define the metric using left-invariant 1-forms σ₁, σ₂, σ₃ on S³, with a radial function k(r) that modifies the S³ geometry and ensures asymptotic Kaluza-Klein structure.
• Construct the gauge potentials Aⁱ and scalar fields Xᵢ using parameters μ, δᵢ (related to charges), and l (related to angular momentum), preserving the action's invariance under the deformation.
• Derive the metric and field equations by substituting the deformed forms into the Einstein equations and solving the resulting ordinary differential equation for k(r).
• Perform coordinate transformations to analyze the asymptotic behavior, showing that at r → r∞, the spacetime approaches a four-dimensional Minkowski spacetime fibered over an S¹.
• Compute conserved charges (mass, angular momentum, electric charges) via boundary integrals at spatial infinity, expressing them in terms of parameters μ, δᵢ, l, and r∞.

Experimental results

Research questions

• RQ1 How can stationary, rotating black hole solutions in five-dimensional U(1)³ ungauged supergravity be compactified to exhibit Kaluza-Klein asymptotics with a single compactified dimension?
• RQ2 What is the role of the squashing transformation in modifying the S³ geometry of the Cvetič-Youm solution to yield a Kaluza-Klein black hole?
• RQ3 How do the conserved charges (mass, angular momentum, electric charges) depend on the deformation parameters μ, δᵢ, l, and r∞ in the asymptotically Kaluza-Klein limit?
• RQ4 What limits of the solution reproduce known solutions such as the minimal supergravity black hole or asymptotically flat Cvetič-Youm solutions?
• RQ5 What is the structure of the extremal limit, and does it yield known extremal black hole solutions in Taub-NUT space?

Key findings

• The squashing transformation successfully generates a new class of nonextremal Kaluza-Klein black hole solutions with three electric charges and angular momentum along the compactified fifth dimension.
• At spatial infinity (r → r∞), the metric asymptotes to a four-dimensional Minkowski spacetime with a compact S¹ fiber, confirming Kaluza-Klein behavior.
• The mass, electric charges Qᵢ, and angular momentum Jw are explicitly computed in terms of μ, δᵢ, l, and r∞, with Jϕ = 0 due to symmetry.
• In the limit r∞ → ∞, the solution reduces to the five-dimensional asymptotically flat Cvetič-Youm solution with equal angular momenta.
• In the limit μ → 0, l → 0, δᵢ → -∞ with μsᵢcᵢ and l(cᵢ - sᵢ) finite, the solution reduces to the extremal three-charge black hole in Taub-NUT space, matching known results.
• The solution reproduces the nonextremal Kaluza-Klein black hole in five-dimensional minimal supergravity when δ₁ = δ₂ = δ₃, confirming consistency with known limits.