[Paper Review] Higher dimensional vacuum $p-$brane solutions with trans-spherical symmetry

Jungjai Lee, G. Kang+1 more|arXiv (Cornell University)|Jan 3, 2008

Black Holes and Theoretical Physics1 citations

TL;DR

This paper studies higher-dimensional vacuum p-brane solutions in D=n+p+3 spacetime dimensions with trans-spherical symmetry, showing that geometric properties like the area of S^{n+1} and total spatial volume depend only on three parameters: mass density, sum of tension densities, and sum of squared tension densities—not individual tension values. The causal structure mirrors the 5D case, with naked singularities appearing except for Schwarzschild p-branes and Kaluza-Klein bubbles.

ABSTRACT

We investigate the geometrical properties of static vacuum $p$-brane solutions of Einstein gravity in $D=n+p+3$ dimensions, which have spherical symmetry of $S^{n+1}$ orthogonal to the $p$-directions and are invariant under the translation along them. % The solutions are characterized by mass density and $p$ tension densities. % The causal structure of the higher dimensional solutions is essentially the same as that of the five dimensional ones. Namely, a naked singularity appears for most solutions except for the Schwarzschild black $p$-brane and the Kaluza-Klein bubble. % We show that some important geometric properties such as the area of $S^{n+1}$ and the total spatial volume are characterized only by the three parameters such as the mass density, the sum of tension densities and the sum of tension density squares rather than individual tension densities. These geometric properties are analyzed in detail in this parameter space and are compared with those of 5-dimensional case.

Motivation & Objective

• To analyze the geometric and causal structure of static vacuum p-brane solutions in higher-dimensional Einstein gravity.
• To identify which geometric quantities depend on collective tension parameters rather than individual tension densities.
• To generalize insights from 5D p-brane solutions to higher dimensions (D=n+p+3).
• To characterize the spatial volume and S^{n+1} area using only three key parameters: mass density, sum of tension densities, and sum of squared tension densities.

Proposed method

• Constructing static, translation-invariant p-brane solutions in D=n+p+3 dimensions with orthogonal S^{n+1} symmetry.
• Applying the Einstein equations to derive the metric structure under the assumption of trans-spherical symmetry.
• Parameterizing the solution using mass density and tension densities along the p-dimensions.
• Reducing geometric observables (e.g., S^{n+1} area, spatial volume) to functions of three aggregate parameters: mass density, sum of tension densities, and sum of squared tension densities.
• Analyzing the causal structure via the behavior of the metric components and curvature invariants.
• Comparing the higher-dimensional results with the well-known 5D case to identify structural similarities and differences.

Experimental results

Research questions

• RQ1 How do geometric properties like the area of S^{n+1} and total spatial volume depend on the individual tension densities in higher-dimensional p-brane solutions?
• RQ2 What is the role of collective parameters—sum of tension densities and sum of squared tension densities—in determining the geometry of p-brane solutions?
• RQ3 Does the causal structure of higher-dimensional p-brane solutions remain similar to that of 5D solutions, particularly regarding the presence of naked singularities?
• RQ4 Can the full geometric behavior of the p-brane be captured using only three parameters: mass density, sum of tension densities, and sum of squared tension densities?
• RQ5 How do the geometric features of higher-dimensional p-branes compare quantitatively with those in the 5-dimensional case?

Key findings

• The area of the S^{n+1} sphere and the total spatial volume of the p-brane solution depend solely on the mass density, the sum of tension densities, and the sum of squared tension densities, not on individual tension values.
• The causal structure of the higher-dimensional solutions is qualitatively identical to the 5D case, with naked singularities present in most configurations except for the Schwarzschild p-brane and Kaluza-Klein bubble.
• The geometric properties are fully characterized in a three-parameter space defined by mass density, sum of tension densities, and sum of squared tension densities.
• The parameter space analysis reveals that certain geometric behaviors observed in 5D p-branes extend to higher dimensions, indicating structural universality.
• The results confirm that individual tension densities do not independently affect key geometric observables, simplifying the classification of p-brane solutions.