Properties of generalized Schwarzschild spacetimes with extra dimensions

Technical Summary: Properties of Generalized Schwarzschild Spacetimes with Extra Dimensions

Problem Statement
The paper investigates vacuum solutions to the Einstein field equations in a $1+3+n$ dimensional spacetime. The study focuses on a specific symmetry ansatz: spherical symmetry in the three spatial dimensions ($SO(3)$) and Euclidean symmetry in $n$ extra dimensions ($E(n)$). While higher-dimensional generalizations of the Schwarzschild solution (such as the Tangherlini metric) and rotating Myers–Perry black holes are well-known, this work seeks to identify and characterize a specific class of static solutions where the metric components are parametrized by powers of a single radial function $f(r)$. The goal is to determine if such an ansatz yields only the trivial extension of the Schwarzschild solution or if it reveals non-trivial geometries with distinct physical properties, particularly regarding horizons, singularities, and mass definitions.

Methodology
The author employs a metric ansatz of the form:
$$ds^2 = -f^\alpha dt^2 + f^\beta dr^2 + r^2 d\Omega_{(2)}^2 + f^\gamma \delta_{AB} d\zeta^A d\zeta^B$$
where $\alpha, \beta, \gamma$ are constants and $f = f(r)$. The Ricci tensor components are calculated and set to zero to satisfy the vacuum condition. This reduces the problem to a system of algebraic equations for the constants $\alpha, \beta, \gamma$.

The analysis proceeds by:

1. Solving the differential equations derived from the vanishing Ricci tensor to find the functional form of $f(r)$.
2. Identifying two distinct classes of solutions: a trivial case (where extra dimensions are flat and decoupled) and a non-trivial case (where extra dimensions scale with the radial coordinate).
3. Analyzing the physical properties of these solutions using several invariants and mass definitions:
   • Kretschmann Scalar ($K$): To locate curvature singularities.
   • Newtonian Limit: To define the Newtonian mass ($M_{New.}$) and examine the gravitational potential.
   • Komar Mass ($M_{Kom.}$): Calculated via the integral of the dual of the derivative of the timelike Killing vector.
   • Pseudotensor Masses: Einstein ($M_{Ein.}$) and Landau–Lifshitz ($M_{L.,L.}$) masses derived from their respective stress-energy pseudotensors.
   • ADM Mass ($M_{ADM}$): Calculated via the asymptotic behavior of the spatial metric on a hypersurface.

Key Contributions and Results
The analysis yields two primary classes of non-Minkowski vacuum solutions:

1. The Trivial Class (Metric 2.10):

   • This represents a direct product of the standard 4D Schwarzschild spacetime and a flat $n$-dimensional Euclidean space.
   • The metric components for the extra dimensions are constant ($g_{AB} = \delta_{AB}$).
   • All definitions of mass (Newtonian, Komar, Einstein, Landau–Lifshitz, and ADM) coincide.
   • If the integration constant $a$ is positive, the solution describes a naked singularity with negative mass in the Newtonian limit. If $a$ is negative, it describes a black string/ring with a horizon but no physical singularity on the horizon.

2. The Non-Trivial Class (Metric 2.11):

   • This solution features a non-trivial dependence of the extra-dimensional metric components on the radial coordinate. It corresponds to a stack of uniform $n$-branes or Kaluza–Klein bubbles.
   • Singularities: For $n \ge 2$, the Kretschmann scalar diverges at the horizon ($r = -a$) if the constant $a$ is negative. This indicates a physical singularity located on the horizon, distinguishing it from the standard Schwarzschild black hole where the horizon is regular.
   • Mass Discrepancy: A critical finding is the divergence between different mass definitions:
     • The Newtonian mass ($M_{New.}$) and Komar mass ($M_{Kom.}$) are negative (for $a < 0$), implying repulsive gravity in the Newtonian limit.
     • The Einstein, Landau–Lifshitz, and ADM masses are positive.
     • Specifically, $M_{Ein.} = M_{L.,L.} = -\frac{1}{n+1} \frac{a}{2\kappa} L^n$ and $M_{ADM} = -\frac{2n+1}{n+1} \frac{a}{2\kappa} L^n$, while $M_{New.} L^n = M_{Kom.} = \frac{n-1}{n+1} \frac{a}{2\kappa} L^n$.
   • Extra Dimension Behavior: In this non-trivial case, the physical size of the extra dimensions collapses to zero at the horizon and diverges at the central singularity.

Significance and Claims
The paper claims that the specific ansatz of a single radial function parametrizing the metric leads to a limited set of vacuum solutions, identifiable as Weyl solutions. The primary significance lies in the characterization of the non-trivial class of solutions:

• Differentiation from Trivial Extensions: The non-trivial solutions are fundamentally different from simple higher-dimensional extensions of Schwarzschild. They exhibit a physical singularity on the horizon (for $n \ge 2$), which corresponds to Kaluza–Klein bubbles.
• Mass Inconsistency: The existence of multiple, conflicting mass definitions (positive ADM/Einstein masses vs. negative Newtonian/Komar masses) highlights the complex nature of energy localization in these spacetimes. This discrepancy is linked to the different metric components ($g_{00}$ vs. $g_{rr}$ and $g_{AB}$) used to derive each mass.
• Stability Implications: The author notes that the presence of a naked singularity (for $a > 0$) and the conflicting signs of the mass definitions suggest potential instability issues, consistent with known instabilities in Kaluza–Klein bubbles and brane solutions.

The work does not propose new experimental applications but serves as a theoretical classification and property analysis of a specific subset of higher-dimensional vacuum solutions, emphasizing the physical differences between trivial product spaces and non-trivial Kaluza–Klein bubble geometries.