[Paper Review] Spherically-symmetric solutions with a chain of n internal Ricci-flat spaces

В. Д. Иващук|arXiv (Cornell University)|Jun 23, 2010

Black Holes and Theoretical Physics2 references3 citations

TL;DR

This paper generalizes the Schwarzschild solution to higher-dimensional spacetimes with $ n $ internal Ricci-flat manifolds, deriving an exact spherically-symmetric vacuum solution. The key result is that a horizon exists in the four-dimensional section only when all internal space scale factors are constant, reducing the solution to the standard Schwarzschild metric; otherwise, no horizon forms due to divergent light ray propagation near $ R = L $. The scalar-vacuum extension includes a minimally coupled scalar field, modifying the curvature constraint but preserving the horizon condition.

ABSTRACT

The Schwarzschild solution is generalized for the case of n internal Ricci-flat spaces. It is shown that in the four-dimensional section of the metric a horizon exists only when the internal space scale factors are constant. The scalar-vacuum generalization of the solution is also presented. [This paper is the English translation of the part of Chapter. 2.4 of the author's PhD dissertation (Moscow, 1989).]

Motivation & Objective

• To extend the Schwarzschild solution to spacetimes with $ n $ internal Ricci-flat manifolds.
• To determine conditions under which a horizon forms in the four-dimensional section of the metric.
• To derive a scalar-vacuum generalization by including a minimally coupled scalar field.
• To analyze the role of scale factor evolution in horizon formation and causal structure.

Proposed method

• Derive a spherically-symmetric metric ansatz with $ n $ internal Ricci-flat manifolds and a radial variable $ u $, using harmonic radial coordinates.
• Solve the vacuum Einstein equations $ R_{MN} = 0 $ by reducing them to a system of ODEs involving $ \beta_\nu(u) $, $ \alpha_0(u) $, and $ \gamma(u) $.
• Introduce a new radial coordinate $ R $ via a transformation involving $ f(\bar{u}, B) $, leading to a manifestly coordinate-invariant form of the solution.
• Apply the transformation $ R = e^{-\sum_{\nu \neq 0} D_\nu N_\nu} \times \text{function}(\bar{u}, B) $ to express the metric in terms of $ R $, $ L $, and constants $ a, a_i $.
• Derive the scalar-vacuum extension by adding a minimally coupled scalar field $ \varphi $, leading to modified Einstein equations with a source term $ \kappa^2 \partial_M \varphi \partial_N \varphi $.
• Solve the modified system by assuming $ \varphi = Qu + \bar{\varphi}_0 $, leading to a logarithmic solution $ \varphi = \frac{1}{2}q \ln(1 - L/R) + \varphi_0 $, and derive the new constraint (20).

Experimental results

Research questions

• RQ1 Under what conditions does a horizon form in the four-dimensional section of a spacetime with $ n $ internal Ricci-flat manifolds?
• RQ2 How does the evolution of internal space scale factors affect the causal structure and horizon formation?
• RQ3 What is the role of the scalar field in modifying the curvature constraint and horizon conditions?
• RQ4 How does the solution reduce to the standard Schwarzschild metric in special cases?
• RQ5 What is the necessary and sufficient condition for horizon existence in the scalar-vacuum extension?

Key findings

• A horizon exists at $ R = L $ in the four-dimensional section only when all internal space scale factors are constant, i.e., $ a_1 = \cdots = a_n = 0 $ and $ a = 1 $, reducing the solution to the standard Schwarzschild metric.
• When internal scale factors vary ($ a_i \neq 0 $), the integral for radial light rays converges at $ R = L $, but the metric does not describe a horizon due to non-constant redshift behavior.
• The condition for horizon existence is equivalent to $ |a + \frac{1}{2}\sum a_i N_i| < 1 $, which is violated when $ a = \pm 1 $ and $ a_i = 0 $, but only $ a = 1 $ yields a horizon.
• In the scalar-vacuum case, the horizon condition is preserved only when $ q = a - 1 = a_1 = \cdots = a_n = 0 $, meaning the scalar field must be trivial for a horizon to form.
• The solution is trivial (flat 4D and constant internal scale factors) when $ L = 0 $, regardless of the values of $ a, a_i $, indicating no gravitational collapse.
• The constraint equation (20) generalizes (10) to include the scalar charge $ q $, with $ \kappa^2 q^2 $ contributing to the curvature balance, but not altering the horizon condition unless $ q = 0 $.