[Paper Review] Do Black Holes Exist in a Finite Universe Having the Topology of a Flat 3-Torus?

Frank Steiner|arXiv (Cornell University)|Aug 10, 2016

Galaxies: Formation, Evolution, Phenomena22 references3 citations

TL;DR

This paper proposes that black holes may exist in a finite, flat 3-torus universe by deriving the first-order solution to Einstein’s equations for a static, non-rotating star using Appell and Epstein zeta functions to regularize the divergent image method. The solution reveals a topological dark energy term Λ_top > 0, leading to a repulsive force and anisotropic gravitational field, violating Birkhoff’s theorem and suggesting 'hairy' black holes with 6 parameters: mass, volume, and 5 Teichmüller parameters.

ABSTRACT

Based on perturbation theory, we present the exact first-order solution to the Einstein equations for the exterior static gravitational field of an isolated non-rotating star in a spatially finite universe having the topology of a flat 3-torus. Since the method of images leads to a divergent Poincare' series, one needs a regularization which we achieve by using the Appell respectively the Epstein zeta function. The solution depends on a new positive constant which is completely fixed by the mass of the star and the spatial volume of the universe. The physical interpretation is that a stable or metastable equilibrium requires a topological dark energy which fills the whole universe with positive energy density and negative pressure. The properties of the gravitational field are discussed in detail. In particular, its anisotropy is made explicit by deriving an exact multipole expansion which shows that in this case Birkhoff's theorem does not hold. While the monopole describes the Newtonian potential, there is no dipole but always a non-vanishing quadrupole which leads to a repulsive force experienced by a planet at rest. Finally, we put forward the conjecture that black holes exist in a toroidal universe and that their gravitational field is in the weak-field limit well approximated by the first-order field.

Motivation & Objective

• To investigate whether black holes can exist in a spatially finite universe with the topology of a flat 3-torus, challenging the standard assumption of infinite flat space.
• To resolve the divergence of the method of images in a compact universe by applying Appell and Epstein zeta functions for regularization.
• To derive the first-order exterior gravitational field of a non-rotating star in such a topology and determine its physical implications.
• To examine whether Birkhoff’s theorem holds in a toroidal universe and explore the possibility of 'hairy' black holes with topological hair.
• To propose a mechanism for topological dark energy arising from global topology, with measurable anisotropic gravitational effects.

Proposed method

• Using first-order perturbation theory in general relativity to solve Einstein’s field equations for a star in a finite, flat 3-torus universe.
• Applying the method of images to model the gravitational potential, but regularizing the resulting divergent Poincaré series using Appell’s and Epstein’s zeta functions.
• Deriving the metric solution that satisfies R_μν = -Λ_top g_μν, where Λ_top is a positive constant determined by mass M and spatial volume |Ω|.
• Performing a multipole expansion of the gravitational potential to reveal the absence of a dipole and the presence of a non-vanishing quadrupole term.
• Defining a traceless quadrupole tensor D̂_kl to isolate the anisotropic part of the field and analyze its force contributions.
• Conjecturing that full black hole solutions in a toroidal universe would be well-approximated by this weak-field solution, especially in the far-field regime.

Experimental results

Research questions

• RQ1 Does the method of images yield a finite gravitational potential in a compact, flat 3-torus universe, or does it diverge?
• RQ2 Can a consistent first-order solution to Einstein’s equations be derived for a star in a finite, topologically non-trivial universe?
• RQ3 Does the resulting gravitational field violate Birkhoff’s theorem due to topological anisotropy?
• RQ4 What is the physical origin and nature of the effective cosmological constant Λ_top in this context?
• RQ5 Could black holes in a toroidal universe possess additional parameters (hair) beyond mass, due to the topology?

Key findings

• The gravitational field solution satisfies R_μν = -Λ_top g_μν with Λ_top > 0, indicating a topological dark energy density ε_top ∼ Λ_top and negative pressure p_top = -ε_top.
• The monopole term corresponds to the Newtonian potential, while the first non-vanishing multipole is a quadrupole field Φ_Q(x) = - (Λ_top c² / 6) r² - GM ∑_{k,ℓ} D̂_kℓ x_k x_ℓ.
• A repulsive central force ∼ (Λ_top c² / 3) r is present, independent of the star’s mass, and arises solely from the topological constant Λ_top.
• The anisotropic force due to the traceless tensor D̂_kℓ is generally non-zero and depends on the shape of the torus, vanishing only in symmetric cases like a cubic torus.
• The solution explicitly breaks global rotational symmetry, demonstrating that Birkhoff’s theorem does not hold in a 3-torus universe.
• The authors conjecture that black holes in such a universe would be 'hairy,' characterized by mass M, spatial volume |Ω|, and 5 Teichmüller parameters, making them fundamentally different from Schwarzschild black holes.