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On the Geometric Interpretation of Non-Singular Black Hole Interiors

This paper proposes a teleparallel gravity framework utilizing geometric drift vectors to replace the central singularity of black holes with a smooth de Sitter-like core, where gravitational effects arise from torsion rather than curvature, allowing for analytic extensions of trajectories across event horizons.

Original authors: Melih Gumus, Bilgehan Baris Oner

Published 2026-07-07✓ Author reviewed ⓘ

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CC BY 4.0

Original authors: Melih Gumus, Bilgehan Baris Oner

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Problem: The "Infinite Crunch"

Imagine a star collapsing under its own weight. In our current understanding of gravity (General Relativity), if the star gets too heavy, it collapses into a black hole. But there's a scary problem at the very center: a singularity.

Think of a singularity like a point where the rules of physics break down completely. It's as if you are driving a car toward a cliff, and the road just ends abruptly into an infinite drop. The math says the forces become infinite, and the car (or any particle) is crushed into nothingness. This creates a "dead end" in the universe where time and space stop making sense.

The New Idea: Gravity as a "River"

The authors of this paper propose a different way to look at gravity. Instead of thinking of gravity as a curved road (which is how Einstein usually described it), they use Teleparallel Gravity.

The Analogy:
Imagine spacetime isn't a curved sheet, but a flat, calm lake. Gravity isn't a dip in the water; instead, it's a current or a drift flowing through the water.

The "Drift Field": The authors call this the "Geometric Drift Vector." Imagine a river flowing toward a waterfall. The water moves faster and faster as it gets closer to the edge.
The Black Hole: In this view, a black hole is just a region where this "drift" becomes incredibly strong. The event horizon (the point of no return) is simply where the river flows as fast as the speed of light.

How They Fix the "Dead End"

In the standard view, as you get closer to the center of the black hole, the "river" speeds up infinitely, and the math explodes (the singularity).

The authors suggest that in extreme conditions (like the center of a black hole), the "river" behaves differently, similar to how non-Newtonian fluids (like oobleck or cornstarch and water) act.

The Analogy: If you push a spoon slowly through cornstarch, it moves easily. But if you hit it hard and fast, the mixture suddenly becomes as hard as concrete.
The Paper's Claim: The authors say that in the weak gravity of our solar system, the "drift" behaves normally (like water). But deep inside a black hole, where the "drift" is super strong, a non-linear effect kicks in. The "fluid" of space stiffens up.

Instead of the drift speed becoming infinite and crushing everything, this stiffening effect changes the flow. The center of the black hole stops being a "dead end" and becomes a smooth, expanding bubble (like a de Sitter core).

The Result:

No Crunch: The "river" doesn't end at a cliff. Instead, it smoothly transitions.
No Dead End: If you were to fall into this black hole, you wouldn't hit a point of infinite destruction. You would pass through the center and continue moving, because the "road" never actually ends. The math shows that particles can travel through the center and keep going forever without hitting a wall.

Why This Matters (According to the Paper)

No Magic Tricks: Many other theories try to fix the singularity by inventing new, imaginary particles or "exotic matter" to hold the center open. This paper claims they don't need that. They fixed it just by changing how the "drift" of space behaves mathematically.
No Information Loss: Because the center isn't a dead end where information gets destroyed, the "Information Loss Paradox" (a major puzzle in physics about whether things disappear forever) might be solved. The information can pass through the center.
It's Real, Not Just a Trick: The authors spent a lot of time proving that this "drift" isn't just a trick of how we measure things (a coordinate choice). They showed that this drift creates real, physical effects (torsion) that cannot be erased by changing your point of view.

Summary

The paper suggests that black holes aren't cosmic trash compactors that crush everything into an infinite point. Instead, they are regions where space flows like a river. Deep inside, the flow changes its nature (like a fluid getting stiff), turning the "infinite drop" into a smooth, round tunnel. This allows the universe to remain whole and continuous, even at the heart of a black hole.

Technical Summary: On the Geometric Interpretation of Non-Singular Black Hole Interiors

Problem Statement
The paper addresses the persistent challenge of gravitational singularities within black hole physics. In standard General Relativity (GR), while the event horizon is a coordinate singularity, the central region ( $r \to 0$ ) represents a physical singularity where the Kretschmann scalar diverges ( $r^{-6}$ ), leading to infinite tidal forces and geodesic incompleteness. This central singularity is inextricably linked to the "information loss" paradox, where the evaporation of a black hole seemingly destroys quantum information. Existing solutions, such as regular black hole models, often rely on ad-hoc metric modifications or the introduction of exotic matter that violates energy conditions, lacking a derivation from a fundamental gravitational theory.

Methodology
The authors employ Teleparallel Gravity, specifically the Teleparallel Equivalent of General Relativity (TEGR), augmented by Geometric Drift Vectors (GDVs).

Geometric Framework: Instead of describing gravity via spacetime curvature (Riemannian geometry), the theory encodes gravitational effects in the torsion tensor of a flat spacetime (Weitzenb¨ock connection). Gravity is interpreted as a geometric effect of frame dragging via a dynamical drift field.
Tetrad Formalism: The study utilizes a comoving tetrad framework where the gravitational field is represented by a drift vector field $W$ . In the weak-field regime, this reproduces the Schwarzschild solution using Painlev´e–Gullstrand (PG) coordinates, where the event horizon corresponds to the drift velocity reaching the speed of light.
Nonlinear Lagrangian Extension: The core methodological innovation is the extension of the standard TEGR Lagrangian ( $L_{TEGR} \propto T$ ) to include nonlinear torsion invariants. Drawing an analogy to non-Newtonian fluids and nonlinear optics, the authors propose a total Lagrangian $L_{Total} = \frac{e}{2\kappa}f(T)$ that incorporates higher-order terms (specifically quadratic invariants of the velocity gradient tensor) which become dominant in strong-field regimes.
Field Equations: The authors derive field equations from this nonlinear Lagrangian to solve for the drift vector field $W(r)$ in a spherically symmetric vacuum, avoiding the assumption of a pre-defined metric ansatz.

Key Contributions and Results

Derivation of a Non-Singular Core: Solving the nonlinear field equations yields a drift vector field that transitions from the standard Schwarzschild behavior ( $W \sim r^{-1/2}$ ) in the exterior to a linear behavior ( $W \sim Hr$ ) in the interior. This interior solution corresponds to a smooth de Sitter-like core, effectively replacing the central singularity with a region of finite, constant torsion.
Geodesic Completeness: The paper explicitly demonstrates that the resulting spacetime is geodesically complete. By solving the autoparallel equations for both null and timelike trajectories, the authors show that particles and light rays can be analytically extended across the horizon and through the central region ( $r=0$ ) to infinite affine parameters. The effective force term (contortion) remains bounded everywhere.
Physicality of the Drift Field: The authors rigorously test the physical nature of the $W$ field against diffeomorphism and local Lorentz gauge transformations. They conclude that the field is not a pure gauge artifact; it generates non-vanishing torsion invariants and directly influences geodesic motion and the action, distinguishing it from coordinate-induced effects.
Resolution of Energy Condition Violations: Unlike many regular black hole models that require exotic matter violating energy conditions, this approach resolves the singularity purely through geometric nonlinearities (torsion invariants). Consequently, no exotic stress-energy tensor is introduced, and classical energy conditions are rendered inapplicable rather than violated.
Horizon Interpretation: The event horizon is reinterpreted as a "drift horizon," a causal boundary where the geometric drift velocity approaches the speed of light. The paper notes a structural equivalence between the Hawking temperature and the Unruh temperature in this framework, linking the black hole horizon to the Rindler horizon in flat space.

Significance and Claims
The paper claims to offer a conceptual framework rather than just another regular black hole model. Its primary significance lies in:

Theoretical Economy: It eliminates the need for ad-hoc metric engineering or new scalar fields, deriving the regular core directly from the nonlinear dynamics of the teleparallel connection.
Geometric Resolution: It posits that the singularity is an artifact of the linear approximation of gravity in strong fields. By incorporating nonlinear torsion contributions, the theory naturally prevents the divergence of invariants.
Causality: The model avoids the causality issues often associated with inner horizons in regular black hole models by imposing a physical bound on the drift velocity (preventing superluminal flows).

The authors conclude that this approach provides a consistent, non-singular description of black hole interiors within a teleparallel geometric framework, potentially offering new avenues for understanding quantum field theory in curved spacetime and the information paradox without invoking unknown matter sources.