Horizon Threshold Project
The Horizon Threshold Project is a research programme dedicated to investigating the limits of classical spacetime description in regions of extreme curvature associated with black-hole interiors.
The central element of the project is an original method for determining a transition scale directly from invariant properties of the exterior Schwarzschild geometry. This approach links a fundamental curvature scale to the macroscopic mass of the object without introducing an arbitrary and independent regularization parameter for the interior.
In the first completed study within the project, this method was applied to a known exponential regular black-hole metric. The analysis demonstrated that the resulting geometry possesses a global curvature bound that is independent of the asymptotic mass. The work also established an exact classification of horizon regimes, examined the position of the inner horizon relative to the transition region, and determined the behaviour of the inner-horizon surface gravity. The result was supported by an analytic derivation and a computer-assisted verification of the global curvature bound.
The project is currently being developed along several interconnected research directions.
Rotating Geometries
One of the principal objectives is to extend the curvature-threshold method to Kerr geometry. In rotating black holes, the transition region is no longer expected to preserve spherical symmetry and must instead be described as a deformed surface related to the rotational structure of spacetime.
This research direction examines the relation between such a transition surface, the inner-horizon structure, and the limits of dynamical completeness of the geometric description.
Comparison of Regular Black-Hole Models
A further direction involves the systematic comparison of different regular interior profiles subjected to the same curvature-based scale-setting procedure.
The purpose of this work is to determine which results are structural consequences of the relation between exterior curvature and the transition scale, and which depend on the specific form of the selected regularization profile.
Dynamical and Semiclassical Stability
The project also includes a programme devoted to the stability of inner horizons and regular interiors.
This part of the research concerns perturbative dynamics, amplification mechanisms near inner horizons, and the possible influence of semiclassical effects on curvature-regular geometries. The objective is to determine which static configurations remain physically meaningful once dynamical processes and backreaction are taken into account.
Geodesic Completeness and Maximal Extensions
A separate research direction focuses on geodesic completeness and the global structure of the effective spacetime.
The analysis includes maximal extensions of regular geometries and the behaviour of causal and geodesic trajectories in different horizon regimes. These studies are intended to determine whether curvature regularity is accompanied by global completeness of the spacetime description.
Research Programme
The Horizon Threshold Project is not limited to the construction of a single regular metric. Its purpose is to develop a coherent research framework for determining:
- where classical geometry remains an adequate physical description;
- how the boundary of its applicability may be defined invariantly;
- which properties of regular black-hole interiors are universal;
- which results depend on a particular regularization profile;
- how static curvature regularity is modified by perturbative, dynamical, and semiclassical effects.
Detailed methods, complete derivations, computational materials, and quantitative results are released progressively through preprints and scientific publications associated with the project.
Research Outputs
Each study developed within the Horizon Threshold Project is presented in a common format:
- a concise summary of the scientific question;
- the principal result;
- a short explanatory video;
- a link to the complete publication or preprint;
- supplementary computational material, where applicable;
- the current publication status.
This structure will be retained for all subsequent works in the project.
Study 1
Planck-Curvature Thresholds and the Limits of Geometric Description in Regular Black Hole Interiors
Status: Completed preprint
This study introduces a curvature-based prescription for determining the transition scale of a regular black-hole interior directly from the exterior Schwarzschild geometry.
The analysis shows that the resulting scale depends on the asymptotic mass and leads, within the exponential regularization model, to a global curvature bound independent of that mass. The work also derives the exact horizon classification, examines the relation between the transition region and the inner horizon, and proves the monotonic behaviour of the inner-horizon surface gravity.
The computer-assisted component verifies the global monotonicity of the dimensionless curvature profile and supports the exact global bound derived in the paper.
Publication:
[Zenodo link]
Research report:
A structured, source-based summary of the paper and its principal results
Explanatory video:
Supplementary material:
Available within the Zenodo record.
Study 2
[Title of the next study]
Status: In preparation
Research focus:
[Short public description of the scientific question.]
Planned scope:
[General description without disclosing unpublished derivations or detailed methodology.]
Publication:
To be added after release.
Explanatory video:
To be added after release.
Study 3
[Title of the next study]
Status: Planned
Research focus:
[Short public description.]
Publication:
To be added after release.
Explanatory video:
To be added after release.
Study 4
[Title of the next study]
Status: Planned
Research focus:
[Short public description.]
Publication:
To be added after release.
Explanatory video:
To be added after release.
Planned Research Areas
The following studies are currently planned within the Horizon Threshold Project:
- a direct comparison between the exponential regularization profile and the Hayward profile under the same exterior curvature-threshold prescription;
- a full analysis of geodesic completeness and maximal spacetime extensions;
- a perturbative stability study of regular inner horizons;
- an investigation of semiclassical stress-energy and horizon stability;
- an extension of the curvature-threshold framework to rotating Kerr geometries;
- an analysis of deformed curvature-transition surfaces in axisymmetric spacetimes.
The detailed mathematical procedures and unpublished intermediate results remain part of the active research programme and will be disclosed only through completed scientific outputs.