Effective and Phenomenological Models

Regular metrics and theory-agnostic deformations are useful because they make assumptions testable. They are also easy to overinterpret. This directory groups five effective model families by what is actually specified: a line element, a matter or gravity completion, a controlled approximation, or only a phenomenological parameterization. None is presented as an observed alternative to an astrophysical black hole.

The comparison is object-first: each page fixes one construction before assessing what it can and cannot establish.

Choosing an effective model

Start from the scientific question, not from the promise of a nonsingular center. A metric ansatz can answer geometric questions once its coordinate and parameter domains are fixed. It does not, by itself, supply field equations, a formation channel, a well-posed perturbation problem, or an observational prior.

Model family	What is defined most sharply	Essential caution
Bardeen regular model	A static spherical mass function with a finite-curvature core and a known nonlinear-electrodynamics realization	The metric and the specified matter theory are distinct claims
Hayward regular model	A compact de Sitter-core interpolation with an analytically tractable horizon threshold	The commonly used static metric is not automatically the exterior of a dynamical collapse solution
Parametrically deformed Kerr metrics	A theory-agnostic null-test framework for stationary strong-field geometry	Coordinate coefficients are not invariant multipoles, and fitted bounds are ansatz-dependent
Black-bounce models	A smooth interpolation among black-hole, null-throat, and traversable-wormhole causal regimes	Curvature regularity does not establish ordinary matter support, stability, or formation
Polymerized black-hole models	Symmetry-reduced quantum-geometry modifications and candidate transition surfaces	Results can depend on polymerization, lapse, gauge, and fiducial choices unless covariance is demonstrated

Four questions every page answers

What is the defining object? For a static spherical model this means the areal-radius metric and mass function. For a Kerr deformation it means the complete parameterized tensor, its asymptotic frame, and its regularity domain. For a polymerized model it means the phase-space variables, constraint or effective equations, and prescription for the quantum scale.

What has actually been checked? Finite scalar curvature invariants are valuable, but they are not synonymous with geodesic completeness. A horizon root is not a stability result. An effective stress tensor obtained by moving the Einstein tensor to the right-hand side is not proof of a fundamental matter action. Each page therefore separates curvature, extension, energy conditions, dynamics, and evidence.

Which limit anchors the model? The Bardeen and Hayward metrics must recover Schwarzschild as their short-distance scale vanishes. A deformed-Kerr family must recover Kerr when every deformation is set to zero. A black bounce has distinct Schwarzschild and wormhole limits. A polymerized model needs a controlled classical limit at fixed physical data, not merely a formal replacement parameter sent to zero.

What could observations constrain? Data constrain a forward model that includes emission, matter, detector response, selection, and nuisance parameters. They do not directly constrain a line element. The appropriate logic is developed in Null Tests and Alternative-Model Tests and From Constraint to Physical Conclusion.

A practical comparison

For a static spherical line element

$$ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_2^2,$$

three calculations provide a first diagnostic: solve $f(r)=0$ and locate double roots, expand $f$ and curvature invariants at the center, and take every advertised large-distance or zero-deformation limit. These checks determine horizon branches and local regularity. They do not determine maximal extension, geodesic completeness, or mode stability; those require separate arguments.

Across all five families, the most useful comparison is therefore not “regular versus singular.” It is a chain:

$$\text{defined variables} \longrightarrow \text{equations or ansatz} \longrightarrow \text{admissible domain} \longrightarrow \text{controlled observable} \longrightarrow \text{data model}.$$

A broken link is a limitation to state, not a reason to fill the gap with interpretation. Readers interested in exact metric documentation should begin with Specifying an Exact Spacetime Solution; readers comparing horizonless alternatives should use Discriminating Exotic Objects from Black Holes.

What this directory cannot establish

No entry here, by itself, supplies a realistic formation history, a population model, or evidence that nature realizes the construction. A finite-curvature core does not imply geodesic completeness or stability; a fitted deformation does not identify a fundamental theory; a quantum-gravity motivation does not make a prescription unique. Those conclusions require the canonical dynamical, theoretical, and empirical analyses linked from each page.