Black-Bounce and Wormhole–Black-Hole Interpolating Models

Black-bounce geometries interpolate continuously between a regular black-hole spacetime and a traversable wormhole by replacing the vanishing areal radius with a nonzero throat. The interpolation is mathematically explicit, but its branches are physically different objects. Some contain horizons, one is a limiting null throat, and others are horizonless; none follows from ordinary vacuum general relativity.

Black-bounce geometry and causal branches

Property	Statement
Object class	Effective model
Theory	Phenomenological metric; selected matter-supported completions
Dimension and symmetry	Four-dimensional, static, spherical baseline
Parameters	Mass length $m=GM$ and bounce/throat scale $a\geq0$
Areal radius	$R(r)=\sqrt{r^2+a^2}$ with $r\in\mathbb R$
Physical status	Parameter-dependent phenomenological geometry; mathematical comparison model with no established astrophysical realization

The Simpson–Visser 2019 representative, §§2–5 is

$$ds^2=-F(r)dt^2+\frac{dr^2}{F(r)}+(r^2+a^2)d\Omega_2^2, \qquad F(r)=1-\frac{2m}{\sqrt{r^2+a^2}}.$$

The nonzero minimum $R(0)=a$ removes the Schwarzschild areal-radius origin when $a>0$. Horizons satisfy $R=2m$, hence

$$r_h=\pm\sqrt{4m^2-a^2}.$$

For $0<a<2m$, the extension contains a regular black-hole/black-bounce regime with Killing horizons. At $a=2m$, the minimum surface is null. For $a>2m$, $F>0$ everywhere and $r=0$ is a timelike traversable throat connecting two asymptotic ends. Calling all three regimes “black holes” would erase the central physical distinction.

Regularity, limits, and matter

Because $R\geq a>0$, scalar curvature invariants of the displayed metric remain finite. Smooth components in the $r$ chart and finite invariants support a regular extension through $r=0$. A complete claim must still inspect causal geodesics and inextendibility; Coordinate, Curvature, and Quasiregular Singularities explains why finite scalars alone are insufficient in general.

Three limits check the family. For fixed $r\ne0$, $a\to0$ recovers the Schwarzschild form on each side of the classical center. For $m\to0$ with $a>0$, one obtains a two-ended, ultrastatic wormhole geometry rather than Minkowski space in ordinary spherical coordinates. Finally, $a\to2m$ from either side tracks the merger of the two horizon roots into a null throat. These limits test geometry; they do not establish continuity of a supporting matter theory.

In Einstein gravity the effective stress tensor violates the null energy condition near the throat, as in the general traversable-wormhole analysis of Morris and Thorne 1988, §§II.B–II.C, pp. 397–400. That is expected for a static traversable wormhole, but it also means that a metric can be regular while its proposed material source is problematic. Explicit scalar-plus-nonlinear-electrodynamics realizations studied by Bronnikov 2022, §§II–IV illustrate that the supporting-field model is additional data; in those solutions the scalar can be phantom only in a bounded strong-field region. Modified gravity can move terms between “geometry” and “effective matter,” so energy-condition language must specify the physical matter metric and field equations. Constructing Alternative-Theory Solution Branches gives the checks required of a covariant completion.

The horizon mechanics is similarly branch-specific. For $0<a<2m$, the outer-horizon surface gravity in the asymptotic time normalization is

$$\kappa_+=\frac12\left.\frac{dF}{dr}\right|_{r_+} =\frac{\sqrt{4m^2-a^2}}{8m^2}.$$

It approaches the Schwarzschild value $1/(4m)$ as $a\to0$ and vanishes as the null-throat limit is approached. This is a controlled geometric result. It does not provide a first law for varying $a$, which depends on a supporting action and its charges.

Strongest result and principal obstruction

The strongest established result is the exact causal interpolation within the specified metric family: the horizon equation, throat, regularity, and asymptotic ends can all be analyzed without a numerical ansatz. This makes the model valuable for separating observables controlled by an exterior potential from those sensitive to a surface, throat, or second asymptotic region.

The principal obstruction is dynamical viability. A background supported by exotic effective stresses is not known to form from generic regular initial data. Linear stability depends on how the supporting degrees of freedom are perturbed; perturbing only the metric while freezing its source can give a nonconserved or incomplete system. Lobo, Simpson, and Visser 2020, §§II–III show how even thin-shell dynamics requires an explicit surface stress prescription. Inner-horizon blueshift, throat modes, and nonlinear evolution require their own analyses.

Accordingly, “regular” must be qualified. The representative has finite scalar invariants and a smooth areal-radius minimum. Geodesic completeness requires the full extension and affine analysis; acceptable energy conditions require a declared matter theory; mode stability requires perturbing every dynamical field; and nonlinear stability requires evolution from finite disturbances. These are independent tests, not synonyms.

Observable distinctions and evidence cutoff

At the 20 August 2026 evidence cutoff, no observation establishes a black bounce. Proposed signatures include altered shadows, tidal response, absorption, echoes, lensing through a second exterior, and branch-dependent quasinormal spectra. Each is degenerate with mass, spin, environment, compact-object structure, and waveform or emission modeling. A late-time feature is not evidence for a throat unless its phase, amplitude, population behavior, and competing explanations are jointly tested.

A controlled inference should vary the bounce scale together with source and detector nuisance parameters, compare against ordinary material surfaces and environmental echoes, and demonstrate that the conclusion survives changes in the supporting-field perturbation model. Exterior-metric agreement alone cannot locate or characterize the throat.

Discriminating Exotic Objects from Black Holes supplies the multipole, tide, absorption, ringdown, image, and population comparison. Formation and current empirical viability remain unresolved.

References

• Bronnikov, Kirill A. “Black Bounces, Wormholes, and Partly Phantom Scalar Fields.” Physical Review D 106 (2022), 064029. doi:10.1103/PhysRevD.106.064029.
• Lobo, Francisco S. N., Alex Simpson, and Matt Visser. “Dynamic Thin-Shell Black-Bounce Traversable Wormholes.” Physical Review D 101 (2020), 124035. doi:10.1103/PhysRevD.101.124035.
• Morris, Michael S., and Kip S. Thorne. “Wormholes in Spacetime and Their Use for Interstellar Travel: A Tool for Teaching General Relativity.” American Journal of Physics 56 (1988), 395–412. doi:10.1119/1.15620.
• Simpson, Alex, and Matt Visser. “Black-Bounce to Traversable Wormhole.” Journal of Cosmology and Astroparticle Physics 2019, 042 (2019). doi:10.1088/1475-7516/2019/02/042.