Black Bounce Spacetimes: Regular Black Holes & Wormholes

• Black bounce spacetimes are nonsingular, spherically symmetric geometries that replace the Schwarzschild singularity with a finite throat.
• They interpolate between regimes—a regular black hole, one-way wormhole, and two-way traversable wormhole—controlled by a bounce parameter affecting horizon structure and causal properties.
• They are supported by diverse matter sources such as phantom scalars and nonlinear electrodynamics, exhibiting unique features in geodesics, optical signatures, and thermodynamic behavior.

Black-bounce spacetimes are nonsingular spherically symmetric geometries in which the two-sphere area never shrinks to zero, so the Schwarzschild singularity is replaced by a throat of finite minimal area. In the standard Simpson–Visser family, a single bounce parameter interpolates between Schwarzschild at zero deformation, a regular black hole when the throat is hidden behind horizons, a one-way wormhole at the extremal value, and a two-way traversable wormhole when no horizon remains. The same idea has been developed in static spherical symmetry, spherical symmetry with dynamics, and stationary axisymmetry, and extended to charged, rotating, and generalized mass-function constructions (Lobo et al., 2020, Simpson, 2021).

1. Geometric definition and canonical metrics

A common starting point is the static, spherically symmetric line element in Buchdahl or Schwarzschild-like coordinates,

$$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+\Sigma(r)^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right),$$

with

$$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$

In generalized static models one takes

$$M(r)=\frac{m\,\Sigma(r)\,r^k}{\left(r^{2n}+a^{2n}\right)^{\frac{k+1}{2n}}},$$

where $m>0$ is the ADM mass at infinity, $a\ge 0$ is the bounce or throat scale, and $(k,n)$ control the radial fall-off of the mass profile (Moreira et al., 27 Jul 2025).

The special case $n=1$, $k=0$ gives the Simpson–Visser geometry,

$f(r)=1-\frac{2m}{\sqrt{r^2+a^2}},$

which is the standard black-bounce or regularized Schwarzschild metric. The same generalized family also contains the Bardeen-like bounce at $n=1$, $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$0, and a quartic deformation at $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$1, $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$2 (Moreira et al., 27 Jul 2025). In a closely related notation, the original Simpson–Visser construction introduces $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$3 and writes

$$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$4

with $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$5 playing the role of the bounce scale (Simpson, 2021).

The same regularization recipe extends to charged and rotating geometries. In black-bounce–Reissner–Nordström and black-bounce–Kerr–Newman spacetimes, one replaces $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$6 by $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$7 in the Reissner–Nordström or Kerr–Newman metric functions. In the rotating case,

$$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$8

and the resulting spacetime is everywhere regular for $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$9 (Franzin et al., 2021).

2. Regularity, horizons, and causal classification

The defining geometric property of a black bounce is that the areal radius remains strictly positive. In the static analysis of Lobo et al., regularity is guaranteed by demanding $$M(r)=\frac{m\,\Sigma(r)\,r^k}{\left(r^{2n}+a^{2n}\right)^{\frac{k+1}{2n}}},$$0 together with finite $$M(r)=\frac{m\,\Sigma(r)\,r^k}{\left(r^{2n}+a^{2n}\right)^{\frac{k+1}{2n}}},$$1, $$M(r)=\frac{m\,\Sigma(r)\,r^k}{\left(r^{2n}+a^{2n}\right)^{\frac{k+1}{2n}}},$$2, $$M(r)=\frac{m\,\Sigma(r)\,r^k}{\left(r^{2n}+a^{2n}\right)^{\frac{k+1}{2n}}},$$3, $$M(r)=\frac{m\,\Sigma(r)\,r^k}{\left(r^{2n}+a^{2n}\right)^{\frac{k+1}{2n}}},$$4, and $$M(r)=\frac{m\,\Sigma(r)\,r^k}{\left(r^{2n}+a^{2n}\right)^{\frac{k+1}{2n}}},$$5. For $$M(r)=\frac{m\,\Sigma(r)\,r^k}{\left(r^{2n}+a^{2n}\right)^{\frac{k+1}{2n}}},$$6, the throat occurs at $$M(r)=\frac{m\,\Sigma(r)\,r^k}{\left(r^{2n}+a^{2n}\right)^{\frac{k+1}{2n}}},$$7 with $$M(r)=\frac{m\,\Sigma(r)\,r^k}{\left(r^{2n}+a^{2n}\right)^{\frac{k+1}{2n}}},$$8, and the causal character of that throat is controlled by the sign of $$M(r)=\frac{m\,\Sigma(r)\,r^k}{\left(r^{2n}+a^{2n}\right)^{\frac{k+1}{2n}}},$$9: timelike for $m>0$0, null for $m>0$1, and spacelike for $m>0$2 (Lobo et al., 2020).

For the Simpson–Visser metric, the horizon equation is

$m>0$3

This yields the standard three-way classification (Simpson, 2021):

Condition	Geometry	Throat character
$m>0$4	regular black hole	spacelike
$m>0$5	one-way wormhole	null
$m>0$6	two-way traversable wormhole	timelike

The same horizon-versus-throat logic persists in generalized black-bounce families. In the Bardeen-type model, two horizons may appear and coalesce at an extremal parameter value, while higher-order Fan–Wang-type models and oscillatory mass profiles can generate several simple horizons or extremal cases (Lobo et al., 2020). A separate construction with modified Schwarzschild/Simpson–Visser functions exhibits multiple horizons, throats, and anti-throats, with asymptotically Schwarzschild behavior and a regular Kretschmann scalar throughout the manifold (Rodrigues et al., 1 Feb 2025).

Regularity is not only local at the throat. In the generalized thermal study, the Ricci scalar

$m>0$7

and the Kretschmann scalar remain finite at $m>0$8 because $m>0$9, and remain finite as $a\ge 0$0 as well (Moreira et al., 27 Jul 2025). Lobo et al. further prove a general static-regularity theorem: in a strictly static region, finiteness of the Kretschmann scalar forces every orthonormal component of the Riemann tensor to be finite, so the spacetime is everywhere curvature-regular (Lobo et al., 2020).

3. Matter sources, stress tensors, and energy conditions

Within classical general relativity, black-bounce geometries are typically supported by exotic matter. Rodrigues and Silva construct the Simpson–Visser and Bardeen-type black-bounce solutions from Einstein gravity coupled to a phantom scalar field and nonlinear electrodynamics. Their action is

$a\ge 0$1

with $a\ge 0$2 for a phantom scalar, and they show that the phantom scalar is linked to the violation of the null energy condition (Rodrigues et al., 2023). For the Simpson–Visser case,

$a\ge 0$3

while the nonlinear electromagnetic sector supplies the remaining stress-energy required by Einstein’s equations (Rodrigues et al., 2023).

A general diagnostic is the radial null energy condition. In the anisotropic-fluid interpretation,

$a\ge 0$4

Since $a\ge 0$5 for the standard black-bounce profile, $a\ge 0$6 is violated wherever $a\ge 0$7; accordingly, all standard pointwise energy conditions are violated off any trapping horizon in these models (Lobo et al., 2020). This is consistent with the source reconstructions in which the phantom scalar carries the NEC-violating sector, while the nonlinear electromagnetic contribution may satisfy one or both null-energy combinations in parts of the manifold (Silva et al., 24 Jun 2025).

Several source constructions refine this picture. An electrically charged reconstruction in nonlinear electrodynamics plus scalar field shows that electric sources can produce branching in $a\ge 0$8 because $a\ge 0$9 need not be monotonic, in contrast to the purely magnetic case (Alencar et al., 2024). A nonminimal scalar–electrodynamic coupling model,

$(k,n)$0

determines $(k,n)$1 from the ansatz $(k,n)$2 and shows that both Simpson–Visser-type and Bardeen-type black bounces admit a linear-electrodynamics limit $(k,n)$3 when $(k,n)$4 (Cordeiro et al., 28 Sep 2025).

Not all constructions use phantom scalar plus nonlinear electrodynamics. A T-duality-inspired model takes an anisotropic effective fluid with density

$(k,n)$5

derives a regular black-bounce geometry directly from Einstein’s equations, and interprets $(k,n)$6 as a minimal length or zero-point length induced by path-integral duality (Alencar et al., 4 Mar 2026). This suggests that black-bounce metrics admit genuine matter-source ambiguity: the same background geometry can be supported by inequivalent source models, a point that later becomes dynamical in ringdown studies.

4. Dynamical interiors, junction conditions, and collapse

A dynamical black-bounce interior can be constructed by matching a regularized exterior to a regularized FLRW-like interior. In the Israel–Darmois treatment of Pal, Pal, and Sarkar, the exterior is the Simpson–Visser spacetime written in $(k,n)$7 coordinates, while the interior takes the form

$(k,n)$8

Matching is performed across a timelike hypersurface $(k,n)$9, with the usual continuity conditions on the induced metric and extrinsic curvature (Pal et al., 2024).

The main result is negative for smooth matching: a regularized FLRW-like interior geometry cannot be matched smoothly with the exterior black-bounce spacetime through a timelike hypersurface, because for $n=1$0 the second junction condition cannot be satisfied on both sides. The mismatch is interpreted as a thin shell with surface stress-energy tensor

$n=1$1

and perfect-fluid surface variables

$n=1$2

(Pal et al., 2024).

Imposing an equation of state on the shell closes the system. For the choice

$n=1$3

and after choosing the integration constant appropriately and setting $n=1$4, the interior scale factor obeys

$n=1$5

When $n=1$6, this reduces to the usual Oppenheimer–Snyder solution for homogeneous dust. The same analysis identifies interior trapped surfaces through

$n=1$7

and discusses event-horizon formation for collapsing initial data (Pal et al., 2024).

The collapse dynamics differs qualitatively from singular dust collapse. The regularization parameter removes the central singularity; neither the Ricci scalar nor the Kretschmann invariant diverges even when $n=1$8. For reasonable shell equations of state, such as $n=1$9, the collapsing boundary proceeds smoothly through $k=0$0 into a bounce region connected to the second universe of the Simpson–Visser wormhole (Pal et al., 2024).

A distinct dynamical extension promotes the mass to a null-coordinate-dependent function $k=0$1 in Eddington–Finkelstein coordinates,

$k=0$2

with $k=0$3 for advanced time and $k=0$4 for retarded time. In this Vaidya-type black-bounce, the dynamical horizon is

$k=0$5

so accretion can turn a wormhole into a regular black hole, while evaporation can turn a regular black hole into a wormhole remnant (Simpson, 2021).

5. Geodesics, circular orbits, and tidal structure

The geodesic sector of the Simpson–Visser geometry is analytically tractable. In the equatorial plane, the point-particle Lagrangian yields conserved energy and angular momentum,

$k=0$6

and the radial equation

$k=0$7

For null geodesics, the unstable circular photon orbit satisfies

$k=0$8

valid for $k=0$9. For timelike geodesics, the marginally stable circular orbit satisfies

$f(r)=1-\frac{2m}{\sqrt{r^2+a^2}},$0

valid for $f(r)=1-\frac{2m}{\sqrt{r^2+a^2}},$1 (Silva et al., 2024).

As $f(r)=1-\frac{2m}{\sqrt{r^2+a^2}},$2 grows, both $f(r)=1-\frac{2m}{\sqrt{r^2+a^2}},$3 and $f(r)=1-\frac{2m}{\sqrt{r^2+a^2}},$4 shrink. The same geodesic analysis shows that at $f(r)=1-\frac{2m}{\sqrt{r^2+a^2}},$5, corresponding to the wormhole regime, multiple photon rings may appear, including an inner stable ring and an outer unstable one (Silva et al., 2024). When the background is interpreted as arising from nonlinear electrodynamics, photons propagate in an effective metric,

$f(r)=1-\frac{2m}{\sqrt{r^2+a^2}},$6

and the light-bending law acquires $f(r)=1-\frac{2m}{\sqrt{r^2+a^2}},$7 corrections (Silva et al., 2024).

Tidal forces are one of the clearest local signatures of black-bounce geometry. In the instantaneous rest frame of an infalling timelike geodesic, the tidal tensor has eigenvalues $f(r)=1-\frac{2m}{\sqrt{r^2+a^2}},$8 in the radial sector and $f(r)=1-\frac{2m}{\sqrt{r^2+a^2}},$9 in the angular sector. For the general metric

$n=1$0

these are

$n=1$1

$n=1$2

In all black-bounce models studied, $n=1$3 and $n=1$4 remain finite throughout the spacetime, including at the wormhole throat (Crispim et al., 30 Jun 2025).

The sign structure of the tidal eigenvalues differs qualitatively from Schwarzschild. In Schwarzschild, radially infalling matter is stretched radially and compressed angularly, with both effects diverging as the singularity is approached. In black-bounce models all divergences disappear, and sufficiently large bounce parameters can induce sign flips in $n=1$5 or $n=1$6, so that bodies undergo radial compression and angular stretching. The Bardeen-type black bounce exhibits the richest pattern, with both $n=1$7 and $n=1$8 capable of changing sign twice (Crispim et al., 30 Jun 2025). This suggests that finite, sign-alternating tidal sectors are not incidental regularizations but characteristic features of bounce geometries.

6. Shadows, ringdown, thermodynamics, and broader theoretical realizations

Optically, black-bounce spacetimes interpolate between ordinary black-hole shadows and horizonless wormhole lensing. In the generalized $n=1$9–$$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$00 family, photon motion is governed by

$$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$01

and the critical impact parameter is

$$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$02

For fixed mass, three regimes appear: $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$03 gives a regular black hole with a standard shadow; $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$04 gives a horizonless geometry with two light rings and a double-ring image; and $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$05 gives no horizons and no circular null orbits, so there is no formal shadow, only lensing caustics near the throat (Nascimento et al., 27 Oct 2025). Numerical ray tracing with GYOTO and a geometrically thin, optically thick Page–Thorne disk finds two brightness rings whenever $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$06: an inner photon ring at $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$07 and an outer lensing ring at $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$08 (Nascimento et al., 27 Oct 2025).

Several ringdown studies probe the same transition through wave dynamics. In the quartic “novel black-bounce” spacetime,

$$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$09

echoes appear only in the traversable-wormhole regime $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$10; as $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$11 increases further, the echo train is replaced by the fundamental wormhole quasinormal mode (Yang et al., 2021). In the charged case, echoes occur for

$$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$12

while a regular black hole with two horizons requires

$$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$13

and $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$14 gives a two-way traversable wormhole (Wu et al., 2022).

Matter-source ambiguity leaves an additional imprint on ringdown. For axial perturbations of the Simpson–Visser metric, an anisotropic-fluid interpretation leads to a single-channel Regge–Wheeler-type equation, whereas a nonlinear-electrodynamics-plus-scalar interpretation yields an irreducibly coupled two-channel system. Time-domain extractions show branch-dependent damping: in the black-hole branch $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$15, the nonlinear-electrodynamics interpretation damps faster than the anisotropic-fluid model; in the wormhole branch $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$16, the nonlinear-electrodynamics coupled system produces longer-lived fundamental modes (Yang et al., 21 Mar 2026).

Thermodynamically, generalized static black-bounce models admit a standard tunneling derivation of Hawking temperature,

$$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$17

and a positive Hernandez–Misner–Sharp quasi-local mass,

$$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$18

In the models studied, the curvature invariants remain finite, the quasi-local mass approaches $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$19 at the bounce and $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$20 at infinity, and the temperature profile can exhibit maxima, zero-temperature extremal points, and monotonic suppression as $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$21 or deformation parameters increase (Moreira et al., 27 Jul 2025). This suggests stable, cold remnants in some parameter regions, though the detailed phase structure depends on the chosen mass function.

Beyond minimally deformed Schwarzschild, black-bounce geometries arise in several quantum-gravity-motivated frameworks. A T-duality-inspired effective source produces a regular metric with horizons for $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$22, an extremal null throat at $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$23, and a traversable wormhole for $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$24; comparison with the Event Horizon Telescope shadow of Sgr A* constrains the model to

$$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$25

at the $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$26 level (Alencar et al., 4 Mar 2026). In quasi-topological gravity, an infinite tower of higher-curvature terms yields

$$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$27

or exactly

$$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$28

with a minimal radius $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$29 and a black-to-white-hole bounce rather than a singular $$\Sigma(r)=\sqrt{r^2+a^2},\qquad f(r)=1-\frac{2M(r)}{\Sigma(r)}.$$30 region (Ling et al., 29 Aug 2025). Rotating and charged black-bounce geometries extend the same regularization idea to Kerr–Newman-like spacetimes; the rotating case retains a rank-2 Killing tensor and a Carter-like constant, but not the full Killing tower of principal tensor and Killing–Yano tensor (Franzin et al., 2021).

Taken together, these results place black-bounce spacetimes at the intersection of regular black holes, traversable wormholes, thin-shell constructions, anisotropic fluids, nonlinear electrodynamics, and effective quantum-gravity corrections. The common geometric core is the replacement of the central singularity by a nonzero minimal sphere; the main open discriminants are then dynamical and observational, including tidal sign flips, double-ring optical structure, echo production, source-dependent ringdown, and thermodynamic endpoint behavior.