Generalized Black-Bounce Spacetimes

• Generalized black-bounce spacetimes are regular geometries that replace central singularities with a finite minimal-area 2-sphere, interpolating between regular black holes and wormholes.
• They incorporate generalized metrics with nontrivial mass functions and multiple horizons, showcasing dynamic, rotating, and charged extensions that enrich causal structures.
• Supporting these spacetimes typically requires exotic matter—often a phantom scalar combined with nonlinear electrodynamics—though modified gravity can redistribute this exoticity.

Generalized black-bounce spacetimes are regular geometries in which the classical central singularity is replaced by a finite minimal-area 2-sphere, so that a single parametric family can describe a regular black hole, a one-way wormhole, or a traversable wormhole depending on how horizons sit relative to that minimum surface. In the literature, the original Simpson–Visser construction plays the role of the seed model, while later work broadened the framework to include nontrivial mass functions, multiple horizons, multiple throats and anti-throats, dynamical Vaidya-type evolutions, rotating and charged analogues, and modified-gravity or effective-quantum realizations (Simpson, 2021, Lobo et al., 2020).

1. Seed geometry and basic classification

The standard static seed metric is obtained by replacing the Schwarzschild areal radius by a bounced radius,

$ds^{2}=-\left(1-\frac{2m}{\sqrt{r^{2}+a^{2}}}\right)dt^{2} +\frac{dr^{2}}{1-\frac{2m}{\sqrt{r^{2}+a^{2}}} +\left(r^{2}+a^{2}\right)\left(d\theta^{2}+\sin^{2}\theta\, d\phi^{2}\right),$

with global radial coordinate $r\in(-\infty,+\infty)$, mass parameter $m$, and bounce parameter $a$ (Simpson, 2021). The areal radius is

$R(r)=\sqrt{r^{2}+a^{2}},$

so the area of the symmetry spheres is

$A(r)=4\pi(r^2+a^2),$

and the minimal 2-sphere sits at $r=0$, where $A(0)=4\pi a^2$ (Simpson, 2021).

Horizons occur where

$$\sqrt{r^2+a^2}=2m, \qquad r_\pm=\pm\sqrt{(2m)^2-a^2}.$$

This yields the basic taxonomy: $a=0$ gives Schwarzschild; $r\in(-\infty,+\infty)$0 gives a regular black hole or black-bounce; $r\in(-\infty,+\infty)$1 gives a one-way wormhole with a null throat; and $r\in(-\infty,+\infty)$2 gives a two-way traversable wormhole (Simpson, 2021). In this sense, black-bounce spacetimes are not a separate category from wormholes and regular black holes so much as a unifying interpolation between them.

The geometric distinction is controlled by the causal character of the minimal sphere. In the traversable regime the throat is timelike; at the critical value it is null; and in the black-bounce regime it is spacelike and hidden behind horizons (Simpson, 2021). This is why the term “black-bounce” is often reserved for the case in which the regularized core lies in a black-hole-type causal region, so that infalling worldlines encounter a bounce into another future region rather than a curvature singularity (Simpson, 2021).

2. Static generalizations beyond Simpson–Visser

A widely used static generalization is written in Buchdahl form,

$r\in(-\infty,+\infty)$3

with

$r\in(-\infty,+\infty)$4

Equivalently,

$r\in(-\infty,+\infty)$5

Here $r\in(-\infty,+\infty)$6 is the asymptotic mass, $r\in(-\infty,+\infty)$7 the bounce scale, and $r\in(-\infty,+\infty)$8 deformation parameters; the Simpson–Visser metric is recovered for $r\in(-\infty,+\infty)$9, and Schwarzschild is recovered as $m$0 (Moreira et al., 27 Jul 2025).

This family already produces qualitatively different causal structures. The case $m$1 has

$m$2

with horizons determined by $m$3, while the Bardeen-type black-bounce case $m$4 has

$m$5

and admits the extremal threshold

$m$6

For $m$7, that Bardeen-type model has four real horizons, interpreted as two event horizons and two Cauchy horizons, and its throat remains timelike even in the black-hole sector (Lobo et al., 2020). This is one of the clearest examples in which generalized black-bounces develop a much richer interior causal structure than the seed metric.

A different generalization uses a quasiglobal coordinate $m$8 and an oscillatory deformation,

$m$9

with

$a$0

The new parameter $a$1 introduces oscillations in both the lapse and the areal radius, thereby allowing multiple horizons together with multiple throats and anti-throats, where throats are local minima of $a$2 and anti-throats are local maxima (Rodrigues et al., 1 Feb 2025). In the limit $a$3, the metric reduces to the Simpson–Visser form. This suggests that “generalized black-bounce” is not just a label for one extra parameter but a broader geometric program in which the minimal-radius replacement of the center is combined with nontrivial internal radial structure.

3. Matter sources and energy-condition structure

In standard GR, the supporting matter sector is typically exotic. For static spherically symmetric black-bounce geometries of the form

$a$4

a general source-reconstruction scheme uses Einstein gravity coupled to a scalar field and nonlinear electrodynamics,

$a$5

With a purely magnetic ansatz

$a$6

the crucial reconstruction identity is

$a$7

For the usual black-bounce profiles, $a$8, so reality of the scalar requires $a$9, i.e. a phantom scalar (Rodrigues et al., 2023). In this class, the exoticity is therefore directly tied to the flare-out or bounce structure of the areal radius.

For the Simpson–Visser and Bardeen-type black-bounces, that program yields explicit phantom-scalar-plus-NED sources. In both examples the scalar profile is

$R(r)=\sqrt{r^{2}+a^{2}},$0

while the electromagnetic sector is nonlinear and magnetic rather than Maxwellian (Rodrigues et al., 2023). This places black-bounces in continuity with the broader GR literature on wormholes, regular black holes, phantom scalars, and nonlinear electrodynamics.

A more general theorem for static Buchdahl-coordinate models states that if $R(r)=\sqrt{r^{2}+a^{2}},$1, $R(r)=\sqrt{r^{2}+a^{2}},$2, and $R(r)=\sqrt{r^{2}+a^{2}},$3, then the standard pointwise energy conditions are violated (Lobo et al., 2020). For the canonical choice

$R(r)=\sqrt{r^{2}+a^{2}},$4

this makes the violation of $R(r)=\sqrt{r^{2}+a^{2}},$5 generic away from horizons (Lobo et al., 2020). A common misconception is therefore that regularity alone makes these spacetimes ordinary-matter solutions of GR; in the GR constructions surveyed here, the bounce structure is usually paid for by NEC violation.

Metric $R(r)=\sqrt{r^{2}+a^{2}},$6 gravity softens this conclusion but does not erase it. For

$R(r)=\sqrt{r^{2}+a^{2}},$7

the scalar-sector relation becomes

$R(r)=\sqrt{r^{2}+a^{2}},$8

which allows $R(r)=\sqrt{r^{2}+a^{2}},$9 to change sign across spacetime (Silva et al., 26 Feb 2025). In this framework, black-bounce solutions can be supported by nonlinear electrodynamics and a partially phantom scalar field, and in some regions of spacetime all energy conditions can be satisfied (Silva et al., 26 Feb 2025). Related $A(r)=4\pi(r^2+a^2),$0 analyses of the Simpson–Visser geometry also constructed a zero-density model with horizons and a positive-density model with

$A(r)=4\pi(r^2+a^2),$1

features explicitly contrasted with the GR case (Fabris et al., 2023). The modified-gravity sector does not eliminate exoticity uniformly, but it redistributes part of the burden from matter to curvature.

4. Dynamical, rotating, and theory-derived extensions

The simplest dynamical extension imports the Vaidya prescription into the black-bounce metric. In Eddington–Finkelstein form, one promotes $A(r)=4\pi(r^2+a^2),$2 and obtains

$A(r)=4\pi(r^2+a^2),$3

with $A(r)=4\pi(r^2+a^2),$4 or $A(r)=4\pi(r^2+a^2),$5 (Simpson et al., 2019). The minimal sphere remains at $A(r)=4\pi(r^2+a^2),$6, and its induced metric is

$A(r)=4\pi(r^2+a^2),$7

Hence the same fixed minimal sphere is timelike if $A(r)=4\pi(r^2+a^2),$8, null if $A(r)=4\pi(r^2+a^2),$9, and spacelike if $r=0$0 (Simpson et al., 2019). Apparent horizons occur at

$r=0$1

so accretion can drive a wormhole-to-black-bounce transition, while evaporation or negative-energy influx can reverse it (Simpson et al., 2019).

The review literature extends the same regularization logic to charged and rotating cases by keeping $r=0$2 unchanged and replacing explicit occurrences of $r=0$3 by $r=0$4 in the metric functions (Simpson, 2021). In the rotating charged black-bounce Kerr–Newman geometry,

$r=0$5

the singular Kerr–Newman core is replaced by a smooth bounce while horizons and ergosurfaces remain controlled by $r=0$6 and $r=0$7 in the usual way (Simpson, 2021). The same review also emphasizes that regularity, the extended radial range $r=0$8, and the smooth interpolation between regular black holes and traversable wormholes persist across static spherical, dynamical spherical, and stationary axisymmetric sectors (Simpson, 2021).

A stronger generalization derives black-bounce metrics from modified gravity rather than postulating them. In quasi-topological gravity, the static spherical solution obeys

$r=0$9

and in $A(0)=4\pi a^2$0 becomes

$A(0)=4\pi a^2$1

This construction introduces a minimal allowed radius

$A(0)=4\pi a^2$2

so the singular center is excluded and replaced by a bounce surface (Ling et al., 29 Aug 2025). A different theory-derived example uses a $A(0)=4\pi a^2$3-duality-inspired effective anisotropic fluid with $A(0)=4\pi a^2$4, and yields a black-bounce geometry with regimes $A(0)=4\pi a^2$5 regular black hole, $A(0)=4\pi a^2$6 extremal null throat, and $A(0)=4\pi a^2$7 traversable wormhole (Alencar et al., 4 Mar 2026). These theory-driven constructions show that generalized black-bounces can arise from higher-curvature or string-inspired effective dynamics, not only from reverse-engineered exotic matter.

5. Free-fall, tidal dynamics, and causal subtleties

For radial timelike geodesics in the general metric

$A(0)=4\pi a^2$8

one has the Killing energy

$A(0)=4\pi a^2$9

and in the infalling orthonormal tetrad the tidal tensor is diagonal,

$$\sqrt{r^2+a^2}=2m, \qquad r_\pm=\pm\sqrt{(2m)^2-a^2}.$$0

with

$$\sqrt{r^2+a^2}=2m, \qquad r_\pm=\pm\sqrt{(2m)^2-a^2}.$$1

Positive $$\sqrt{r^2+a^2}=2m, \qquad r_\pm=\pm\sqrt{(2m)^2-a^2}.$$2 means radial stretching, negative $$\sqrt{r^2+a^2}=2m, \qquad r_\pm=\pm\sqrt{(2m)^2-a^2}.$$3 radial compression; positive $$\sqrt{r^2+a^2}=2m, \qquad r_\pm=\pm\sqrt{(2m)^2-a^2}.$$4 means angular stretching, negative $$\sqrt{r^2+a^2}=2m, \qquad r_\pm=\pm\sqrt{(2m)^2-a^2}.$$5 angular compression (Crispim et al., 30 Jun 2025).

A central result is that, for the representative black-bounce models analyzed, the tidal components remain finite at the throat or minimal-radius surface (Crispim et al., 30 Jun 2025). In the Simpson–Visser case,

$$\sqrt{r^2+a^2}=2m, \qquad r_\pm=\pm\sqrt{(2m)^2-a^2}.$$6

while in the Bardeen-type black-bounce,

$$\sqrt{r^2+a^2}=2m, \qquad r_\pm=\pm\sqrt{(2m)^2-a^2}.$$7

This replaces the Schwarzschild tidal divergence by finite free-fall tidal fields all the way to the minimal sphere (Crispim et al., 30 Jun 2025).

The same analysis also reveals a distinctive non-Schwarzschild effect: sign reversals of the tidal components can create regions of radial compression together with angular stretching. The Bardeen-type black-bounce shows the clearest instance of this “reversal of spaghettification,” especially in inner regions associated with its richer event-horizon/Cauchy-horizon structure (Crispim et al., 30 Jun 2025). A plausible implication is that regularity alone is not the only dynamical novelty; the deformation pattern experienced by infalling bodies can be qualitatively different from that of singular black holes.

Related thin-shell bounce analogues underscore a causal caution. In matched Schwarzschild-to-Schwarzschild or Schwarzschild-to-de Sitter models, bounded radial geodesics can be continued across a static spacelike shell, but a single global conformal chart regular both at the shell and at horizons generally does not exist, except for special cases (Lin et al., 2023). This does not identify a pathology in smooth black-bounce metrics themselves, but it suggests that conformal compactifications of bounce geometries can be subtler than their nonsingular curvature invariants alone might indicate.

6. Optical, lensing, and thermodynamic phenomenology

Null geodesics in generalized $$\sqrt{r^2+a^2}=2m, \qquad r_\pm=\pm\sqrt{(2m)^2-a^2}.$$8-$$\sqrt{r^2+a^2}=2m, \qquad r_\pm=\pm\sqrt{(2m)^2-a^2}.$$9 black-bounce metrics are controlled by

$a=0$0

with

$a=0$1

The critical impact parameter is

$a=0$2

where the light-ring radius satisfies the extremum condition for $a=0$3 (Nascimento et al., 27 Oct 2025). This family includes the Simpson–Visser case at $a=0$4 and reduces to Schwarzschild as $a=0$5 (Furtado et al., 28 Apr 2025).

Strong-field optics can be organized by two thresholds: a horizon threshold $a=0$6 and an optical threshold $a=0$7. For $a=0$8, the spacetime is black-hole-like and supports a genuine shadow; for $a=0$9, it is horizonless but still has two light rings, producing nearly concentric thin rings; for $r\in(-\infty,+\infty)$00, it has neither horizons nor light rings, so a dark central cavity, if present, is not a geometric shadow in the strict sense (Nascimento et al., 27 Oct 2025). This directly addresses a common misconception in black-hole imaging: bright rings or dark cavities are not automatically evidence for an event horizon.

Weak-field lensing is comparatively less discriminating. For generalized $r\in(-\infty,+\infty)$01-$r\in(-\infty,+\infty)$02 black-bounces, the deflection angle to second order is

$r\in(-\infty,+\infty)$03

so $r\in(-\infty,+\infty)$04 and $r\in(-\infty,+\infty)$05 do not enter at that order (Furtado et al., 28 Apr 2025). By contrast, strong-field lensing and shadows do depend sensitively on the generalized structure through the photon-sphere equation and through

$r\in(-\infty,+\infty)$06

The same work emphasizes that generalized models can possess multiple horizons or multiple photon spheres, unlike the seed Simpson–Visser metric (Furtado et al., 28 Apr 2025).

Thermodynamics is likewise model-dependent. In the generalized static family with $r\in(-\infty,+\infty)$07, the Hawking temperature is computed from

$r\in(-\infty,+\infty)$08

and the surveyed models are described as free of curvature singularities, endowed with positive Hernandez–Misner–Sharp quasi-local masses, and capable of exhibiting multiple horizons, extremal configurations, and asymmetric cases (Moreira et al., 27 Jul 2025). For the Simpson–Visser case specifically,

$r\in(-\infty,+\infty)$09

so increasing $r\in(-\infty,+\infty)$10 lowers the temperature and drives it to zero at extremality (Moreira et al., 27 Jul 2025). Several generalized families show zero-temperature points or secondary thermal peaks, which the literature interprets as extremal or quasi-stable states (Moreira et al., 27 Jul 2025).

String-inspired effective-fluid black-bounces add an observational link. In the $r\in(-\infty,+\infty)$11-duality model, the shadow analysis yields compatibility with Event Horizon Telescope data provided

$r\in(-\infty,+\infty)$12

within the quoted $r\in(-\infty,+\infty)$13 range, while the heat capacity exhibits a divergence interpreted as a second-order phase transition (Alencar et al., 4 Mar 2026). This suggests that, in generalized black-bounce phenomenology, minimal-length parameters can simultaneously control causal structure, shadow size, and thermodynamic stability.