Simpson–Visser Spacetime Overview

• Simpson–Visser spacetime is a regularized black hole metric that replaces the central singularity with a nonzero minimal areal radius.
• It unifies Schwarzschild, black-bounce, and traversable wormhole geometries by varying the regularization parameter relative to mass.
• Extensions to charged, rotating, and AdS scenarios reveal modified thermodynamics, lensing, and accretion dynamics with practical observational implications.

The Simpson–Visser spacetime is a static, spherically symmetric regularization of Schwarzschild in which the singular areal radius is replaced by a nonvanishing function with a regular minimum, so that the same metric family interpolates among Schwarzschild, regular black holes or black-bounces, one-way wormholes, and traversable wormholes. In the literature surveyed here, the regularization parameter is denoted $a$, $l$, or $\ell$ in the static case, while the rotating literature عادة uses $a$ for spin and $l$ or $\ell$ for the regularizing length scale. The construction has also been extended to Reissner–Nordström-like, Kerr-like, AdS, braneworld, and other generalized settings (Bronnikov et al., 2021).

1. Geometric definition and coordinate realizations

A common static form of the Simpson–Visser geometry is

$$ds^2 = A(x)\, dt^2 - \frac{dx^2}{A(x)} - r^2(x)\,d\Omega^2, \qquad r(x)=\sqrt{x^2+a^2},$$

with $d\Omega^2=d\theta^2+\sin^2\theta\,d\varphi^2$. The essential replacement is

$r\mapsto \sqrt{x^2+a^2}, \qquad dr\mapsto dx,$

so the areal radius never reaches zero and instead satisfies $r(x)\ge a>0$. In the Schwarzschild-like case,

$l$0

Equivalently, many papers write the metric in coordinates $l$1 as

$l$2

or in the notation of recent Bondi-accretion work,

$l$3

with

$l$4

In all such forms, the central geometric distinction is the same: the area of the symmetry 2-spheres is $l$5, $l$6, or $l$7, so the would-be center is replaced by a minimal 2-sphere (Bronnikov et al., 2021, Gambino et al., 29 Jul 2025).

The same prescription extends directly to charged geometries. The Reissner–Nordström-like Simpson–Visser metric is obtained from

$l$8

or, in the $l$9-notation,

$\ell$0

Thus $\ell$1 plays the role of electric charge while the regularization parameter still removes the vanishing-areal-radius center (Bronnikov et al., 2021, Gambino et al., 29 Jul 2025).

2. Causal regimes, horizons, and regularity

The regime structure is controlled by the relative size of the regularization parameter and the mass scale. In the neutral case the literature states, equivalently,

$\ell$2

for a regular black hole or black-bounce with horizons,

$\ell$3

for the critical case, and

$\ell$4

for a traversable wormhole. At the critical value, the terminology differs across papers: the same limiting geometry is described as an “extremal black hole with a double horizon,” a “one-way wormhole with a null throat,” or a “black throat.” This is a difference of description, not of the underlying metric family (Bambhaniya et al., 2021, Bronnikov et al., 2021).

For the Schwarzschild-like SV metric, the horizon condition is

$\ell$5

hence

$\ell$6

In coordinates where the areal radius itself is denoted by $\ell$7, one often sees

$\ell$8

These formulas are consistent once the coordinate choice is kept fixed. A useful caution appears in recent accretion work: when the metric is written with auxiliary coordinate $\ell$9 and areal radius $a$0, the natural horizon locations are

$a$1

while the corresponding areal radius at the horizon is

$a$2

This distinction matters when comparing papers that use $a$3 and papers that use $a$4 as the non-areal radial coordinate (Gambino et al., 29 Jul 2025).

Regularity is encoded directly in the geometry. Because the minimum areal radius is nonzero, the center $a$5 is removed from the manifold. In the RN-like case the curvature invariants at $a$6 are explicitly finite for any $a$7: $a$8

$a$9

$l$0

They diverge as $l$1, confirming that the regularization parameter is precisely what removes the singularity (Bronnikov et al., 2021).

3. Matter sources and energy-condition structure

A major line of work treats Simpson–Visser metrics not merely as kinematic ansätze but as exact solutions of Einstein equations coupled to matter. One reconstruction result states that any static, spherically symmetric SV metric can be sourced by a minimally coupled phantom scalar field with nonzero potential $l$2 together with a magnetic nonlinear electrodynamics sector $l$3, where

$l$4

For the Schwarzschild-like and RN-like SV geometries, explicit source functions are given. The scalar profile is

$l$5

and the scalar must be phantom, $l$6, because the geometry requires null-energy-condition violation near the throat or bounce (Bronnikov et al., 2021).

The reason is geometric. For static spherical metrics with areal radius $l$7, the combination

$l$8

shows that near any minimum of $l$9 one has $\ell$0, so NEC violation is unavoidable. For the SV choice

$\ell$1

and one work states that NEC is violated throughout the spacetime except at horizons where $\ell$2 (Bronnikov et al., 2021).

Later papers keep the same overall source structure but generalize the geometry. One study of “black bounces as magnetically charged phantom regular black holes” embeds the original Simpson–Visser metric in Einstein gravity with nonlinear electrodynamics and a self-interacting phantom scalar, and identifies it as part of a broader class of static, spherically symmetric, asymptotically flat black bounces of Simpson–Visser type (Cañate, 2022). Another constructs oscillatory deformations with multiple horizons, throats, and anti-throats, generated by a scalar field coupled to nonlinear electrodynamics and characterized by a partially phantom scalar field plus a magnetic monopole; in that class the energy conditions can be satisfied or violated depending on the spacetime region (Rodrigues et al., 1 Feb 2025). A further extension uses the Simpson–Visser prescription as a generic method for geodesically completing regular but geodesically incomplete black-hole spacetimes, again with a phantom scalar field plus nonlinear electrodynamics as exact source matter (Pal et al., 2023).

4. Rotating Simpson–Visser geometry and orbital dynamics

The rotating Simpson–Visser spacetime is a Kerr-like regularization in Boyer–Lindquist-type coordinates. In the notation used in the rotating literature,

$\ell$3

with

$\ell$4

Here $\ell$5 is the ADM mass, $\ell$6 is the spin parameter, and $\ell$7 is the regularization parameter. The metric reduces to Kerr for $\ell$8 and to Schwarzschild when $\ell$9 (Patel et al., 2022).

The horizon radii are

$$ds^2 = A(x)\, dt^2 - \frac{dx^2}{A(x)} - r^2(x)\,d\Omega^2, \qquad r(x)=\sqrt{x^2+a^2},$$0

and the static limit surface is

$$ds^2 = A(x)\, dt^2 - \frac{dx^2}{A(x)} - r^2(x)\,d\Omega^2, \qquad r(x)=\sqrt{x^2+a^2},$$1

The rotating literature distinguishes several regimes: a two-horizon “Regular Black Hole-2” branch, a one-horizon “Regular Black Hole-1” branch, an extremal regular black hole for $$ds^2 = A(x)\, dt^2 - \frac{dx^2}{A(x)} - r^2(x)\,d\Omega^2, \qquad r(x)=\sqrt{x^2+a^2},$$2 and $$ds^2 = A(x)\, dt^2 - \frac{dx^2}{A(x)} - r^2(x)\,d\Omega^2, \qquad r(x)=\sqrt{x^2+a^2},$$3, and horizonless wormhole-like branches when $$ds^2 = A(x)\, dt^2 - \frac{dx^2}{A(x)} - r^2(x)\,d\Omega^2, \qquad r(x)=\sqrt{x^2+a^2},$$4 is sufficiently large or $$ds^2 = A(x)\, dt^2 - \frac{dx^2}{A(x)} - r^2(x)\,d\Omega^2, \qquad r(x)=\sqrt{x^2+a^2},$$5. Geometrically, increasing $$ds^2 = A(x)\, dt^2 - \frac{dx^2}{A(x)} - r^2(x)\,d\Omega^2, \qquad r(x)=\sqrt{x^2+a^2},$$6 shrinks the horizon and ergoregion radii and can eliminate horizons entirely (Patel et al., 2022).

Despite these geometric changes, the maximum Penrose-process efficiency at the event horizon is unchanged from Kerr and independent of $$ds^2 = A(x)\, dt^2 - \frac{dx^2}{A(x)} - r^2(x)\,d\Omega^2, \qquad r(x)=\sqrt{x^2+a^2},$$7. The quoted values are the standard Kerr ones, reaching

$$ds^2 = A(x)\, dt^2 - \frac{dx^2}{A(x)} - r^2(x)\,d\Omega^2, \qquad r(x)=\sqrt{x^2+a^2},$$8

at extremality. This makes the rotating SV spacetime a Kerr mimicker in the restricted sense of horizon-level Penrose efficiency, even though its horizon and ergoregion structure are $$ds^2 = A(x)\, dt^2 - \frac{dx^2}{A(x)} - r^2(x)\,d\Omega^2, \qquad r(x)=\sqrt{x^2+a^2},$$9-dependent (Patel et al., 2022).

A separate timing study analyzes orbital and epicyclic frequencies of test particles in the rotating SV background for high-frequency QPO phenomenology. It uses the dimensionless deformation parameter

$d\Omega^2=d\theta^2+\sin^2\theta\,d\varphi^2$0

and tests eleven HFQPO models against six black-hole sources. The reported conclusion is that the observationally favored models cannot discriminate between the Kerr and Simpson–Visser scenarios: Kerr corresponds to $d\Omega^2=d\theta^2+\sin^2\theta\,d\varphi^2$1, nonzero $d\Omega^2=d\theta^2+\sin^2\theta\,d\varphi^2$2 is often allowed, and model degeneracy remains dominant (Dasgupta et al., 19 Sep 2025).

5. Null geodesics, lensing, and optical structure

In the static SV spacetime, null geodesics encode several characteristic transitions. One thin-disk study states that the photon circular orbit exists for $d\Omega^2=d\theta^2+\sin^2\theta\,d\varphi^2$3 at

$d\Omega^2=d\theta^2+\sin^2\theta\,d\varphi^2$4

with critical impact parameter

$d\Omega^2=d\theta^2+\sin^2\theta\,d\varphi^2$5

so the shadow size is the same as in Schwarzschild and does not depend on $d\Omega^2=d\theta^2+\sin^2\theta\,d\varphi^2$6. The same work gives the marginally stable circular orbit as

$d\Omega^2=d\theta^2+\sin^2\theta\,d\varphi^2$7

For $d\Omega^2=d\theta^2+\sin^2\theta\,d\varphi^2$8, there is no photon sphere and the wormhole throat contributes instead (Bambhaniya et al., 2021).

Strong-deflection lensing refines that picture. In the notation of that analysis, the critical impact parameter remains

$d\Omega^2=d\theta^2+\sin^2\theta\,d\varphi^2$9

for all $r\mapsto \sqrt{x^2+a^2}, \qquad dr\mapsto dx,$0, so the photon-ring diameter alone does not distinguish Schwarzschild from the regular black-bounce and wormhole branches as long as $r\mapsto \sqrt{x^2+a^2}, \qquad dr\mapsto dx,$1. For $r\mapsto \sqrt{x^2+a^2}, \qquad dr\mapsto dx,$2, the throat is an antiphoton sphere; at the special value

$r\mapsto \sqrt{x^2+a^2}, \qquad dr\mapsto dx,$3

two photon spheres and the throat antiphoton sphere merge into a marginally unstable throat orbit, and the deflection angle diverges nonlogarithmically: $r\mapsto \sqrt{x^2+a^2}, \qquad dr\mapsto dx,$4 For $r\mapsto \sqrt{x^2+a^2}, \qquad dr\mapsto dx,$5, the throat itself becomes the outer photon sphere and

$r\mapsto \sqrt{x^2+a^2}, \qquad dr\mapsto dx,$6

This makes Simpson–Visser an especially explicit model of how photon spheres, antiphoton spheres, and wormhole throats reorganize the strong-field lensing problem (Tsukamoto, 2020).

If the same background geometry is interpreted dynamically as a solution sourced by nonlinear electrodynamics plus a phantom scalar, physical photons need not follow the background metric’s null geodesics. One orbital study introduces the effective optical metric

$r\mapsto \sqrt{x^2+a^2}, \qquad dr\mapsto dx,$7

with effective line element

$r\mapsto \sqrt{x^2+a^2}, \qquad dr\mapsto dx,$8

In that formulation the photon effective potential differs from the ordinary massless-particle potential by an overall factor $r\mapsto \sqrt{x^2+a^2}, \qquad dr\mapsto dx,$9, so the light-ring radii are unchanged but the critical impact parameter and absorption cross section are reduced. The same work states that, with its sign conventions, photons are spacelike with respect to the background metric except for purely radial propagation, which is why the effective metric is physically necessary in the NED interpretation (Silva et al., 2024).

6. Accretion, disk emission, and tidal disruption

Accretion studies probe Simpson–Visser geometry through both thin-disk and spherical flows. For Novikov–Thorne thin disks, one analysis finds that the direct images are “almost similar compared to the Schwarzschild geometry,” so visual morphology alone is strongly degenerate. The electromagnetic properties are less degenerate: increasing the regularization parameter $r(x)\ge a>0$0 increases the temperature, emitted radiative flux, and observed luminosity of the disk, while the accretion efficiency remains fixed at the Schwarzschild value

$r(x)\ge a>0$1

The same study argues that thermal and spectral observables can in principle distinguish wormhole geometries from Schwarzschild and from black-bounce geometries, even though the images themselves remain nearly indistinguishable (Bambhaniya et al., 2021).

A recent Bondi-accretion analysis treats both neutral and charged SV geometries under steady-state, spherically symmetric, test-fluid assumptions. In the neutral case the mass flux and Bernoulli-type relations are

$r(x)\ge a>0$2

and the critical-point conditions yield, for the neutral metric,

$r(x)\ge a>0$3

The reported numerical result is that the regularization parameter $r(x)\ge a>0$4 mainly shifts the sonic-point location, while the critical velocity is remarkably robust: $r(x)\ge a>0$5 to numerical precision. For charged SV accretion, the charge $r(x)\ge a>0$6 produces only modest additional shifts in the sonic radius and slightly suppresses the inflow velocity near the horizon in the barotropic case. The paper therefore proposes a hierarchy in which $r(x)\ge a>0$7 is the dominant geometric marker and $r(x)\ge a>0$8 is a smaller perturbation (Gambino et al., 29 Jul 2025).

Tidal-force studies reach a related conclusion. In the $r(x)\ge a>0$9-coordinate form

$l$00

the static-observer tidal components are

$l$01

while for a radially infalling observer the radial component is unchanged but the angular component becomes energy-dependent: $l$02 A subsequent Roche-limit analysis uses the radial tidal force to define the disruption radius of stars through

$l$03

Its central result is that increasing the regularization parameter $l$04 weakens the tidal field and decreases the Roche radius, so stellar disruption can move from outside the horizon to inside the horizon or disappear altogether (Silva, 22 Jan 2026).

7. Thermodynamics and broader extensions

In AdS, the Simpson–Visser prescription produces a regular black hole with modified thermodynamics. For the uncharged SV–AdS geometry, the Hawking temperature and mass–radius relation are

$l$05

Imposing the first law $l$06 yields a modified entropy,

$l$07

The reported phase structure differs qualitatively from Schwarzschild–AdS: there is an extremal $l$08 limit, no Hawking–Page transition, and instead a first-order small/large black-hole transition with a van der Waals-like critical point (Kumar et al., 26 Nov 2025).

The charged SV–AdS extension combines the regularization parameter $l$09, electric charge $l$10, and AdS pressure. Its entropy becomes

$l$11

and the extended-phase-space equation of state is

$l$12

That study reports Van der Waals–like small/large black-hole behavior, non-area-law entropy, and explicit dependence of particle dynamics, shadow size, and thermodynamic stability on the regularization parameter (Ahmed et al., 15 Jan 2026).

A distinct Euclidean-action analysis of an Einstein–phantom-scalar–NLED realization of the SV geometry finds an unusual thermodynamic outcome: the asymptotic boundary contributions cancel so that the thermodynamic mass vanishes,

$l$13

while the magnetic charge, temperature, and entropy remain finite, and the first law takes the modified form

$l$14

The same work compares the regular SV black hole with a scalar-free singular black hole in the same heat bath and finds that the singular configuration always has lower free energy, so the regular SV branch is thermodynamically disfavored in that ensemble (Li et al., 11 Mar 2026).

Beyond asymptotically flat and AdS black holes, several papers use the Simpson–Visser prescription as a construction principle. It has been applied to geodesically complete regular black holes obtained by radial completion of already regular but geodesically incomplete metrics (Pal et al., 2023), to braneworld geometries in which the induced metric on a Randall–Sundrum brane is exactly Simpson–Visser and is generated by bulk deformations rather than brane matter (Crispim et al., 2024), and to oscillatory black-bounce metrics with multiple horizons, throats, and anti-throats that reduce to Simpson–Visser in an appropriate limit (Rodrigues et al., 1 Feb 2025). This suggests that “Simpson–Visser spacetime” now denotes both a specific regularized Schwarzschild family and a broader regularization method used to construct nonsingular black-hole and wormhole geometries across several settings.