Simpson–Visser BB Solution

• Simpson–Visser-type BB solution is a minimal extension of Schwarzschild geometry that replaces the singularity with a bounce function.
• It enables smooth transitions between regular black holes, one-way or two-way traversable wormholes, and Ellis–Bronnikov geometries using two key parameters.
• The solution employs a composite matter source, including a phantom scalar field and nonlinear electrodynamics, yielding distinctive astrophysical and lensing signatures.

The Simpson–Visser-type Black-Bounce (BB) solution is a class of globally regular, static, spherically symmetric spacetimes introduced as a minimal one-parameter extension of the Schwarzschild geometry. The key feature is the regularization of the central singularity by replacing the radial coordinate with a “bounce” function, leading to a metric that smoothly interpolates between regular black holes, traversable wormholes, and special cases such as the Ellis–Bronnikov wormhole. The solution is realized as an exact solution of the Einstein field equations with a composite matter source—typically a phantom scalar field with a self-interaction potential and a magnetic field described by nonlinear electrodynamics. This metric is explicitly constructed to enable a continuous transition between various spacetime structures as a function of two principal parameters, most commonly denoted as $m$ (analogous to mass) and %%%%1%%%% (the bounce, or regularization, parameter).

1. Metric Structure and Parameter Domain

The canonical Simpson–Visser metric is formulated (here in Buchdahl-type coordinates) as: $$ds^2 = -A(r) dt^2 + \frac{dr^2}{A(r)} + (r^2 + a^2)(d\theta^2 + \sin^2\theta\, d\phi^2)$$ with

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

where $m \geq 0$ and $a \geq 0$.

The parameter $a$ modulates the interior structure:

• $a = 0,\ m \neq 0$: Schwarzschild black hole.
• $0 < a < 2m$: Regular black hole ("black-bounce") with a spacelike throat (no singularity).
• $a = 2m$: One-way traversable wormhole; the throat is null.
• $a > 2m$: Two-way traversable wormhole with a timelike throat.
• $a \neq 0,\ m = 0$: Reduces to the Ellis–Bronnikov wormhole (a reflection-symmetric traversable wormhole).

The function $r^2 + a^2$ ensures that the "areal" radius never vanishes, regularizing curvature invariants everywhere, including $r=0$.

2. Field Source Content and Exact Solution

The Simpson–Visser metric is not a vacuum solution. It is supported by a composite, anisotropic matter source:

• Phantom Scalar Field: A real scalar field with negative kinetic energy. For static, spherically symmetric solutions, $\phi(r) = \pm \arctan(r/a) + \mathrm{const}$ (up to normalization).
• Nonlinear Electrodynamics (NED): A magnetic field with field strength $F_{23} = q\sin\theta$, generating an invariant $\mathcal{F} = 2q^2/(r^2 + a^2)^2$.

The total stress-energy tensor incorporates both contributions, with explicit Lagrangians for NED and a self-interaction potential $V(\phi)$ for the scalar field. The key Einstein equation components in these metrics show that at the throat ($r=0$) the null energy condition (NEC) is violated (i.e., $T^t_t - T^r_r < 0$), a feature necessary for supporting traversable wormhole regions.

For the Reissner–Nordström-like generalization (including electric charge $Q$), the metric function becomes

$$A(r) = 1 - \frac{2M}{\sqrt{r^2 + a^2}} + \frac{Q^2}{r^2 + a^2}$$

while maintaining regularity at $r=0$.

3. Horizons, Throats, and Global Structure

Solutions exhibit a rich variety of causal structures depending on parameter choices:

• Event Horizons: Roots of $A(r)=0$. For $a<2m$, two symmetric horizons exist at $r = \pm\sqrt{4m^2 - a^2}$.
• Throat ("Bounce"): The minimum value of the areal radius at $r=0$, which for $a>2m$ becomes the wormhole throat.
• Carter–Penrose Diagrams: Show transitions between R (static) and T (nonstatic) regions and the global structure, including multiple asymptotically flat regions in the wormhole cases.

In extended solutions ($a \neq 0$), the replacement $r \to \sqrt{r^2 + a^2}$ eliminates the $r=0$ singularity, and all curvature scalars remain finite. The regularizing parameter $a$ thus not only controls the presence or absence of horizons but decomposes the spacetime into black hole, wormhole, or black-bounce domains.

4. Lensing, Orbits, and Observational Signatures

Gravitational Lensing

• Strong Deflection Limit: The deflection angle $\alpha(b)$ for impact parameter $b$ near the photon sphere $b_m$ generally exhibits a logarithmic divergence: $$\alpha(b) = -\bar{a} \log[(b/b_m) - 1] + \bar{b} + O( (b/b_m - 1)\log(b/b_m - 1))$$ where $\bar{a}$ and $\bar{b}$ are metric-dependent coefficients.
• Degenerate Case $a=3m$: The photon sphere and antiphoton sphere merge at the throat, generating a "marginally unstable photon sphere." The divergence then becomes nonlogarithmic: $$\alpha(b) = \frac{\tilde{c}}{(b/b_m - 1)^{1/4}} + \bar{d}$$ This leads to correspondingly different gravitational lensing signatures for the relativistic images—relativistic ring spacings and magnification differ fundamentally from both black holes and "generic" wormholes (Tsukamoto, 2020).

Circular Orbits and Tidal Effects

• ISCO and Photon Spheres: The innermost stable circular orbit (ISCO) radius and photon sphere position are both $a$-dependent and, for large enough $a$, may vanish or be replaced by a throat.
• Epicyclic Oscillations: The radial and vertical epicyclic (oscillation) frequencies are analytically computable and are tuned by $a$. The $3:2$ frequency ratio for high frequency quasi-periodic oscillations (HF QPOs), central in accretion physics, shifts outward in the wormhole regime and facilitates matching QPO data for supermassive compact objects (Stuchlík et al., 2021, Dasgupta et al., 19 Sep 2025).
• Tidal Forces: Tidal stresses experienced by infalling observers remain finite everywhere, in contrast to the divergent Schwarzschild case, with distinctive oscillatory profiles for angular tidal deviations near the bounce (Arora et al., 2023).

Shadows

• Photonsphere and Shadow Radius: The photonsphere radius is a function of $a$, but the critical impact parameter and shadow radius match the Schwarzschild value for all $a$: $b_p = 3\sqrt{3}m$ (Jha, 2023).

5. Thermodynamics, Accretion, and Astrophysical Applications

• Accretion Disk Signatures: Simulations of Novikov–Thorne disks show the optical images for $a > 0$ closely resemble Schwarzschild, but accretion disk luminosity, temperature profiles, and radiation flux increase systematically with growing $a$, distinguishing regular black holes, wormholes, and Schwarzschild cases in high-precision spectral data (Bambhaniya et al., 2021).
• Bondi Accretion: The location of the sonic point, accretion rate, and disk luminosity depend sensitively on $a$. Charged generalizations add further, though minor, shifts; joint measurements of $a$ and $Q$ are proposed as observational markers (Gambino et al., 29 Jul 2025).
• Thermodynamics: The entropy and temperature formulas remain Bekenstein–Hawking-like but evaluated at the effective horizon or throat location. For metrics regularized with a cloud of strings, the heat capacity is always negative (thermal instability), and the energy density can be rendered positive with sufficient string density (Rodrigues et al., 2022).
• Phase Transitions: Modifications (e.g., in Verlinde’s emergent gravity context) admit phase transitions, with multiple mass ranges corresponding to different stability regimes, detectable via behavior of heat capacity and temperature as functions of $a$ (Filho, 2023).

6. Generalizations, Matter Couplings, and Regularity

• Nonlinear Electrodynamics Coupling: The most general solutions allow for nonminimal couplings between the phantom scalar field and (nonlinear) electromagnetic fields, i.e., terms $W(\phi)\mathcal{L}(F)$ in the action. The coupling function $W(\phi)$ can be reconstructed from the assumed radial profile of $\mathcal{L}_F = d\mathcal{L}/dF \propto F^n$, leading in special cases to minimal coupling ($W(\phi) = 1$). Regularity is preserved for a large class of such couplings as long as the bounce structure remains intact (Cordeiro et al., 28 Sep 2025).
• Multiple Horizons and Throats: The metric can be further extended to allow for multiple horizons, throats, and anti-throats by introducing oscillatory functions multiplying the original bounce structure (e.g., cosine factors of $a_0 / \sqrt{a^2 + x^2}$). This leads to spacetimes with alternating trapped and anti-trapped regions, supported by an anisotropic fluid (partially phantom scalar + NED), and a regular Kretschmann scalar (Rodrigues et al., 1 Feb 2025).
• Geodesic Completeness: The Simpson–Visser regularization $r \to \sqrt{r^2 + r_0^2}$, when applied generically to any incomplete static, spherically symmetric regular solution, renders the geometry geodesically complete, allowing smooth extension across $r=0$ and symmetry in $r$ (Pal et al., 2023).

7. Astrophysical and Observational Implications

• Gravitational Lensing and Shadows: Accurate modeling of strong field lensing and black hole shadows demonstrates that while certain observables (e.g., angular image separations, flux ratios) differ in theory from the classic Kerr case, the absolute discrepancies are typically at the $\mathcal{O}(\mu$as) level—beyond current EHT capabilities but potentially within reach for next-generation arrays (Islam et al., 2021, Jha et al., 2022).
• Parameter Constraints: For the M87* black hole, EHT data constrain the bounce parameter $\ell$ and, when augmented with noncommutative geometry effects, provide finite observational bounds: $\ell \in [0.827158 M, 0.85739801 M]$, suggesting the viability of this geometry for astrophysical black hole modeling (Jha et al., 2022).
• QPO Modeling: Current data on high-frequency QPOs do not yet allow discrimination between the Kerr and Simpson–Visser scenarios due to parameter degeneracies and uncertainties, though systematic comparison with model predictions can in principle constrain the regularization parameter $\beta = a$ or $l$ (Dasgupta et al., 19 Sep 2025).

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Table: Parameter Regimes and Associated Spacetime Types

Regime	Condition	Geometry
Schwarzschild	$a=0,\ m\neq0$	Classical black hole
Regular Black Hole	$0	Black-bounce, regular BH
One-way Wormhole	$a=2m$	Null throat, one-way traversable
Two-way Wormhole	$a>2m$	Traversable, symmetric wormhole
Ellis-Bronnikov	$m=0,\ a\neq0$	Reflection-symmetric wormhole

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The Simpson–Visser-type BB solution is thus distinguished by its minimal but powerful generalization of classical black holes, the explicit realization as an exact solution with rich field content, the capacity for parameter-controlled transitions between black holes and wormholes, and a robust regularity ensured independently of the detailed matter coupling. Its phenomenology connects gravitational lensing, accretion, thermodynamics, and fundamental issues of regularity and geodesic completeness—a combination that makes it a canonical reference model for nonsingular static spherically symmetric compact objects in general relativity and beyond.