Simpson–Visser Metric: Black Bounce & Wormhole Lensing

• Simpson–Visser Metric is a two-parameter, spherically symmetric solution that smoothly transitions from Schwarzschild black holes to regular (non-singular) black holes and traversable wormholes.
• It employs a mass parameter (m) and a regularization parameter (a) to define the causal structure and photon sphere configurations, affecting gravitational lensing phenomena.
• Strong deflection analyses in this metric reveal distinct logarithmic and power-law divergences, offering a framework to differentiate exotic compact objects in astrophysical observations.

A Simpson–Visser metric (also called black-bounce or SV metric) is a two-parameter, spherically symmetric solution to the Einstein equations that interpolates between the Schwarzschild black hole, regular (non-singular) black holes, and traversable wormholes. It is defined by a mass parameter $m$ and a nonnegative regularization parameter $a$ that governs the causal and geometric structure of the spacetime. The SV metric is important for modeling singularity-free compact objects in gravitational lensing and related strong-field phenomenology.

1. Definition and Geometric Structure

The Simpson–Visser spacetime has the line element in Buchdahl coordinates: $$ds^2 = -A(r)\,dt^2 + B(r)\,dr^2 + C(r)\,(d\theta^2 + \sin^2\theta\,d\varphi^2)$$ with

$$A(r) = 1 - \frac{2m}{\sqrt{r^2 + a^2}},\quad B(r) = \frac1{A(r)},\quad C(r) = r^2 + a^2.$$

Alternatively, redefine the areal (Schwarzschild) radius as $\rho = \sqrt{r^2 + a^2}$ so that the 2-sphere area is $4\pi\rho^2$. The parameters $m\geq 0$, $a\geq 0$ control the mass and the “bounce” (regularizing) scale.

Distinct regimes: | Parameter values | Geometric interpretation | |-------------------------|------------------------------------------| | $a=0$, $m\neq0$ | Schwarzschild black hole | | $0 < a < 2m$ | Regular black hole (“black bounce”) | | $a = 2m$ | One-way traversable wormhole (throat is null) | | $a > 2m$ | Two-way traversable wormhole | | $a\neq0$, $m=0$ | Ellis–Bronnikov wormhole |

The position of the event horizon (when present) is given by solving $A(r_h)=0$: $\rho_h = \sqrt{r_h^2 + a^2} = 2m \implies r_h = \sqrt{(2m)^2 - a^2}\quad \text{(for %%%%14%%%%).}$

2. Photon Sphere and Effective Potential

The photon (and antiphoton) spheres, which determine the strong lensing structure, are located at radii that extremize the effective potential for null geodesics. The condition is: $V'_\text{eff}(r) = 0,$ with

$V_\text{eff}(r) = \frac{\ell^2}{C(r)}A(r)$

where $\ell$ is the (conserved) angular momentum per unit energy.

For Schwarzschild ($a=0$): unique photon sphere at $\rho=3m$.

For $a > 0$: the “radius” entering the photon sphere calculation is $\rho = \sqrt{r^2 + a^2}$. For $a < 2m$ (regular black hole), the photon sphere remains outside the event horizon. As $a$ increases, the causal and topological structure changes:

• When $a > 3m$, the throat itself can be a photon sphere.
• At $a = 3m$ two photon spheres (one on each side of the throat) and the antiphoton sphere merge into a degenerate “marginally unstable” photon sphere at $r=0$ (the throat).

The conditions for this marginal degeneracy are: $$V(r_m) = V'(r_m) = 0,\quad V''(r_m) = V'''(r_m) = 0,\quad V^{(4)}(r_m)<0.$$

3. Strong Deflection Limit and Deflection Angle

In the vicinity of the photon sphere, the gravitational lensing deflection angle, as a function of impact parameter $b$, diverges. The divergent behavior depends crucially on the SV parameters:

• For typical (nondegenerate) cases $a \neq 3m$, the deflection angle for light rays with impact parameter $b$ just above the critical $b_m$ is: $$\alpha(b) = -\bar{a}\ln\left(\frac{b}{b_m} - 1\right) + \bar{b} + O\left(\left(\frac{b}{b_m}-1\right)\ln\left(\frac{b}{b_m}-1\right)\right),$$ where $\bar{a}$ and $\bar{b}$ are constants determined by $m$ and $a$ (in the Schwarzschild case, $\bar{a}=1$ and $\bar{b} = \ln[216(7-4\sqrt{3})] - \pi$).
• For $a=3m$: the divergence switches to a nonlogarithmic power-law: $$\alpha(b) = \frac{\bar{c}}{(b/b_m - 1)^{1/4}} + \bar{d},$$ with $\bar{c}$ and $\bar{d}$ set by the geometry at the (degenerate) throat. This “softer” divergence reflects the quartic nature of the effective potential’s extremum.

4. Lensing Observables: Image Positions and Magnifications

In the strong deflection regime, the precise nature of the deflection angle divergence dictates the positions and amplification of “relativistic images” (images formed by photons orbiting arbitrarily close to the photon sphere before escaping):

• In the standard logarithmic case ($a \neq 3m$):

$$\theta_n^{(0)} = \left[1 + \exp\left(\frac{\bar{b} - 2\pi n}{\bar{a}}\right)\right]\theta_\infty, \quad \theta_\infty = \frac{b_m}{D_\text{OL}}$$

where $n$ is the winding number, $D_\text{OL}$ is the observer–lens distance.

• For $a = 3m$, the distinct $(b/b_m – 1)^{-1/4}$ divergence leads to qualitatively different scaling for the angular positions and separations of the images; the angular separation $s$ between the outermost relativistic image and the shadow edge has an altered dependence that facilitates discrimination between regular black holes and wormhole-like throats.

The character of the divergence thus informs both the angular separation and brightness ratio between images.

5. Physical Interpretation of the Parameters and Regimes

Both $m$ and $a$ control the causal structure of the metric:

Combination	Causal Structure	Central region
$a=0$	Schwarzschild	curvature singularity at $r=0$
$0 < a < 2m$	Regular black hole	smooth bounce replaces singularity
$a = 2m$	One-way traversable wormhole	null throat
$a > 2m$	Two-way traversable wormhole	spacelike throat
$m=0$	Ellis–Bronnikov wormhole	non-singular

Photon (and antiphoton) spheres can exist on one or both sides of the throat, with geometries transitioning from black-hole to wormhole character as $a$ increases.

6. Observational and Theoretical Implications

• The behavior of gravitational lensing in the SV metric encodes information about the deep properties of the spacetime, in particular the presence or absence of a photon sphere at the throat, the number of relativistic images, and the scaling properties of their angular separations.
• The “softer” power-law divergence of the deflection angle at $a=3m$ provides a potentially distinguishable signature of the marginally unstable photon sphere regime, as opposed to standard black holes where the divergence is logarithmic.
• The SV metric’s ability to interpolate between black-hole and wormhole regimes using a single parameter makes it a valuable testbed for examining observational signatures of non-singular, exotic compact objects. These signatures may be detectable in high-resolution strong lensing observations distinguishing black holes from wormhole mimickers (Tsukamoto, 2020).
• The explicit formulas for the metric, deflection angles, photon spheres, and relativistic image positions enable precise modeling of strong-lensing phenomena and shadow structures in both the black hole and wormhole parameter space.

7. Summary Table: Key Formulas and Regimes

Regime	Deflection angle $\alpha(b)$	Photon sphere behavior	Causal structure
$a \neq 3m$	$-\bar{a}\ln[(b/b_m)-1] + \bar{b} + \ldots$	Unique photon sphere	Black hole or wormhole, depending on $a$
$a = 3m$	$\bar{c}/(b/b_m - 1)^{1/4} + \bar{d}$	Marginally unstable at $r=0$	Quartic degeneracy at the throat

This highlights the critical role played by the regularization parameter $a$ in determining both the lensing phenomenology and the spacetime’s global causal structure. The Simpson–Visser metric thus provides a unified framework for probing gravitational lensing, photon orbits, and observational signatures in regular black holes and traversable wormholes (Tsukamoto, 2020).