Black-Bounce-Schwarzschild Deflection Angle

• Black-Bounce-Schwarzschild Deflection Angle is a measure of how gravitational bending in a regularized Schwarzschild metric, with a bounce parameter 'a', eliminates central singularities.
• The analysis applies analytic and perturbative methods in both weak and strong field regimes, incorporating energy, angular momentum conservation, and velocity-dependent corrections.
• Derived lensing observables, including explicit a² corrections, offer potential astrophysical signatures to distinguish between black holes and traversable wormhole geometries.

The Black-Bounce-Schwarzschild deflection angle quantifies the gravitational bending of null and timelike geodesics in the Simpson–Visser "black-bounce-Schwarzschild" spacetime, a regularization of the Schwarzschild solution characterized by a bounce parameter $a$ that eliminates the central singularity and interpolates between a black hole and traversable wormhole geometries. The deflection angle is central to the analysis of gravitational lensing phenomena in these regular geometries and provides a direct probe of deviations from the Schwarzschild metric through corrections induced by $a$. This formalism allows precise analytic and perturbative computations of lensing observables in both the weak and strong deflection regimes, with explicit parameter dependence relevant for astrophysical modeling and for distinguishing black-bounce metrics from their singular counterparts.

1. The Black-Bounce-Schwarzschild Metric and Geodesics

The Simpson–Visser black-bounce-Schwarzschild metric is given by

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

where $M$ is the ADM mass and $a \geq 0$ is the bounce parameter. For $a=0$ this reduces to the Schwarzschild solution; for $a>0$ the central singularity is replaced by a smooth throat, and for $a$ exceeding critical values the geometry may represent a traversable wormhole.

Geodesic motion in the equatorial plane ($\theta = \pi/2$) is considered for both null ($ds^2=0$) and timelike ($ds^2>0$) particles. Conservation of energy and angular momentum leads to an effective potential

$$V_{\text{eff}}(r) = \frac{L^2}{r^2+a^2}\left(1-\frac{2M}{\sqrt{r^2+a^2}}\right),$$

where $L$ is the angular momentum per unit mass. The turning point or radius of closest approach $r_0$ is determined by initial conditions, and the impact parameter $b$ is expressed as $$b^2 = (r_0^2 + a^2)/\left(1 - \frac{2M}{\sqrt{r_0^2 + a^2}}\right)$$ (Furtado et al., 28 Apr 2025, Tsukamoto, 2020).

2. Weak-Field Deflection Angle: Expansion and Bounce Corrections

In the regime $M/b \ll 1$, the weak-field expansion of the deflection angle $\alpha$ for both light and massive particles is derived via post-Minkowskian (PM) or post-Newtonian (PN) expansion and perturbative techniques. For relativistic particles with rest mass $\mu$ and speed $0 < v \leq 1$, the expansion through $\mathcal{O}(M^3/b^3)$ is (He et al., 2024): $$\alpha = 2\left(1+\frac{1}{v^2}\right)\frac{M}{b} + \frac{\pi}{4}\left[ 3\left(1+\frac{4}{v^2}\right)+\frac{a^2}{M^2}\right]\frac{M^2}{b^2} + \frac{2}{3}\left[5+\frac{45}{v^2}+\frac{15}{v^4}-\frac{1}{v^6}+\left(1+\frac{3}{v^2}\right)\frac{a^2}{M^2}\right]\frac{M^3}{b^3} + \mathcal{O}\left( (M/b)^4 \right).$$

For null geodesics ($v=1$), the light deflection admits the simplified expansion (Övgün, 2020, Nascimento et al., 2020, Jia, 2020, Furtado et al., 28 Apr 2025): $$\alpha(b) = \frac{4M}{b} + \frac{15\pi}{4} \frac{M^2}{b^2} + \frac{\pi a^2}{4 b^2} + \frac{128}{3}\frac{M^3}{b^3} + \frac{8a^2}{3M^2}\frac{M^3}{b^3} + \mathcal{O}\left( (M/b)^4, Ma^2/b^3\right).$$

Key features:

• The leading term is the classical Einstein deflection; $a$ enters at $\mathcal{O}(a^2/b^2)$ with a positive coefficient, indicating increased bending for larger $a$.
• Velocity dependence increases the leading $M/b$ term for $v<1$, i.e., slower particles are more strongly bent.
• Comparing to the Schwarzschild case ($a=0$), the corrections are strictly positive and subdominant for $a\ll b$.

Multiple works confirm the universality of the $\pi a^2/(4b^2)$ correction in the weak-field limit via geometric, perturbative, and Gauss–Bonnet approaches (Övgün, 2020, Javed et al., 2023).

3. Strong Deflection Limit and Critical Parameters

For impact parameters approaching the critical value $b_c$, the deflection angle diverges logarithmically, characteristic of lensing near the photon sphere. In the black-bounce-Schwarzschild background, the photon sphere is located at (Tsukamoto, 2020, Furtado et al., 28 Apr 2025, Jia et al., 2020): $$r_{\rm ph} = \sqrt{9 M^2 - a^2}\quad\text{(for }a<3M\text{)},$$ and the corresponding critical impact parameter is

$b_c = 3\sqrt{3}\, M,$

independent of $a$ for $a < 3M$. The strong field deflection admits the expansion: $$\alpha(b) = -\bar{a}(a,M)\, \ln\left( \frac{b}{b_c} - 1 \right) + \bar{b}(a,M) + \mathcal{O}( (b/b_c-1)\ln(b/b_c-1) ).$$ where $\bar{a}(a,M) = 3M / \sqrt{9M^2-a^2}$ (Tsukamoto, 2020, Furtado et al., 28 Apr 2025, Nascimento et al., 2020). The logarithmic slope increases with $a$, enhancing the sharpness of the divergence.

For $a \to 3M$ (the "marginal case"), the divergence becomes non-logarithmic, replacing the logarithm with a power law $(b/b_c-1)^{-1/4}$, corresponding to a degenerate photon sphere (Tsukamoto, 2020).

For $a>3M$, the critical impact parameter and the structure of the photon sphere change, with new regimes relevant to wormhole lensing.

4. Lensing of Massive Particles and Velocity-Dependent Effects

Extension to massive, neutral test particles with speed $v<1$ alters both the weak- and strong-field structure of the bending angle. The leading term for massive particles is enhanced to $2(1+1/v^2)M/b$ (He et al., 2024), and higher-order velocity-dependent terms are nontrivial. The particle sphere radius and critical impact parameter become functions of both $a$ and $v$, with the general form (He et al., 4 Feb 2026): $$r_c = M \sqrt{ \frac{(4v^2 -1 + \xi)^2}{4v^4} - \frac{a^2}{M^2}},\quad \xi \equiv \sqrt{1+8v^2},$$

$$u_c(v) = \frac{2(3+\xi)^{3/2}}{(1+\xi)\sqrt{\xi-1}}\, M.$$

In the ultrarelativistic limit $v\to 1$, these reduce to the photon sphere and impact parameter of the null case.

In the strong field regime, the standard logarithmic form persists, but the "slope" and "offset" coefficients $\bar{a}(v,a),\,\bar{b}(v,a)$ acquire explicit velocity dependence, increasing the phenomenological richness compared to the null case (He et al., 4 Feb 2026).

5. Topological and Physical Interpretation

The positive-definite $a^2$ corrections in the deflection angle reflect the regularized geometry’s throat, confirming that lensing can probe global, nonlocal features of the spacetime. The Gauss–Bonnet theorem provides a robust interpretation: the integrated effect of curvature encodes global topological information, appearing as an additional contribution to the bending (Övgün, 2020, Javed et al., 2023).

Physically, the bounce parameter $a$ "smears out" the central curvature singularity, broadening the photon sphere and softening the gravitational potential near the core. In the weak field, this produces $\mathcal{O}(a^2)$ increments to the bending; in the strong field, the divergence of the deflection angle as $b\to b_c$ is enhanced (for $a < 3M$).

6. Observational Significance and Astrophysical Applications

The black-bounce-Schwarzschild deflection angle underpins lensing observables including image positions, magnification ratios, time delays, and shadow radii for black holes and wormholes (He et al., 2024, He et al., 4 Feb 2026, Furtado et al., 28 Apr 2025). Applying these results to supermassive black holes such as Sgr A* enables explicit computation of the bounce- and velocity-induced effects on practical lensing observables.

Corrected shadow radii are given as $r_{\text{sh}}(a) = r_m / \sqrt{f(r_m)}$, decreasing monotonically with $a$ (Furtado et al., 28 Apr 2025, Jia et al., 2020). For $a \to 3M$, the shadow radius vanishes, indicating the collapse of the photon sphere on the throat.

Although $a$-dependent modifications are typically subdominant for astrophysically plausible $a\ll b$, detection of such effects through high precision lensing or shadow observations could in principle probe the regularization scale of the compact object.

7. Summary Table of Deflection Angle Expansions

Regime	Deflection Angle $\alpha(b)$	$a$-dependence
Weak-field, null	$$\frac{4M}{b} + \frac{15\pi M^2}{4b^2} + \frac{\pi a^2}{4b^2}$$	First order $a^2$ correction at $1/b^2$
Weak-field, massive	$$2(1 + 1/v^2)\frac{M}{b} + \cdots + (\pi/4)a^2 M^0 b^{-2}$$	Strong velocity corrections; $a^2$ enters at $1/b^2$
Strong-field	$$-\bar{a}(a)\ln\left(\frac{b}{b_c} - 1\right) + \bar{b}(a)$$	$\bar{a}(a)=3M/\sqrt{9M^2-a^2}$; sharper divergence for larger $a$

These results are consistent across analytic, geometric, and perturbative approaches (He et al., 2024, Tsukamoto, 2020, Övgün, 2020, Jia, 2020, Furtado et al., 28 Apr 2025, Jia et al., 2020).

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Principal references: (He et al., 2024, He et al., 4 Feb 2026, Tsukamoto, 2020, Övgün, 2020, Nascimento et al., 2020, Furtado et al., 28 Apr 2025, Jia, 2020, Jia et al., 2020, Javed et al., 2023).