Black-bounce spacetime and galactic rotation curves: from singularity resolution to observable gravitational effects

Technical Summary: Black-bounce spacetime and galactic rotation curves

Problem Statement
General Relativity (GR) successfully describes gravitational phenomena across vast scales but inevitably predicts spacetime singularities within compact objects, where curvature scalars diverge and predictive power breaks down. While various regular black hole models (e.g., Bardeen, Hayward) have been proposed to resolve these singularities by modifying the interior geometry, many rely on non-standard matter fields or lack systematic confrontation with precision observational data. The black-bounce spacetime, introduced by Simpson and Visser, offers a geometric resolution by replacing the radial coordinate $r$ with an effective radius $\sqrt{x^2 + a^2}$, where $a$ is a regularization parameter. This framework eliminates the central singularity and allows for a continuous transition between a regular black hole ($a < 2M$), an extremal wormhole ($a = 2M$), and a traversable wormhole ($a > 2M$). However, prior to this work, the black-bounce metric had not been systematically tested against precision data at Solar System scales or applied to galactic dynamics.

Methodology
The authors investigate the observational implications of the black-bounce metric across three distinct regimes using geometrized units ($G=c=1$) unless otherwise noted:

1. Metric Analysis and Horizon Structure: The study reviews the line element $ds^2 = -f(x)dt^2 + f(x)^{-1}dx^2 + (x^2+a^2)(d\theta^2 + \sin^2\theta d\phi^2)$ with $f(x) = 1 - 2M/\sqrt{x^2+a^2}$. It analyzes the horizon condition $f(x)=0$ to define the three geometric phases based on the ratio of the bounce parameter $a$ to the ADM mass $M$.
2. Solar System Precision Tests: The authors derive leading-order corrections to three canonical GR observables:
   • Perihelion Precession: Using geodesic perturbation theory for Mercury, they calculate the precession angle $\Delta\phi$ as a function of $a$.
   • Gravitational Lensing: They compute the light deflection angle for null geodesics in the equatorial plane.
   • Shapiro Time Delay: They integrate the null geodesic equation for radar signals passing near a massive body to determine the time delay.
3. Galactic Rotation Curve Fitting: For the first time, the black-bounce metric is applied to galactic dynamics. In the weak-field limit, the metric yields an effective potential $\Phi_{BB}(r) = -GM/\sqrt{r^2+a^2}$. The authors construct a four-component mass decomposition model to fit the observed rotation curve of the spiral galaxy NGC 7331:
   • A black-bounce central potential.
   • A Freeman exponential stellar disc.
   • A Hernquist stellar bulge.
   • A Navarro–Frenk–White (NFW) dark matter halo.
     The total circular velocity is the quadrature sum of these components. The model parameters are constrained by minimizing the weighted $\chi^2$ statistic against high-resolution HI and H$\alpha$ rotation curve data for NGC 7331.

Key Contributions and Results

• Solar System Constraints: The analysis reveals that the bounce parameter $a$ introduces small but qualitatively distinct corrections to GR predictions:
  • Perihelion Precession: The precession angle decreases monotonically as $a$ increases.
  • Gravitational Lensing: The deflection angle decreases as $a$ increases.
  • Shapiro Time Delay: Unlike the other two tests, the time delay increases with $a$.
    These trends provide complementary observational signatures; current high-precision measurements of Mercury and the Cassini experiment imply that $a$ must be significantly smaller than the solar radius.
• Galactic Kinematics Constraint: The fitting of the NGC 7331 rotation curve yields the first observational constraint on the bounce parameter from galaxy kinematics. The best-fit model (with $\chi^2_{red} = 1.56$) determines:
  $$a = 1.035 \pm 0.119 \text{ kpc}$$
  This value is significantly smaller than the disc scale radius ($h \approx 3.13$ kpc) and the NFW scale radius, indicating that the bounce effect is confined to the innermost galactic core.
• Physical Interpretation: The fitted parameter $a$ acts as a softening length that regularizes the central cusp, causing the circular velocity to rise linearly ($v \propto r$) at small radii rather than following a Keplerian decline. However, the black-bounce potential alone (scaling as $r^{-1}$ at large distances) cannot sustain the flat outer rotation curves; the NFW dark matter halo remains necessary to explain the dynamics at large radii.

Significance and Claims
The paper claims to establish the black-bounce spacetime as a "compelling testbed" for singularity-free alternatives to classical black holes. Its primary significance lies in:

1. Observational Validation: Providing the first quantitative, observational constraints on the bounce parameter $a$ using both Solar System precision tests and galactic rotation curves.
2. Regularization without Dark Matter Replacement: Demonstrating that the black-bounce metric can successfully regularize the inner galactic core (resolving the "cusp" issue) while remaining fully consistent with standard dark matter halo models at large radii. It does not claim to replace dark matter but rather acts as a complementary geometric modification.
3. Scale Dependence: The authors note a significant discrepancy between the Solar System constraint ($a \sim 10^3$ km) and the galactic constraint ($a \sim 1$ kpc). They interpret this not as a failure of the model, but as an expected feature where $a$ acts as an effective, system-dependent regularization length rather than a universal constant, analogous to how structural parameters (like core radii) vary across different galaxies.

The work concludes that the black-bounce metric offers a physically transparent, single-parameter framework that bridges black hole and wormhole geometries, offering distinct, testable deviations from GR that are currently consistent with existing data.