Convexity criterion and radial-profile response for off-shell Kerr geometries: a fuzzy-dark-matter profile as an analytic benchmark

Technical Summary: Convexity Criterion and Radial-Profile Response for Off-Shell Kerr Geometries

Problem Statement
The nature of dark matter (DM) remains a central open problem, with ultralight bosonic fields (fuzzy dark matter, FDM) offering a compelling alternative that modifies galactic structures via macroscopic de Broglie wavelengths. A critical challenge in modeling black holes (BHs) embedded in such environments is distinguishing between the asymptotic total mass (ADM mass) and the local radial mass distribution. Most existing approaches either treat DM perturbatively on a fixed Kerr background or introduce phenomenological metric modifications without rigorous consistency checks. Furthermore, constructing exact stationary, axisymmetric solutions for rotating black holes supported by realistic scalar configurations is generally intractable. The Newman–Janis algorithm is frequently used to generate rotating metrics from static seeds, but the complexification step is not unique, and the resulting "off-shell" geometries may not solve the same matter equations as their static seeds. This paper addresses the need for a controlled, analytic benchmark to isolate the effects of a finite radial mass profile on strong-field observables (horizons, photon spheres, shadows) without claiming a self-consistent rotating Einstein–Klein–Gordon solution.

Methodology
The authors construct an effective Kerr-like geometry based on a static, spherically symmetric seed derived from a specific FDM-inspired density profile:
$$\rho_{\text{FDM}}(r) = \rho_c \left[ 1 + \alpha \left( \frac{r}{r_c} \right)^2 \right]^{-8}$$
where $\alpha = 2^{1/8} - 1$.

1. Static Seed Construction: The enclosed mass $m(r)$ is integrated exactly from the density profile. The static metric is defined using a one-function closure condition ($g_{tt}g_{rr} = -1$), which fixes the radial pressure as $p_r = -\rho$. This yields a static anisotropic fluid solution.
2. Rotating Extension: The Newman–Janis algorithm is applied to the static seed to generate a stationary, axisymmetric metric. Crucially, the radial function $\Delta(r)$ is defined as $\Delta(r) = r^2 - 2r m(r) + a^2$, ensuring it depends only on $r$. This preserves the separability of the Hamilton–Jacobi equation for null geodesics (Carter separability).
3. Analytic Benchmarks: The authors derive a sufficient condition for the convexity of $\Delta(r)$ to classify horizon structures globally. They then perform a first-order perturbative expansion in the soliton mass fraction $f_{\text{sol}} = M_{\text{sol}}/M_{\text{ADM}}$ to derive response functions for horizons, extremal branches, and photon orbits.
4. Comparative Analysis: Results are compared against:
   • The full radial-profile calculation (exact within the ansatz).
   • A first-order perturbative expansion.
   • A constant-ADM-mass replacement (where $m(r) \to M_{\text{ADM}}$), which recovers the standard Kerr geometry.
   • Various completion uncertainties in the slow-rotation limit.

Key Contributions

1. Convexity Criterion for Off-Shell Kerr Geometries:
   The paper establishes a sufficient condition for the strict convexity of the radial function $\Delta(r) = r^2 - 2r m(r) + a^2$. For a positive, nondecreasing $C^2$ mass profile $m(r)$, the inequality:
   $$1 - 2m'(r) - r m''(r) > 0$$
   ensures that $\Delta(r)$ is strictly convex on $r > 0$.

   • Implication: This guarantees that for $a \neq 0$, $\Delta(r)$ has at most two positive roots (defining inner and outer horizons) or none. For $a=0$, there is exactly one positive root. This theorem provides a global classification of horizon topology, ruling out secondary extremal branches or additional horizons for a broad class of monotone mass functions, provided the derivative bound holds.

2. Analytic FDM-Inspired Benchmark:
   The authors demonstrate that the specific FDM profile satisfies the convexity criterion over a domain significantly wider than the parameter space used for numerical scans ($f_{\text{sol}} / (r_c/M_{\text{ADM}}) < 0.69$). They provide closed-form expressions for the enclosed mass, the extremal branch conditions, and the weak energy condition (WEC) boundaries.

3. Profile-Functional Response Theory:
   The paper derives first-order response formulas for observables under general deformations $m(r) = M_{\text{ADM}}(1 + \epsilon h)$.

   • Horizons and Extremality: The shifts in the outer horizon and extremal spin depend on local combinations of $h$ and $h'$.
   • Photon Region: The photon sphere radius depends on both the enclosed mass $m(r)$ and the mass gradient $m'(r)$.
   • Shadow: The shadow area and shape are shown to be sensitive to the radial distribution of mass, not just the total ADM mass.

4. Distinction Between Robust and Conditional Responses:
   A critical finding is the separation of responses into two categories:

   • Profile-Controlled (Robust): Static horizon location, photon sphere radius, and shadow area (to leading order) are determined solely by the radial mass function $m(r)$.
   • Completion-Dependent (Conditional): Spin-odd shadow displacements depend on the specific choice of the rotating completion (e.g., the frame-dragging function $\omega(r)$). The Newman–Janis prescription is not unique; alternative completions can produce comparable or larger displacements in the shadow, indicating that spin-dependent observables are not robust predictions of the radial profile alone.

Results

• Horizon Structure: The extremal spin $j_{\text{ext}}$ decreases as the environmental mass fraction $f_{\text{sol}}$ increases. The horizon radius moves inward compared to a Kerr black hole of the same ADM mass.
• Weak Energy Condition (WEC): The rotating effective source satisfies the WEC in the exterior region only if the outer horizon $r_+$ lies outside a critical radius $r_{\text{WEC}}$ where the transverse energy density becomes negative. This defines a "sign-based consistency domain" distinct from the horizon existence domain.
• Approximation Validity:
  • The constant-ADM-mass approximation (replacing $m(r)$ with $M_{\text{ADM}}$) fails to capture any radial-profile effects, returning the standard Kerr results regardless of $f_{\text{sol}}$.
  • The first-order perturbative expansion is accurate (sub-percent errors) for $f_{\text{sol}} \lesssim 0.1$ and moderate spins ($j_{\text{ADM}}/j_{\text{ext}} \lesssim 0.7$). However, it becomes non-uniform and inaccurate near the extremal limit ($j_{\text{ADM}} \to j_{\text{ext}}$), particularly for the horizon location.
• Shadow Observables: The environmental mass distribution modifies the shadow's size and shape. However, the horizontal displacement of the shadow is highly sensitive to the choice of the rotating completion, rendering it a non-robust indicator of the FDM profile in the absence of a fundamental rotating scalar solution.

Significance and Claims
The paper explicitly frames its contributions as formal analytic benchmarks rather than astrophysical predictions or self-consistent physical solutions.

• Not a Fundamental Solution: The authors emphasize that the constructed geometry is an "effective" metric. It does not claim to be a solution to the Einstein–Klein–Gordon equations for a rotating scalar field. The pressure closure and frame-dragging functions are chosen for analytic tractability (Newman–Janis) rather than derived from a fundamental matter Lagrangian.
• Benchmark Utility: The primary value lies in isolating the geometric consequences of a finite radial mass profile. It demonstrates that strong-field observables are controlled by local derivatives of the mass function ($m', m''$), which are lost when replacing the distributed mass with a point mass.
• Limitations: The paper notes that for realistic astrophysical FDM cores, the strong-field gradient effects are negligible ($\ll 10^{-20}$) unless the soliton is artificially compressed. The "compact production box" used for numerical scans is a formal parameter space, not a realization of an unperturbed FDM core.
• Conclusion: The work provides a rigorous mathematical framework (the convexity theorem) to classify horizon structures in off-shell Kerr geometries and quantifies the information loss incurred by approximating distributed dark matter as a constant mass. It highlights that while static observables are profile-dependent, spin-dependent observables require a source-derived rotating completion to be physically meaningful.