Composite BH–Halo Spacetimes

• Composite BH–halo spacetimes are exact solutions to Einstein’s equations representing black holes embedded in dark matter halos with anisotropic, extended matter distributions.
• They utilize specific halo density profiles and anisotropic stress–energy tensors to compute modified horizon, photon sphere, and shadow properties.
• Recent studies reveal that these models can alter quasinormal mode frequencies and thermodynamic relations, offering measurable effects in gravitational lensing and ringdown signals.

A composite black hole–halo spacetime is an exact or approximate solution to the Einstein equations representing a central black hole (BH) embedded within a non-vacuum environment modeled by a galactic-scale dark matter (DM) halo. Such configurations capture essential astrophysical features of supermassive black holes at galactic centers, where DM is dynamically significant on parsec-to-kiloparsec scales. Their properties depend crucially on the specific halo density profile, the degree of anisotropy in the stress–energy tensor, and the interplay between the relativistic horizon and extended matter distribution. The composite spacetime supports a broad range of physical phenomena, including modified photon spheres and shadows, altered quasinormal mode (QNM) spectra, perturbed geodesic structure, and distinctive thermodynamic signatures.

1. Relativistic Construction and Halo Modeling

Composite BH–halo metrics are derived by solving Einstein’s equations for a central BH plus a halo modeled as an anisotropic fluid or collisionless system. The standard spherically symmetric ansatz is: $$ds^2 = -f(r)\,dt^2 + g(r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\theta\,d\phi^2),$$ where $f(r)$ and $g(r)$ generally differ due to anisotropies in the matter source (Bolokhov, 7 Dec 2025). For a given halo density profile $\rho_{\text{halo}}(r)$, the Misner–Sharp mass function is constructed as

$$m(r) = 4\pi\int_0^r \rho_{\text{halo}}(\xi)\xi^2 d\xi + M_0,$$

with $M_0$ the BH mass. The equations of motion then yield $g(r) = 1 - 2m(r)/r$ and a generalized TOV structure for $f(r)$, with pressure components determined by stress–energy conservation. Imposing the restrictive condition $f(r) = g(r)$ is only valid for unphysical equations of state $P_r = -\rho$ (Bolokhov, 7 Dec 2025).

Leading profiles in the recent literature include:

• Dehnen (1,4,2): $\rho(r) = \rho_s (r/r_s)^{-2}[1 + r/r_s]^{-2}$ (Pathrikar, 4 Nov 2025, Uktamov et al., 26 May 2025).
• Hernquist: $\rho(r) = M_{\text{halo}}\,a/(2\pi r (r+a)^3)$ (Macedo et al., 20 Feb 2024, Tan et al., 26 Sep 2024).
• Burkert: $\rho(r) = \rho_0 r_0^3/[(r + r_0)(r^2 + r_0^2)]$ (Yang et al., 11 Nov 2025).
• Pseudo-isothermal: $\rho(r) = \rho_0/(1 + (r/r_c)^2)$ (Yang et al., 2023).
• Extensions to charged/rotating cases and inclusion of cosmological constant or galactic environmental effects have also been developed (Konoplya et al., 3 Sep 2025, Uktamov et al., 30 Aug 2025, Xu et al., 2018).

2. Curvature, Horizon Structure, and Energy Conditions

The BH event horizon location, $r_h$, is determined by the largest root of $g(r_h) = 0$ or $f(r_h) = 0$ where appropriate. The presence of the halo generally pushes $r_h$ outward compared to the Schwarzschild case, with the shift depending on the enclosed DM mass within $r_h$ (Uktamov et al., 26 May 2025, Lobo et al., 7 May 2025). For the Dehnen (1,4,2) profile,

$$f(r) = 1 - \frac{2M}{r} - 8\pi \rho_s r_s^2 (1 + r_s/r)^3 \ln(1 + r_s/r)$$

quantifies the negative correction due to the halo (Pathrikar, 4 Nov 2025).

Curvature invariants such as the Kretschmann scalar,

$$K = R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma} = K_{\rm Schw}(r) + \Delta K_{\rm halo}(r),$$

exhibit enhanced sensitivity to the halo in the near-horizon region, with all invariants diverging at $r\to0$ as in vacuum BHs, and effects scaling with DM density (Lobo et al., 7 May 2025).

The energy–momentum tensor typically takes an anisotropic form, $T^\mu{}_\nu = \text{diag}[-\rho, P_r, P_t, P_t]$, with halo parameters set so that all standard energy conditions (null, weak, strong, dominant) are satisfied for physical ranges (Uktamov et al., 26 May 2025, Xiong et al., 16 Nov 2025).

3. Photon Spheres, Null Geodesics, and Shadows

Photon spheres in composite spacetimes obey: $$r_{\text{ph}} f'(r_{\text{ph}}) - 2f(r_{\text{ph}}) = 0,$$ with the shadow radius for a far observer given by

$$R_{\text{sh}} = r_{\text{ph}} / \sqrt{f(r_{\text{ph}})}$$

in units where $c=G=1$.

The inclusion of DM generically shifts the photon sphere and shadow radius outward, but for parameter regimes relevant to realistic galaxy centers, the fractional correction is typically at the $\sim 10^{-2}$ or per-mille level (Macedo et al., 20 Feb 2024, Lobo et al., 7 May 2025). In the limit of extreme compactness or high DM concentration, novel features arise; for example, multiple light rings (unstable and stable) and nontrivial deviations in lensing observables (Xavier et al., 2023).

The ISCO (innermost stable circular orbit) and null circular orbits are likewise shifted, altering the orbital dynamics of both non-spinning and spinning test particles (Tan et al., 26 Sep 2024). For spinning particles, the ISCO moves inwards, and pericenter precession is enhanced, with implications for EMRI (extreme-mass-ratio inspiral) waveforms (Tan et al., 26 Sep 2024).

4. Linear Perturbations, Ringdown, and Quasinormal Modes

Scalar, electromagnetic, and gravitational field perturbations in composite BH–halo spacetimes obey generalized Regge–Wheeler–type master equations with effective potentials of the form: $$V_{s,\ell}(r) = f(r)\left[\frac{\ell(\ell+1)}{r^2} + \text{spin-dependent terms}\right]$$ (Pathrikar, 4 Nov 2025, Yang et al., 11 Nov 2025).

Quasinormal mode frequencies are typically computed using the high-order WKB method enhanced by Padé approximants or the continued fraction (Leaver) method. Key trends:

• Increasing halo density ($\rho_s$) or scale radius ($r_s$) in Dehnen-halo models: both the real and imaginary parts of QNM frequencies decrease, yielding longer ringdown times and lower oscillation frequencies (Pathrikar, 4 Nov 2025).
• For cored (Burkert) halos: both $\omega_R$ and $|\omega_I|$ increase with core radius or central density, leading to faster, more strongly damped oscillations (Yang et al., 11 Nov 2025).
• Consistency of extraction methods: WKB and Leaver approaches agree to within $0.1\%$–level for low overtones and $\ell \ge 1$ (Yang et al., 11 Nov 2025, Pathrikar, 4 Nov 2025).

Greybody factors and associated Hawking emission are enhanced due to the lower effective potential barrier, resulting in increased transmission probabilities compared to vacuum (Pathrikar, 4 Nov 2025, Hamil et al., 24 May 2025). The “sparsity” of Hawking radiation is likewise decreased in the presence of an extended halo (Hamil et al., 24 May 2025).

5. Thermodynamics and Stability

The presence of a DM halo alters standard thermodynamic relations. The horizon radius and entropy shift by percent-level amounts depending on halo parameters. For example, the explicit mass–entropy relation in halo models calibrated to NGC 4649 is: $$M(S, V_c, a) = \frac{\sqrt{S} \left[\pi a^2 + S - 2S V_c^2\right]}{2\sqrt{\pi} (\pi a^2 + S)}$$ (Lobo et al., 7 May 2025).

The Hawking temperature generally decreases (or even vanishes for extremal configurations), and the presence of a finite tangential pressure at the horizon in the extremal limit signals a non-trivial extension of BH thermodynamics (Lobo et al., 7 May 2025). The specific heat remains negative, demonstrating the persistence of thermodynamic instability (Lobo et al., 6 May 2025).

6. Observational Signatures and Constraints

Composite BH–halo spacetimes predict modifications to shadow size, gravitational lensing, light deflection, and QNM spectra, all of which are testable observables (Pathrikar, 4 Nov 2025, Lobo et al., 6 May 2025, Macedo et al., 20 Feb 2024). The Event Horizon Telescope constraints on the Sgr A* shadow, for example,

$4.55\,M \lesssim R_{\text{sh}} \lesssim 5.22\,M,$

place upper bounds on allowed halo density and scale radius in composite metrics (Pathrikar, 4 Nov 2025).

Ringdown frequency shifts of order $10^{-3}$–$10^{-2}$ due to DM are below current GW detector sensitivity but may become accessible to future LISA-class missions. Imaging constraints are tighter for models predicting multiple light rings or for compact halos. The interplay between string clouds (if present), halo properties, and shadow radius further provides a means of disentangling environmental parameters with next-generation facilities (Ahmed et al., 23 Jun 2025).

7. Theoretical Limits and Consistency Issues

Recent analysis emphasizes that several constructions widely used in the literature, in which the static spherically symmetric metric is assumed to satisfy $f(r) = g(r)$, produce unphysical solutions unless the radial pressure equals $-\rho$ (Bolokhov, 7 Dec 2025). The correct procedure requires solving the Einstein system for independent $f(r)$ and $g(r)$ given an input density profile and physically justified anisotropies. When properly constructed, all standard energy conditions are satisfied, and the resulting metrics correspond to legitimate BH–halo spacetimes suitable for strong-field and phenomenological analyses (Bolokhov, 7 Dec 2025, Xiong et al., 16 Nov 2025).

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References

• "Signatures of a Schwarzschild-like Black Hole Immersed in Dark Matter Halo" (Pathrikar, 4 Nov 2025)
• "Static Black Hole Solution with a Dark Matter Halo" (Uktamov et al., 26 May 2025)
• "Ringdown of a black hole embedded in a Burkert dark matter halo" (Yang et al., 11 Nov 2025)
• "Revisiting black holes in dark-matter halos: on consistent solutions to the Einstein equations" (Bolokhov, 7 Dec 2025)
• "Thermodynamics and Null Geodesics of a Schwarzschild Black Hole Surrounded by a Dehnen Type Dark Matter Halo" (Gohain et al., 3 Jul 2024)
• "Motion of spinning particles around black hole in a dark matter halo" (Tan et al., 26 Sep 2024)
• "Black hole spacetimes with dark matter spikes: Energy-momentum tensor and backreaction effects" (Xiong et al., 16 Nov 2025)
• "Shadows of black holes with dark matter halo" (Xavier et al., 2023)
• "Relativistic structure of a supermassive black hole embedded in the dark matter halo of NGC 4649 (M60)" (Lobo et al., 7 May 2025)
• "Observable signatures of Black Holes with Hernquist Dark Matter Halo having a cloud of strings" (Ahmed et al., 23 Jun 2025)
• "Supermassive black hole in NGC 4649 (M60) with a dark matter halo: Impact on shadow measurements and thermodynamic properties" (Lobo et al., 6 May 2025)
• "Optical appearance of black holes surrounded by a dark matter halo" (Macedo et al., 20 Feb 2024)
• "Hyperboloidal approach for static spherically symmetric spacetimes" (Macedo, 2023)
• "Black Hole Space-time In Dark Matter Halo" (Xu et al., 2018)
• "Black hole surrounded by the pseudo-isothermal dark matter halo" (Yang et al., 2023)