PFDM Effects on Black Hole Spacetimes

• PFDM is modeled via a logarithmic metric correction that modifies horizon locations, shadow dimensions, and photon dynamics.
• PFDM alters black hole thermodynamics by redefining entropy, temperature, and stability, leading to complex phase transitions.
• PFDM induces observable changes in gravitational wave ringdowns and shadow imaging, offering novel opportunities for astrophysical tests.

Perfect fluid dark matter (PFDM) effects on black hole spacetimes encompass a range of modifications to the classical theory of gravity, introducing new phenomenology for horizon structure, photon and particle dynamics, shadow properties, quantum probes, and black hole thermodynamics. PFDM is consistently modeled via a logarithmic correction term in the metric, characterized by parameters such as $k$, $\alpha$, $\lambda$, or $\beta$ depending on context, and represents the density/intensity of dark matter in the local environment of the black hole. The following sections provide an in-depth technical review of these effects, reflected in solutions generalizing Schwarzschild, Kerr, Hayward, Bardeen, and their AdS/dS and regularized counterparts.

1. Spacetime Structure and PFDM-Corrected Metrics

PFDM enters the metric via a characteristic term $\sim ({\rm parameter})/r \ln(r/|{\rm parameter}|)$, acting as an anisotropic fluid with density $\rho_{DM} \propto (\mathrm{parameter})/r^3$. For example, the Hayward black hole in PFDM (Ma et al., 2020) has

$$f(r) = 1 - \frac{2 M r^2}{r^3 + Q^3} + \frac{k}{r} \ln\left(\frac{r}{|k|}\right)$$

where $k$ is the PFDM intensity and $Q$ the nonlinear magnetic charge.

For rotating solutions (after Newman-Janis), the effective mass function acquires a $k$-dependent correction: $$m(r) = M - \frac{k}{2}\ln\left(\frac{r}{|k|}\right)$$ with the black hole structure, horizon locations, and ergoregion boundaries all sensitive to $k$.

For charged or regular Bardeen-type black holes with PFDM (Narzilloev et al., 2020), the metric is similarly constructed, and the event horizon $r_h$ and photon sphere $r_{ps}$ are solutions to transcendental equations involving PFDM. Increased PFDM density ($|k|$, $\alpha$, $\lambda$) can either shrink or (for certain parameter domains) expand horizons, alter the causal structure, and shift the positions of photon trapping surfaces.

2. Shadow Morphology, Photon Dynamics, and Observational Probes

Shadow Shape and Size

The shadow radius, calculated for an observer at infinity on the equatorial plane, is determined by effective impact parameters $\xi,\eta$ arising in the geodesic equations: $\alpha = -\xi, \qquad \beta = \pm \sqrt{\eta}$ The shadow of a static spacetime ($a=0$) is always circular; rotation ($a\neq0$), magnetic charge $Q$, and PFDM intensity $k$ induce deformation:

• Increasing $|k|$ (for $k<0$ as in (Ma et al., 2020)): enlarges the shadow, $R_s \uparrow$.
• Increasing $Q$: shrinks $R_s$, increases distortion ($\delta_s$).
• Rotation: distorts the shadow toward a D-shape, with extremal points shifting nontrivially with $a$ and $k$.

Combined analytic formulas allow direct calculation of the shadow observables (radius $R_s$, distortion parameter $\delta_s$) in terms of PFDM parameters, $a$, $Q$: $$R_s = \frac{(\alpha_t - \alpha_r)^2 + \beta_t^2}{2 |\alpha_r - \alpha_t|}, \qquad \delta_s = \frac{|\alpha_l - \alpha_a|}{R_s}$$ where the points $(\alpha_t,\beta_t), (\alpha_r,0), (\alpha_l, 0)$ are extremal shadow coordinates.

Summary Table: PFDM Effects on the Sgr A* Shadow ((Ma et al., 2020), Table 1)

$k$	$R_s$	$\delta_s$ (%)	$\theta_s$ ($\mu$as)
$-0.3$	7.2316	0.0790	36.981
$-0.6$	8.3992	0.0758	42.953
$-0.9$	9.2450	0.0812	47.278

Increase in $|k|$ yields a larger angular shadow size $\theta_s$.

Photon Sphere and Impact Parameters

PFDM generically modifies the photon sphere radius and critical impact parameters controlling strong-lensing observables (e.g., Bozza's formalism): $$x_m = \frac{3}{2} \beta \,\mathrm{ProductLog}\left(\cdots\right), \qquad b_m = \sqrt{\frac{H(x_m)}{F(x_m)}}$$ with $b_m$ controlling the strong-field deflection and shadow scale.

Energy Emission and Hawking Radiation

The high-frequency absorption cross-section, approximated as the shadow area, controls the (semi)classical emission rate: $$\frac{d^2 E(\omega)}{d\omega dt} = \frac{2\pi^2 \sigma_{\rm lim} \omega^3}{e^{\omega/T} - 1}$$ with Hawking temperature

$$T = \frac{r_+^2 f'(r_+) (r_+^2 + a^2) + 2 a^2 r_+ [f(r_+) - 1]}{4\pi (r_+^2 + a^2)^2}$$

PFDM suppresses the emission rate and shifts peak emission to lower frequencies as $|k|$ is increased (Ma et al., 2020).

Observational Prospects

To distinguish PFDM-induced effects, angular resolutions better than $\sim 1~\mu\mathrm{as}$ are required for $k$, considerably more stringent for spin or charge. With Sgr A* and M87* data, PFDM-induced shadow changes are in principle discernible with near-future VLBI systems ((Ma et al., 2020), Table 1).

3. Thermodynamics, Stability, and Phase Structure in PFDM Black Holes

Thermodynamic Redefinitions

PFDM modifies all fundamental thermodynamic quantities. Using the Misner-Sharp quasi-local energy (Kumar et al., 7 Aug 2025, Liang et al., 2023), for the metric

$$f(r) = 1 - \frac{2M}{r} - \frac{\lambda}{r}\log\left(\frac{r}{\lambda}\right),$$

the horizon thermodynamics reads:

• Entropy (with fluctuation corrections, parameterized by $\beta$):

$S = \pi r_h^2 - \beta \log(\pi r_h^2)$

• Temperature:

$T_H = \frac{r_h - \lambda}{4\pi r_h^2}$

• Internal Energy, Volume, Pressure, Enthalpy, Gibbs Free Energy:

Expressed in terms of $r_h, \lambda, \beta$, with pressure $P = -dF/dV$.

The stability phase landscape is altered: for $\beta > 0$, stability transitions as a function of $r_h$ may be double-valued, leading to rich local and global thermodynamic structures.

Quantity	Expression (with PFDM & Corrections)
Entropy ($S$)	$\pi r_h^2 - \beta \log (\pi r_h^2)$
Temperature ($T_H$)	$\frac{r_h - \lambda}{4 \pi r_h^2}$
Internal Energy ($E$)	$$\frac{1}{2}(r_h - \lambda \log r_h) + \frac{\beta}{2 \pi r_h}(1+\frac{\lambda}{2r_h})$$
Specific Heat ($C_v$)	$$2\pi r_h^2 \frac{r_h - \lambda}{2\lambda - r_h} - 2\beta \frac{r_h - \lambda}{2\lambda - r_h}$$

Evaporation and Long-lived Remnants

The Misner-Sharp framework (Liang et al., 2023) predicts that the evolution of the black hole (via Hawking emission) is arrested as $r_h \rightarrow \lambda$, leading to long-lived, nearly extremal remnants provided the initial ratio $\lambda_0/r^0_h$ is large. The evaporation slows logarithmically, similarly to nearly-extremal Reissner-Nordström black holes, unlike Schwarzschild. In astrophysical scenarios, such remnants might exist if the dark sector remains decoupled from baryonic matter.

4. Quasinormal Modes, Dynamical Stability, and Ringdown

For both scalar and electromagnetic field perturbations, the effective potential is modulated by PFDM: $$V(r) = f(r)\left[\frac{\ell(\ell+1)}{r^2} + (1-s)\frac{f'(r)}{r}\right]$$ Parameter $k$ or $\lambda$ lowers the potential barrier, leading to:

• Lower real QNM frequencies ($\Re \omega$): Oscillations slow as $|k|$ increases.
• Lower magnitude of imaginary part ($|\Im \omega|$): Decay becomes slower, signals persist longer
• Stability is always preserved: $\Im \omega < 0$ throughout the physical parameter domain (Tan et al., 8 Apr 2025, Bécar et al., 2023).

These QNM shifts imprint on GW ringdowns and may be distinguishable by next-generation GW detectors or via joint analysis with shadow observations. For large $|k|$, both real and imaginary parts of $\omega$ drop below the Schwarzschild case, predicting longer-lived, lower-frequency overtones in the presence of PFDM (Bécar et al., 2023).

5. Regular Black Holes, ISCO Degeneracy, and Nonlinear Charges

PFDM in conjunction with magnetic or nonlinear magnetic charges (e.g., Hayward, Bardeen) introduces new complications:

• ISCO (innermost stable circular orbit) radius is strongly reduced by both $g$ (magnetic charge) and PFDM parameter $\alpha$.
• Degeneracy problem: Combinations of $g$ and $\alpha$ can mimic the ISCO of a high-spin Kerr black hole, up to $a/M \sim 0.9$ (Narzilloev et al., 2020).
• Implication: ISCO-based inferences (from accretion flow, pulsar timing, or disk spectra) cannot uniquely diagnose spin when $g$, $\alpha$ are unconstrained; distinguishing spin from (magnetic charge, PFDM) degeneracy demands multimodal or independent measurements.

Astrophysical example: The ISCO for the magnetar PSR J1745-2900 orbiting Sgr A* is consistent with both spinning Kerr and static magnetically charged Bardeen black holes with appropriate PFDM, illustrating this degeneracy.

6. Topological Thermodynamics and Global Phase Structure

PFDM can alter the topological class of black hole thermodynamics only in conjunction with certain nonlinear or AdS-like structures (Rizwan et al., 2023). For Schwarzschild and Kerr, PFDM does not change the winding or topological numbers. However, for static Hayward (nonlinear magnetic-charged) black holes in PFDM, new topological numbers appear, with $W=1$ (compared to $W=-1$ for Schwarzschild), signifying a distinct phase topology.

Black Hole	$W$	PFDM Topological Class Affected?
Schwarzschild (w/wo PFDM)	$-1$	No
Kerr (w/wo PFDM)	$0$	No
Kerr-AdS in PFDM	$1$	Yes (AdS-like)
Static Hayward (in PFDM)	$1$	Yes (nonlinear charge)
Rotating Hayward (in PFDM)	$0$	No (rotation restores class)

This demonstrates that PFDM, combined with horizon regularization or AdS asymptotics, can shift global phase structure and introduce novel thermodynamic behaviors.

7. Quantum Information and Nonlocal Probes

PFDM alters the evolution of quantum entanglement and coherence in fields near black holes (Wu et al., 22 Jul 2025). The density parameter ($\alpha$) enters both the effective Hawking temperature and the event horizon location, thus modulating:

• Bosonic entanglement/coherence: More sensitive to PFDM than fermionic.
• Fermionic coherence: Stronger functional response to PFDM than bosonic coherence. Optimal quantum probes thus differ by field type and quantum resource; this sensitivity opens potential indirect channels for detecting PFDM via quantum information tasks.

	Bosonic Entanglement	Fermionic Entanglement	Bosonic Coherence	Fermionic Coherence
PFDM Sensitivity	High	Lower	Lower	High
Degradation	Strong	Moderate	Moderate	Strong
Best Probe	Yes	-	-	Yes

8. Summary and Astrophysical Implications

PFDM effects pervade all aspects of black hole physics:

• Geometry: Enlarged or shrunken shadow and horizon, modified ISCO, photon sphere.
• Observables: Shadow distortion, QNM spectrum shift and GW ringdown modifications, ISCO degeneracy, modified lensing observables, and possible long-lived remnant creation.
• Thermodynamics: Altered entropy, temperature, stability, and possible new critical points. Nontrivial phase structure arises, with PFDM able to stabilize or destabilize regimes depending on parameter choices.
• Detection: PFDM-induced effects are, in principle, discernible via high-precision black hole shadow imaging, ringdown GW observation, accretion disk modeling, and, potentially, relativistic quantum information protocols.

The parameter space for PFDM is increasingly constrained by electromagnetic and gravitational-wave observations (e.g., EHT, LISA, pulsar timing), with multiple independent signatures—shadow size, phase transition locations, QNM frequencies, energy emission spectra—offering cross-consistency tests for the presence and distribution of dark matter around astrophysical black holes.