PFDM-Corrected Metrics in Astrophysics

• PFDM-corrected metrics are spacetime models modified by a perfect fluid dark matter component, introducing logarithmic terms that adjust gravitational potentials.
• They impact key physical properties such as horizon structure, ISCO location, and photon orbits, often requiring numerical methods or special functions like the Lambert W for solution.
• These metrics have broad applications ranging from black hole shadow analysis to quantum gravity studies and even data-driven performance assessments.

PFDM-corrected metrics, or perfect fluid dark matter–corrected metrics, are a class of general relativistic and quantum field theoretical metrics in which the standard geometric structures of spacetime are modified by the presence of a perfect fluid dark matter (PFDM) component. These corrections have been studied in a variety of contexts: black hole physics, quantum gravity, relativistic astrophysics, and data-driven systems such as preference-based performance measures. The unifying signature of PFDM corrections is the appearance of terms proportional to $$(\text{PFDM parameter}) \times r \ln (r/|\text{PFDM parameter}|)$$ in the relevant metric functions or potentials, leading to distinctive departures from classical solutions.

1. Mathematical Definition of PFDM Corrections

A PFDM-corrected metric is most generally constructed by introducing a perfect fluid dark matter density and pressure profile as a source in Einstein’s equations: $$T^\mu{}_\nu = \mathrm{diag}\left[ -\rho(r),\,p_r(r),\,p_\theta(r),\,p_\phi(r) \right]$$ with equations of state that enforce $\rho = -p_r = \alpha/(8\pi r^3)$ and %%%%2%%%%, where $\alpha$ is the PFDM density parameter (Wu et al., 22 Jul 2025, Bécar et al., 2023).

In practical metrics (static or rotating), this source induces a logarithmic correction to the “Newtonian potential” component: $$f(r) = 1 - \frac{2M}{r} + \frac{\alpha}{r} \ln\left(\frac{|\alpha|}{r}\right)$$ or, in rotating metrics (Boyer–Lindquist coordinates), modifies $\Delta$ via: $$\Delta(r) = r^2 - 2Mr + a^2 + \lambda r \ln\left(\frac{r}{|\lambda|}\right)$$ where $M$ is the black hole mass, $a$ the rotational parameter, and $\lambda$ the PFDM parameter (Rodriguez et al., 22 Jul 2024, Narzilloev et al., 10 Aug 2024, Anjum et al., 2023, Atamurotov et al., 2021).

Quantum gravity analogues also exist, with the hallmark structure $s \mapsto \sqrt{s^2 + 4 L^2}$ in the Feynman propagator, where $L$ plays the role of a zero-point length or PFDM scale (Cordoba et al., 2023).

2. Structural Effects on Black Hole and Spacetime Metrics

PFDM-corrected metrics profoundly modify fundamental geometric quantities:

• Horizon Structure: For Schwarzschild-like metrics, the event horizon equation becomes transcendental:

$$1 - \frac{2M}{r_h} + \frac{\alpha}{r_h} \ln\left(\frac{|\alpha|}{r_h}\right) = 0$$

or with rotation and charge:

$$\Delta(r) = r^2 + a^2 - 2 M r + Q^2 + \lambda r \ln\left(\frac{r}{|\lambda|}\right) = 0$$

In general, horizons must be computed numerically or with special functions (e.g., Lambert $W$) (Wu et al., 22 Jul 2025, Bécar et al., 2023, Rodriguez et al., 22 Jul 2024).

• Critical PFDM Parameter ($\lambda_c$): The transition between black holes and naked singularities (with or without extremal horizons) depends on the critical value $\lambda_c$, found by solving

$$\Delta(r_c) = 0,\quad \frac{d\Delta}{dr}\bigg|_{r_c} = 0$$

The existence and number of horizons depends on whether $\lambda < \lambda_c$, $\lambda = \lambda_c$, or $\lambda > \lambda_c$ (Rodriguez et al., 22 Jul 2024).

• Asymptotic Structure: PFDM tails decay logarithmically, producing behavior such as $f(r) \sim 1 - (2M/r) + (\alpha/r)\ln r$ as $r \to \infty$, thus modifying long-range potentials, lensing properties, and shadow sizes (Tan et al., 8 Apr 2025, Jha, 27 Jun 2025).
• Curvature Invariants: Ricci scalar, Ricci tensor squared, and Kretschmann scalar all acquire additive log-corrections, with divergences at $r=0$ generically strengthened by PFDM (Jha, 27 Jun 2025).

3. PFDM Effects on Geodesics, Orbits, and Observables

PFDM-corrected metrics impact the dynamics of geodesics, energy extraction, photon orbits, and observational signatures:

• Circular Geodesics and ISCO: The effective potential, energy, and angular momentum for massive or massless test particles receive explicit PFDM-dependent corrections. For rotating spacetimes, the innermost stable circular orbit (ISCO) shifts outward with increasing PFDM parameter, decreasing Novikov–Thorne efficiency (radiative efficiency) (Narzilloev et al., 10 Aug 2024, Rodriguez et al., 22 Jul 2024).
• Ergosurfaces and Shadows: The shadow radius $R_s$, distortion $\delta_s$, and ergoregion boundaries are all shrunken or shifted by PFDM; larger PFDM parameter reduces $R_s$ and enhances $\delta_s$ nonmonotonically (Anjum et al., 2023, Atamurotov et al., 2021).
• Photon Orbits and Lensing: Spherical photon orbits, deflection angles, and Einstein ring radii all acquire log-corrections with PFDM, modifying both strong and weak lensing phenomena (Atamurotov et al., 2021, Jha, 27 Jun 2025).
• Jet Power and Extracted Energy: In Kerr+PFDM, the Blandford–Znajek jet power $P_{\mathrm{BZ}} \propto \Omega_H^2$, with $\Omega_H$ suppressed by increasing PFDM, so maximal jet power is reduced at fixed spin (Narzilloev et al., 10 Aug 2024).
• Quantum Signatures: PFDM influences quantum entanglement/coherence differently for fermionic and bosonic fields, affecting the selection of quantum probes for dark matter detection (Wu et al., 22 Jul 2025).

4. Analytical and Numerical Properties; Special Functions

The transcendental nature of PFDM-corrected horizon equations necessitates the use of special functions and numerical analysis:

• Lambert W Function: Used for analytic inversion of horizon radius equations, e.g.,

$r_h = k\, W\left(e^{2M/k}\right)$

for Schwarzschild+PFDM (Bécar et al., 2023, Jha, 27 Jun 2025).

• Perturbative Expansions: For small PFDM parameter, approximate expressions for horizon shifts, ISCO location, and other quantities can be derived to leading order in the parameter (Narzilloev et al., 10 Aug 2024, Jha, 27 Jun 2025).
• Critical Points: Horizon radius $r_h(k)$ exhibits minima at analytically determined $k_\text{min}$, with monotonicity breaking at this value (Bécar et al., 2023).
• No Closed-Form Solution for $\Delta(r)=0$ in Rotating, Charged Cases: Roots and critical values must be found numerically, emphasizing the necessity for computational methods in PFDM-corrected contexts (Rodriguez et al., 22 Jul 2024).

5. PFDM in Observational and Quantum Gravity Contexts

• Event Horizon Telescope Constraints: EHT measurements of shadow deviations for M87* and Sgr A* furnish upper bounds on PFDM parameters: typically $k \lesssim 0.08 M$–$0.06 M$ to ensure consistency with observed shadow sizes (Anjum et al., 2023).
• Quantum Gravity Metric Corrections: In the bitensor approach, quantum gravity (zero-point-length)–induced PFDM corrections replace $s \to \sqrt{s^2 + 4L^2}$ in Feynman propagators, regularizing ultraviolet behavior and enforcing a core metric correction inside Planck-scale balls while leaving macroscopic geometry unchanged (Cordoba et al., 2023).
• Data/Performance Applications: The term “PFDM-corrected metric” also appears in the analysis of preference-based performance measures in data-driven model assessment. Here, “PFDM-corrected metrics” are constructed to weight classifier outputs according to downstream consequences (e.g., preference for accurate detection of bad/good data), using probability-theoretic mappings to define tailored performance measures (e.g., $E_{W \rightarrow W}$, $E_{R \rightarrow R}$) (Lan et al., 2017).

6. Domain-Specific Consequences and Use Cases

• Tailored Metric Construction: By explicit choice of PFDM parameter, one can engineer spacetimes for desired observational properties, such as reduced shadow size or modified lensing signatures.
• Model Selection in Data Science: When performance metrics are strongly monotonic in a particular underlying recall parameter (e.g., $Q_3$ for dissimilarity recall), PFDM-corrected metrics permit efficient model selection and hyperparameter tuning, optimizing exactly user-preferred outcome balances (Lan et al., 2017).
• Distinguishing Black Hole Types: Differences in the PFDM-corrected QNM spectra for scalar vs electromagnetic perturbations can, in principle, discriminate between field disturbances in a given astrophysical environment (Tan et al., 8 Apr 2025, Bécar et al., 2023).
• Quantum Probing of Dark Matter: Quantum information protocols (entanglement/coherence) experience PFDM-driven enhancements or suppressions depending on the quantum field type, promoting careful selection of probe systems for indirect PFDM detection (Wu et al., 22 Jul 2025).

7. Summary Table of PFDM Metric Corrections in Representative Contexts

Context	Metric/Potential Correction	Analytical Structure / Key Impact
Schwarzschild/PFDM Black Hole	$$f(r) = 1 - \frac{2M}{r} + \frac{\alpha}{r}\ln(\frac{|\alpha|}{r})$$	Horizon shift, altered asymptotics, curvature invariants acquire log terms
Kerr-Newman/PFDM Black Hole	$$\Delta(r) = r^2 + a^2 - 2Mr + Q^2 + \lambda r \ln(r/|\lambda|)$$	Horizons/ergosurfaces' location, ISCO/energy extraction efficiency modified
Quantum Gravity Bitensor Metric	$s \rightarrow \sqrt{s^2 + 4L^2}$ in $G(s)$	UV regularization, effective metric inside $L$-ball
Preference-based Performance Measure	$E_{\alpha \rightarrow \beta}$ reweighted via PFDM mapping	Nonlinear, task-oriented classifier performance estimation

This table enumerates the mathematical form taken by PFDM corrections, the affected geometric and data-theoretic quantities, and the ensuing phenomenological impacts across research domains.

• Energy extraction through magnetic reconnection from a Kerr-Newman black hole in perfect fluid dark matter (Rodriguez et al., 22 Jul 2024)
• Observed jet power and radiative efficiency of black hole candidates in Kerr + PFDM model (Narzilloev et al., 10 Aug 2024)
• Influence of dark matter on quantum entanglement and coherence in curved spacetime (Wu et al., 22 Jul 2025)
• Testing black holes in a perfect fluid dark matter environment using quasinormal modes (Tan et al., 8 Apr 2025)
• Investigating effects of dark matter on photon orbits and black hole shadows (Anjum et al., 2023)
• Shadow and deflection angle of charged rotating black hole surrounded by perfect fluid dark matter (Atamurotov et al., 2021)
• Black hole surrounded by perfect fluid dark matter with a background Kalb-Ramond field (Jha, 27 Jun 2025)
• Massive Scalar Field Perturbations of Black Holes Surrounded by Dark Matter (Bécar et al., 2023)
• Spacetime metric from quantum-gravity corrected Feynman propagators (Cordoba et al., 2023)
• Preference-based performance measures for Time-Domain Global Similarity method (Lan et al., 2017)