Geodesics and Thermodynamics of a Schwarzschild Black Hole with Hernquist Dark Matter

Abstract: In this work, we investigate the physical and geometrical properties of a Schwarzschild black hole (BH) immersed in a Hernquist dark matter halo. To accomplish our purpose, one builds the BH metric by incorporating the Hernquist dark matter profile into the Schwarzschild geometry. In addition, we verify the null geodesic solutions and the Halo effect on photon dynamics. Within this framework, one examines the corresponding light trajectories to determine the deformation of photon paths generated by the dark matter distribution. Furthermore, the thermodynamic properties of the system are studied by deriving expressions for the black hole mass, the horizon condition, the Hawking temperature, the entropy, the Gibbs free energy, and the heat capacity. Our results show that the dark matter halo modifies the thermal structure and stability conditions of the black hole configuration. Finally, we investigate the scalar perturbations to examine the influence of the Hernquist halo on the dynamical propagation of scalar fields in the BH background. In this framework, the results obtained demonstrate that the dark matter parameters yield nontrivial corrections to the optical, thermodynamic, and perturbative properties of the Schwarzschild black hole, producing deviations from the standard vacuum solution.

• The paper introduces a modified Schwarzschild metric that embeds a Hernquist dark matter halo and quantifies the impact of parameters α, rₛ, and ρₛ on the gravitational field.
• It demonstrates that alterations in null geodesics modify photon trajectories, influencing light lensing and shadow profiles near the event horizon.
• Thermodynamic analysis reveals that dark matter parameters stabilize heat capacity and adjust Hawking temperature and phase transitions.

Geodesics and Thermodynamics of a Schwarzschild Black Hole with Hernquist Dark Matter

Schwarzschild–Hernquist Metric Construction

The study examines a Schwarzschild black hole (BH) embedded in a Hernquist dark matter (DM) halo, introducing a metric that incorporates the Hernquist density profile and an auxiliary parameter $\alpha$. The metric function $f(r)$ explicitly depends on the BH mass $M$, the Hernquist parameters $r_s$ and $\rho_s$, and $\alpha$, yielding a nontrivial deformation of the conventional Schwarzschild spacetime. Specifically, the exponential factor, governed by $(r_s, \rho_s)$, modulates the gravitational field, while $\alpha$ produces a global asymptotic shift in $f(r)$.

Figure 1: Profile of the metric function $f(r)$ vs. $f(r)$0 for $f(r)$1, illustrating modulation by $f(r)$2, $f(r)$3, and $f(r)$4.

Analysis shows that variations in $f(r)$5 shift the entire metric downward, enhancing gravitational attraction; $f(r)$6 controls the spatial extent of the halo’s influence; and $f(r)$7 determines the density-dependent corrections. These three parameters interact to produce significant departures from the Schwarzschild vacuum solution, affecting both the near-horizon structure and the asymptotic regime.

Null Geodesics and Light Propagation

Investigation of null geodesics reveals that the Hernquist parameters and $f(r)$8 alter the effective potential and photon orbital dynamics. The effective potential $f(r)$9 demonstrates parameter-sensitive radial profiles, dictating the existence and stability of photon spheres.

Figure 2: The effective potential $M$0 vs. $M$1 for different Hernquist and deformation parameters at $M$2.

Increasing $M$3 lowers the potential barrier, shifting unstable photon orbits inward. Larger $M$4 broadens the spatial domain of the halo’s gravitational effects, whereas higher $M$5 amplifies the trapping potential for light. Orbital equations show that the lensing and trajectory structure depart from canonical Schwarzschild predictions, especially in the strong-field region.

Figure 3: Light trajectories for variations in $M$6, $M$7, $M$8 with $M$9; halo and geometric deformation deeply affect photon orbits.

Limit analyses demonstrate that for $r_s$0 and/or $r_s$1, the system reduces to a Schwarzschild-like metric with $r_s$2 as a global monopole term, modifying photon sphere radii as $r_s$3. The presence of the Hernquist profile creates a radial-dependent mass correction, affecting both lensing and the shadow structure.

Thermodynamic Properties

The thermodynamics of the Schwarzschild–Hernquist BH feature explicit dependence on the DM halo parameters and $r_s$4. The ADM mass is related to the horizon radius through an exponential term, with $r_s$5 diminishing the mass required for a given event horizon size.

Figure 4: The ADM mass as a function of $r_s$6 for different $r_s$7, with $r_s$8 and $r_s$9 prominently reducing mass for fixed $\rho_s$0.

The Hawking temperature inherits corrections from the modified surface gravity; the exponential halo parameterization and $\rho_s$1 suppress $\rho_s$2, particularly for smaller $\rho_s$3. $\rho_s$4 and $\rho_s$5 have a regularizing effect, smoothing the temperature profile and reducing near-horizon gradients.

Figure 5: Hawking temperature $\rho_s$6 vs. $\rho_s$7; $\rho_s$8 and $\rho_s$9 decrease the temperature, indicating greater thermodynamic stability.

The Bekenstein-Hawking area law remains valid, $\alpha$0, as the horizon’s geometry is preserved.

Detailed examination of Gibbs free energy $\alpha$1 reveals negative $\alpha$2 at large $\alpha$3, signaling stability and possible Hawking–Page transitions. $\alpha$4 further shifts $\alpha$5 downwards at macroscopic horizon sizes.

Figure 6: Gibbs free energy vs. $\alpha$6, showing transition to stable configurations as $\alpha$7 becomes negative for increasing $\alpha$8.

Heat capacity analysis identifies second-order phase transitions, with divergence points shifted by $\alpha$9 and $(r_s, \rho_s)$0. Negative heat capacity regimes are suppressed as $(r_s, \rho_s)$1 increases—showing DM halo regularizes thermal instabilities.

Figure 7: Heat capacity solutions for varying $(r_s, \rho_s)$2; geometric and halo parameters stabilize BH thermodynamics.

Scalar Perturbations and Stability

The scalar field perturbations governed by the Klein–Gordon equation exhibit effective potentials modulated by $(r_s, \rho_s)$3 and $(r_s, \rho_s)$4, causing changes in the quasinormal mode (QNM) spectrum and scattering dynamics. The exponential Hernquist corrections shift the potential barrier height and width, while $(r_s, \rho_s)$5 alters the global normalization.

Figure 8: Effective potential for scalar perturbations ($(r_s, \rho_s)$6), demonstrating parameter-induced modifications in the trapping region.

Large $(r_s, \rho_s)$7 and $(r_s, \rho_s)$8 extend and regularize the region of scalar wave interaction, while vacuum and deformation limits adapt the potential to Schwarzschild and monopole-like cases. In the asymptotic regime, the halo supplies an effective mass correction, further impacting QNM frequencies relevant for gravitational wave observatories.

Practical and Theoretical Implications

Parameter-driven modifications of optical, thermodynamic, and perturbation properties in BHs suggest observational signatures in lensing, shadow imaging (EHT/Sgr A*), and gravitational wave QNM detection. The regularizing thermal effects and stable heat capacity branches may imply extended lifetimes for BHs in dense DM environments. The scalar perturbation analysis points to altered QNM spectra, affecting tests of strong-field gravity and constraints on DM distribution in galactic centers.

Theoretical implications include model generalization for extended dark matter environments, incorporation in precision tests of GR, and potential applications in holography and quantum gravity via modified BH thermodynamics. The interplay between halo structure and spacetime deformation provides a framework for exploring non-vacuum gravitational physics with astrophysical relevance.

Conclusion

This work establishes that a Schwarzschild BH embedded in a Hernquist halo and subject to geometric deformation parameter $(r_s, \rho_s)$9 exhibits significant, quantifiable departures from canonical properties. The combined influence of $\alpha$0 modulates photon effective potentials, light trajectories, thermal profiles, phase transitions, and scalar perturbative dynamics. These corrections are paramount for modeling astrophysical black holes interacting with DM halos, impacting both observational strategies and theoretical investigations within high-energy astrophysics and gravitational physics (2605.24046).