Schwarzschild–Hernquist Black Hole Models

• Schwarzschild–Hernquist black holes are defined as static, spherically symmetric black holes embedded in a finite-mass Hernquist halo that alters key spacetime observables.
• The halo modifies the lapse function, thereby shifting the horizon radius, photon sphere, and geodesic structure, with implications for accretion dynamics and ringdown spectra.
• These models reveal parameter-dependent changes in thermodynamics, imaging shadows, and weak-lensing observables, underscoring their role in environmental strong-field astrophysics.

Searching arXiv for papers on Schwarzschild–Hernquist black holes and closely related constructions. A Schwarzschild–Hernquist black hole denotes a static, spherically symmetric black-hole spacetime in which the Schwarzschild geometry is modified by a surrounding dark-matter halo with a Hernquist density profile. In the recent literature, the term is not attached to a single canonical metric: some works use the direct halo-dressed lapse $f(r)=1-\frac{2M}{r}-\frac{4\pi\rho_s r_s^3}{r+r_s}$ (Ban et al., 7 Jan 2026, Jha, 25 Mar 2025, Heidari et al., 17 Feb 2026, Nieto et al., 18 Jul 2025, Shi et al., 14 Sep 2025), some use the equivalent Hernquist-mass form $f(r)=1-\frac{2M}{r}-\frac{2\alpha r}{(r+\beta)^2}$ with $\alpha=2\pi\rho_s r_s^3$ and $\beta=r_s$ (Ovgun et al., 24 Apr 2026), and some analyze more general or deformed settings in which the Schwarzschild–Hernquist configuration appears as a neutral or undeformed limit (Moreira et al., 21 May 2026, Jha, 31 Dec 2025). Across these formulations, the central idea is the same: a non-rotating, uncharged black hole sits inside a Hernquist halo whose finite total mass and scale radius modify the horizon, photon sphere, geodesics, perturbations, shadow observables, lensing, accretion signatures, and semiclassical emission.

1. Definition and literature usage

The minimal Schwarzschild–Hernquist construction is a Schwarzschild black hole embedded in a Hernquist halo, with line element

$$ds^{2}=-f(r)\,dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$$

and, in several papers,

$$f(r)=1-\frac{2M}{r}-\frac{4\pi\rho_s r_s^3}{r+r_s}.$$

Here $M$ is the black-hole mass, while $\rho_s$ and $r_s$ are the characteristic density and scale radius of the Hernquist halo (Ban et al., 7 Jan 2026, Jha, 25 Mar 2025, Heidari et al., 17 Feb 2026, Nieto et al., 18 Jul 2025). In the halo-free limit $\rho_s\to0$ or $f(r)=1-\frac{2M}{r}-\frac{2\alpha r}{(r+\beta)^2}$0, the metric reduces to Schwarzschild (Ban et al., 7 Jan 2026, Jha, 25 Mar 2025).

A second notational form writes the halo through the enclosed Hernquist mass

$f(r)=1-\frac{2M}{r}-\frac{2\alpha r}{(r+\beta)^2}$1

leading to

$f(r)=1-\frac{2M}{r}-\frac{2\alpha r}{(r+\beta)^2}$2

This is identified explicitly as the Schwarzschild–Hernquist limit of a more general magnetically charged black hole in a Hernquist halo when the magnetic charge is set to zero, $f(r)=1-\frac{2M}{r}-\frac{2\alpha r}{(r+\beta)^2}$3 (Ovgun et al., 24 Apr 2026).

A third strand of the literature introduces an extra deformation parameter $f(r)=1-\frac{2M}{r}-\frac{2\alpha r}{(r+\beta)^2}$4 unrelated to the halo mass parameter above, with

$f(r)=1-\frac{2M}{r}-\frac{2\alpha r}{(r+\beta)^2}$5

and calls the $f(r)=1-\frac{2M}{r}-\frac{2\alpha r}{(r+\beta)^2}$6 case the pure Schwarzschild–Hernquist model (Moreira et al., 21 May 2026). This suggests that the phrase “Schwarzschild–Hernquist black hole” is a family label rather than a uniquely fixed metric ansatz.

A further exact construction based on an anisotropic “Einstein cluster” uses

$f(r)=1-\frac{2M}{r}-\frac{2\alpha r}{(r+\beta)^2}$7

$f(r)=1-\frac{2M}{r}-\frac{2\alpha r}{(r+\beta)^2}$8

$f(r)=1-\frac{2M}{r}-\frac{2\alpha r}{(r+\beta)^2}$9

with ADM mass $\alpha=2\pi\rho_s r_s^3$0 and horizon fixed at $\alpha=2\pi\rho_s r_s^3$1 (Feng et al., 4 Sep 2025). This construction is also called “Schwarzschild–Hernquist black hole” in the QNM and shadow literature.

The literature therefore contains a genuine definitional ambiguity. The shared content is a Schwarzschild central object plus a Hernquist halo; the precise relativistic completion differs by modeling assumptions.

2. Hernquist halo sector

The Hernquist density profile used throughout this literature is

$\alpha=2\pi\rho_s r_s^3$2

or equivalently

$\alpha=2\pi\rho_s r_s^3$3

with inner behavior $\alpha=2\pi\rho_s r_s^3$4 and outer behavior $\alpha=2\pi\rho_s r_s^3$5 (Ovgun et al., 24 Apr 2026, Moreira et al., 21 May 2026, Jha, 25 Mar 2025, Shi et al., 14 Sep 2025). The profile has finite total mass, a property repeatedly emphasized as a distinction from NFW-like halos (Moreira et al., 21 May 2026, Yu et al., 12 Mar 2025).

The enclosed mass is written in equivalent forms: $\alpha=2\pi\rho_s r_s^3$6 or

$\alpha=2\pi\rho_s r_s^3$7

depending on normalization conventions (Ovgun et al., 24 Apr 2026, Ban et al., 7 Jan 2026, Shi et al., 14 Sep 2025). In the $\alpha=2\pi\rho_s r_s^3$8 notation,

$\alpha=2\pi\rho_s r_s^3$9

(Ovgun et al., 24 Apr 2026).

This halo sector enters the metric as an additional attractive term. In the $\beta=r_s$0 convention, the correction is

$\beta=r_s$1

while in the $\beta=r_s$2 convention it is

$\beta=r_s$3

(Jha, 25 Mar 2025, Ovgun et al., 24 Apr 2026).

The asymptotic interpretation depends on the chosen model. For the $\beta=r_s$4 form,

$\beta=r_s$5

so the asymptotic mass is

$\beta=r_s$6

(Ovgun et al., 24 Apr 2026). In the exponential-plus-deformation model,

$\beta=r_s$7

(Moreira et al., 21 May 2026). This suggests that some results depend sensitively on whether comparisons are made at fixed bare mass $\beta=r_s$8 or fixed asymptotic mass $\beta=r_s$9, a distinction explicitly stressed in shadow and QNM analyses (Ovgun et al., 24 Apr 2026).

3. Spacetime structure and geodesic dynamics

The event horizon is determined by the largest positive root of $$ds^{2}=-f(r)\,dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$$0. For the direct Hernquist-dressed Schwarzschild metric, one explicit expression is

$$ds^{2}=-f(r)\,dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$$1

(Jha, 25 Mar 2025). In the vacuum limit, $$ds^{2}=-f(r)\,dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$$2 (Jha, 25 Mar 2025, Heidari et al., 17 Feb 2026). Several works report that increasing halo density or scale radius increases the horizon radius (Jha, 25 Mar 2025, Al-Badawi et al., 2024).

Timelike geodesics obey

$$ds^{2}=-f(r)\,dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$$3

(Ban et al., 7 Jan 2026, Ahmed et al., 23 Jun 2025). For null geodesics,

$$ds^{2}=-f(r)\,dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$$4

(Moreira et al., 21 May 2026, Jha, 25 Mar 2025). In the accretion and EMRI literature, increasing $$ds^{2}=-f(r)\,dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$$5 or $$ds^{2}=-f(r)\,dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$$6 deepens the timelike potential well and shifts characteristic orbits outward (Ban et al., 7 Jan 2026).

The marginally bound orbit and the ISCO are both pushed outward by the halo. For the timelike effective potential, the marginally bound orbit $$ds^{2}=-f(r)\,dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$$7 increases with both $$ds^{2}=-f(r)\,dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$$8 and $$ds^{2}=-f(r)\,dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$$9, and the ISCO radius $$f(r)=1-\frac{2M}{r}-\frac{4\pi\rho_s r_s^3}{r+r_s}.$$0 also increases with both parameters (Ban et al., 7 Jan 2026). A Dehnen-family analysis with Hernquist as a special case similarly reports that halo parameters increase the ISCO radius and weaken the potential barrier (Al-Badawi et al., 2024, Xamidov et al., 17 Jul 2025). This supports the broader interpretation that a Hernquist environment modifies the energy-angular-momentum structure of near-hole circular motion.

Null circular orbits satisfy the standard photon-sphere condition

$$f(r)=1-\frac{2M}{r}-\frac{4\pi\rho_s r_s^3}{r+r_s}.$$1

(Ovgun et al., 24 Apr 2026, Moreira et al., 21 May 2026). The corresponding critical impact parameter is

$$f(r)=1-\frac{2M}{r}-\frac{4\pi\rho_s r_s^3}{r+r_s}.$$2

(Ban et al., 7 Jan 2026, Jha, 25 Mar 2025, Shi et al., 14 Sep 2025). Multiple studies report that the photon sphere and shadow size increase when $$f(r)=1-\frac{2M}{r}-\frac{4\pi\rho_s r_s^3}{r+r_s}.$$3 or $$f(r)=1-\frac{2M}{r}-\frac{4\pi\rho_s r_s^3}{r+r_s}.$$4 increase in the direct Hernquist-dressed metric (Jha, 25 Mar 2025, Shi et al., 14 Sep 2025), whereas the fixed-$$f(r)=1-\frac{2M}{r}-\frac{4\pi\rho_s r_s^3}{r+r_s}.$$5 perturbative analysis of the $$f(r)=1-\frac{2M}{r}-\frac{4\pi\rho_s r_s^3}{r+r_s}.$$6 model finds that the residual halo term reduces the shadow radius relative to Schwarzschild with the same asymptotic mass (Ovgun et al., 24 Apr 2026). This is not a contradiction in the narrow sense; it reflects different comparison schemes.

4. Perturbations, quasinormal modes, and stability

Scalar perturbations in several formulations reduce to

$$f(r)=1-\frac{2M}{r}-\frac{4\pi\rho_s r_s^3}{r+r_s}.$$7

(Ovgun et al., 24 Apr 2026, Moreira et al., 21 May 2026). Electromagnetic and axial gravitational perturbations in the NED–Hernquist framework are governed by

$$f(r)=1-\frac{2M}{r}-\frac{4\pi\rho_s r_s^3}{r+r_s}.$$8

$$f(r)=1-\frac{2M}{r}-\frac{4\pi\rho_s r_s^3}{r+r_s}.$$9

(Ovgun et al., 24 Apr 2026).

For the exact Einstein-cluster Schwarzschild–Hernquist model, axial gravitational perturbations satisfy

$M$0

(Feng et al., 4 Sep 2025). That work computes QNMs using pseudospectral, matrix, and 6th-order WKB/eikonal methods and finds that the modes are redshifted relative to Schwarzschild as halo compactness increases. In the $M$1 limit,

$M$2

for $M$3 (Feng et al., 4 Sep 2025). The same paper reports highly redshifted QNMs for large compactness and identifies these as a key signature of the dark-matter halo.

A complementary analysis in the $M$4 framework derives scalar, electromagnetic, and axial gravitational master equations and computes spectra with high-order WKB plus Padé resummation. There the Hernquist halo lowers $M$5 and modestly decreases $M$6, opposite to the effect of magnetic charge, and the two effects can partially cancel (Ovgun et al., 24 Apr 2026). In the neutral Schwarzschild–Hernquist case this implies slightly lower ringdown frequencies and slightly slower damping than vacuum Schwarzschild at the same bare mass, with fixed-$M$7 comparisons again treated separately (Ovgun et al., 24 Apr 2026).

Scalar-potential analyses without explicit QNM extraction report a single positive barrier outside the horizon and infer dynamical stability under scalar perturbations, while noting halo-induced shifts in barrier height and position (Moreira et al., 21 May 2026). This suggests that the principal perturbative effect of the halo is spectral deformation rather than instability.

5. Shadow, lensing, and multimessenger imaging signatures

In the simplest direct-Hernquist metric, the shadow radius is determined by the critical impact parameter at the photon sphere and is larger than the Schwarzschild value $M$8 for nonzero halo parameters (Jha, 25 Mar 2025, Shi et al., 14 Sep 2025). One study reports explicit parameter-dependent increases ranging from $M$9 to $\rho_s$0 in the photon sphere and shadow size as $\rho_s$1 and $\rho_s$2 increase (Shi et al., 14 Sep 2025). Another derives a deviation parameter

$\rho_s$3

and uses EHT, Keck, and VLTI bounds to constrain Hernquist parameters in the shadow sector (Jha, 25 Mar 2025).

By contrast, the fixed-asymptotic-mass perturbative expansion of the $\rho_s$4 model yields

$\rho_s$5

for $\rho_s$6, implying a reduction relative to Schwarzschild of the same $\rho_s$7 (Ovgun et al., 24 Apr 2026). The same work emphasizes that fixed-$\rho_s$8 and fixed-$\rho_s$9 comparisons lead to opposite qualitative conclusions for the shadow (Ovgun et al., 24 Apr 2026).

Weak-lensing observables also depend on which parameters are held fixed. In the $r_s$0 formulation, the asymptotic deflection angle is

$r_s$1

so the leading term depends only on $r_s$2, while the first subleading term reduces the bending angle relative to Schwarzschild with the same asymptotic mass (Ovgun et al., 24 Apr 2026). In the direct Hernquist-dressed Schwarzschild model, the weak-field deflection angle instead acquires positive halo corrections proportional to $r_s$3 (Jha, 25 Mar 2025). Again, the two statements refer to different parametrizations and reference backgrounds.

Imaging calculations with accretion matter add further structure. A study of thin-disk, static spherical, and infalling spherical accretion reports that direct emission dominates the total observed intensity, while the lensing ring and photon ring occupy increasingly narrow impact-parameter intervals near the critical curve (Shi et al., 14 Sep 2025). In those images, increasing Hernquist parameters enlarges the photon sphere by $r_s$4 to $r_s$5 and decreases measured intensity in the spherical accretion models, with infalling accretion darker than static because of Doppler de-boosting (Shi et al., 14 Sep 2025).

A thin-disk study based on a Novikov–Thorne model in the direct Hernquist metric finds that increasing $r_s$6 or $r_s$7 results in cooler, dimmer disks with modified flux distributions and outward-shifted ISCOs (Ban et al., 7 Jan 2026). Another disk study concludes that the Hernquist halo alters radiative flux, temperature, differential luminosity, and spectral luminosity, and can either increase or decrease radiative efficiency depending on halo parameters (Nieto et al., 18 Jul 2025). These results place the Schwarzschild–Hernquist black hole squarely in the broader program of environmental strong-field astrophysics.

6. Thermodynamics, particle production, and broader interpretation

Thermodynamic analyses of the direct Hernquist-dressed metric derive modified horizon–mass and temperature relations. One study gives

$r_s$8

and

$r_s$9

with a finite endpoint temperature

$\rho_s\to0$0

at the radius where the ADM mass vanishes (Jha, 25 Mar 2025). The same work finds positive-heat-capacity and negative-free-energy regions absent in vacuum Schwarzschild, interpreting them as locally and globally stable phases induced by the halo (Jha, 25 Mar 2025).

A related analysis writes

$\rho_s\to0$1

and defines a remnant radius

$\rho_s\to0$2

from the condition $\rho_s\to0$3, with a corresponding remnant mass obtained by substituting $\rho_s\to0$4 into the mass–radius relation (Nieto et al., 18 Jul 2025). This suggests a halo-induced quenching of evaporation in that semiclassical model.

Quantum-field analyses in the same direct metric study Hawking radiation via Bogoliubov transformations and tunneling. The effective temperature is reported as

$\rho_s\to0$5

which is lower than the Schwarzschild value and decreases as halo parameters increase (Heidari et al., 17 Feb 2026). The corresponding bosonic and fermionic occupation numbers are suppressed by the halo, and the high-frequency evaporation law acquires modified emission rates and longer evaporation times (Heidari et al., 17 Feb 2026).

The literature also contains genuine model-dependent extensions. The neutral limit $\rho_s\to0$6 of a magnetically charged NED black hole immersed in a Hernquist halo reproduces a Schwarzschild–Hernquist metric of the form

$\rho_s\to0$7

and this framework exhibits parameter degeneracies in which halo effects can mimic Schwarzschild values of the horizon radius, shadow radius, or strong-lensing coefficients (Jha, 31 Dec 2025). Another paper adds a cloud of strings and AdS curvature,

$\rho_s\to0$8

and reports that strings enlarge the shadow whereas Hernquist dark matter shrinks it in that composite setting (Ahmed et al., 23 Jun 2025). These are not definitions of the minimal Schwarzschild–Hernquist black hole, but they show how the concept is used as a base sector in broader environmental models.

A central encyclopedic point is therefore that “Schwarzschild–Hernquist black hole” names a class of Schwarzschild-plus-Hernquist spacetimes rather than a single universally standardized geometry. The recurring physical content is robust: a finite-mass Hernquist halo modifies the near-hole lapse, shifts the horizon and photon sphere, alters timelike and null geodesics, changes ringdown and shadow observables, affects lensing and accretion diagnostics, and suppresses semiclassical particle production relative to vacuum Schwarzschild (Ovgun et al., 24 Apr 2026, Ban et al., 7 Jan 2026, Feng et al., 4 Sep 2025, Heidari et al., 17 Feb 2026). The precise sign and magnitude of some observables—especially shadow and weak-lensing corrections—depend on the chosen relativistic completion and on whether comparisons are made at fixed bare mass or fixed asymptotic mass.