---
title: Schwarzschild–Hernquist Black Hole Models
url: https://www.emergentmind.com/topics/schwarzschild-hernquist-black-hole
type: topic
---

# Schwarzschild–Hernquist Black Hole Models

Searching arXiv for recent 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 \(f(r)=1-\frac{2M}{r}-\frac{4\pi\rho_s r_s^3}{r+r_s}\) [2601.03980, 2503.19938, 2602.15420, 2507.14305, 2509.11245], 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 \(\beta=r_s\) [2604.22684], and some analyze more general or deformed settings in which the Schwarzschild–Hernquist configuration appears as a neutral or undeformed limit [2605.24046, 2512.24753]. 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 [2601.03980, 2503.19938, 2602.15420, 2507.14305]. In the halo-free limit \(\rho_s\to0\) or \(r_s\to0\), the metric reduces to Schwarzschild [2601.03980, 2503.19938].

A second notational form writes the halo through the enclosed Hernquist mass
\[
M_H(r)=\frac{\alpha r^2}{(r+\beta)^2},\qquad \alpha\equiv2\pi\rho_s r_s^3,\qquad \beta\equiv r_s,
\]
leading to
\[
f_{\text{SH}}(r)=1-\frac{2M}{r}-\frac{2\alpha r}{(r+\beta)^2}.
\]
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, \(g=0\) [2604.22684].

A third strand of the literature introduces an extra deformation parameter \(\alpha\) unrelated to the halo mass parameter above, with
\[
f(r)=\exp\!\left(-\frac{4\pi r_s^3\rho_s}{r+r_s}\right)-\frac{2M}{r}-\alpha,
\]
and calls the \(\alpha=0\) case the pure Schwarzschild–Hernquist model [2605.24046]. 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
\[
\rho(r)=\frac{M_{\rm DM}\bigl(a_0+2M_{\rm BH}\bigr)\bigl(1-2M_{\rm BH}/r\bigr)}{2\pi\,r(r+a_0)^3},
\]
\[
m(r)=M_{\rm BH}+\frac{M_{\rm DM}r^2}{(a_0+r)^2}\left(1-\frac{2M_{\rm BH}}{r}\right)^2,
\]
\[
f(r)=\left(1-\frac{2M_{\rm BH}}{r}\right)e^{\Upsilon(r)},
\]
with ADM mass \(M_{\rm BH}+M_{\rm DM}\) and horizon fixed at \(r_h=2M_{\rm BH}\) [2509.04001]. 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
\[
\rho_H(r)=\frac{\rho_s}{\displaystyle \frac{r}{r_s}\left(1+\frac{r}{r_s}\right)^3}
\]
or equivalently
\[
\rho(r)=\rho_s\left(\frac{r}{r_s}\right)^{-1}\left(1+\frac{r}{r_s}\right)^{-3},
\]
with inner behavior \(\rho\propto r^{-1}\) and outer behavior \(\rho\propto r^{-4}\) [2604.22684, 2605.24046, 2503.19938, 2509.11245]. The profile has finite total mass, a property repeatedly emphasized as a distinction from NFW-like halos [2605.24046, 2503.09104].

The enclosed mass is written in equivalent forms:
\[
M_H(r)=\frac{2\pi\rho_s r_s^3 r^2}{(r+r_s)^2}
\]
or
\[
M_H(r)=4\pi\rho_c r_s^3\frac{r^2}{(r+r_s)^2},
\]
depending on normalization conventions [2604.22684, 2601.03980, 2509.11245]. In the \((\alpha,\beta)\) notation,
\[
M_H(r)=\frac{\alpha r^2}{(r+\beta)^2},\qquad \alpha=2\pi\rho_s r_s^3,\qquad \beta=r_s
\]
[2604.22684].

This halo sector enters the metric as an additional attractive term. In the \(4\pi\rho_s r_s^3\) convention, the correction is
\[
-\frac{4\pi\rho_s r_s^3}{r+r_s},
\]
while in the \((\alpha,\beta)\) convention it is
\[
-\frac{2\alpha r}{(r+\beta)^2}=-\frac{2M_H(r)}{r}
\]
[2503.19938, 2604.22684].

The asymptotic interpretation depends on the chosen model. For the \((\alpha,\beta)\) form,
\[
f(r)=1-\frac{2(M+\alpha)}{r}+\frac{g^2+4\alpha\beta}{r^2}+\mathcal{O}(r^{-3}),
\]
so the asymptotic mass is
\[
\mathcal{M}=M+\alpha
\]
[2604.22684]. In the exponential-plus-deformation model,
\[
f(r)\simeq 1-\alpha-\frac{2M_{\rm eff}}{r}+\cdots,\qquad M_{\rm eff}=M+2\pi r_s^3\rho_s
\]
[2605.24046]. This suggests that some results depend sensitively on whether comparisons are made at fixed bare mass \(M\) or fixed asymptotic mass \(\mathcal M\), a distinction explicitly stressed in shadow and QNM analyses [2604.22684].

## 3. Spacetime structure and geodesic dynamics

The event horizon is determined by the largest positive root of \(f(r_h)=0\). For the direct Hernquist-dressed Schwarzschild metric, one explicit expression is
\[
r_h=\frac{1}{2}\left[\sqrt{\left(-2M-4\pi r_s^3\rho_s+r_s\right)^2+8Mr_s}+2M+4\pi r_s^3 \rho_s-r_s\right]
\]
[2503.19938]. In the vacuum limit, \(r_h\to2M\) [2503.19938, 2602.15420]. Several works report that increasing halo density or scale radius increases the horizon radius [2503.19938, 2411.01145].

Timelike geodesics obey
\[
\dot r^2+V_{\rm eff}(r)=E^2,\qquad V_{\rm eff}(r)=f(r)\left(1+\frac{L^2}{r^2}\right)
\]
[2601.03980, 2506.18457]. For null geodesics,
\[
V_{\rm eff}(r)=\frac{L^2}{r^2}f(r)
\]
[2605.24046, 2503.19938]. In the accretion and EMRI literature, increasing \(\rho_s\) or \(r_s\) deepens the timelike potential well and shifts characteristic orbits outward [2601.03980].

The marginally bound orbit and the ISCO are both pushed outward by the halo. For the timelike effective potential, the marginally bound orbit \(r_{\rm MBO}\) increases with both \(\rho_s\) and \(r_s\), and the ISCO radius \(r_{\rm ISCO}\) also increases with both parameters [2601.03980]. A Dehnen-family analysis with Hernquist as a special case similarly reports that halo parameters increase the ISCO radius and weaken the potential barrier [2411.01145, 2507.13147]. 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
\[
r_{\rm ph}f'(r_{\rm ph})-2f(r_{\rm ph})=0
\]
[2604.22684, 2605.24046]. The corresponding critical impact parameter is
\[
b_{\rm c}=\frac{r_{\rm ph}}{\sqrt{f(r_{\rm ph})}}
\]
[2601.03980, 2503.19938, 2509.11245]. Multiple studies report that the photon sphere and shadow size increase when \(\rho_s\) or \(r_s\) increase in the direct Hernquist-dressed metric [2503.19938, 2509.11245], whereas the fixed-\(\mathcal M\) perturbative analysis of the \((\alpha,\beta)\) model finds that the residual halo term reduces the shadow radius relative to Schwarzschild with the same asymptotic mass [2604.22684]. 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
\[
\frac{d^{2}\psi}{dr_*^{2}}+\left(\omega^{2}-\mathcal{V}_{\rm scalar}(r)\right)\psi=0,\qquad 
\mathcal{V}_{\rm scalar}(r)=f(r)\left[\frac{\ell(\ell+1)}{r^2}+\frac{f'(r)}{r}\right]
\]
[2604.22684, 2605.24046]. Electromagnetic and axial gravitational perturbations in the NED–Hernquist framework are governed by
\[
V^{(1)}(r)=f(r)\frac{\ell(\ell+1)}{r^2},
\]
\[
V^{(2)}(r)=f(r)\left[\frac{\ell(\ell+1)}{r^2}-\frac{f'(r)}{r}+\frac{2(f(r)-1)}{r^2}\right]
\]
[2604.22684].

For the exact Einstein-cluster Schwarzschild–Hernquist model, axial gravitational perturbations satisfy
\[
\frac{d^2\Psi}{dr_\ast^2}+\left(\omega^2-V(r)\right)\Psi=0,\qquad 
V(r)=\frac{f(r)}{r^2}\Bigl[\ell(\ell+1)-\frac{6m(r)}{r}+m'(r)\Bigr]
\]
[2509.04001]. 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 \(\epsilon\to0\) limit,
\[
\frac{\omega({\cal C},\epsilon)}{\omega(0,0)}=\frac{3\sqrt{3}\,M_{\rm BH}}{b_c}=1-{\cal C}+\frac{{\cal C}^2}{6}+\mathcal{O}({\cal C}^3)
\]
for \({\cal C}\le0.3\) [2509.04001]. 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 \((\alpha,\beta)\) framework derives scalar, electromagnetic, and axial gravitational master equations and computes spectra with high-order WKB plus Padé resummation. There the Hernquist halo lowers \(\omega_R\) and modestly decreases \(\omega_I\), opposite to the effect of magnetic charge, and the two effects can partially cancel [2604.22684]. 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-\(\mathcal M\) comparisons again treated separately [2604.22684].

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 [2605.24046]. 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 \(3\sqrt{3}M\) for nonzero halo parameters [2503.19938, 2509.11245]. One study reports explicit parameter-dependent increases ranging from \(\sim2\%\) to \(\sim30\%\) in the photon sphere and shadow size as \(\rho_c M^2\) and \(r_s/M\) increase [2509.11245]. Another derives a deviation parameter
\[
\delta=\frac{R_s}{3\sqrt{3}M}-1
\]
and uses EHT, Keck, and VLTI bounds to constrain Hernquist parameters in the shadow sector [2503.19938].

By contrast, the fixed-asymptotic-mass perturbative expansion of the \((\alpha,\beta)\) model yields
\[
R_{\rm sh}=3\sqrt{3}\,\mathcal{M}\left[1-\delta\,\frac{\alpha\beta(6\mathcal{M}+\beta)}{\mathcal{M}(3\mathcal{M}+\beta)^2}\right]+\mathcal{O}(\delta^2)
\]
for \(g=0\), implying a reduction relative to Schwarzschild of the same \(\mathcal M\) [2604.22684]. The same work emphasizes that fixed-\(M\) and fixed-\(\mathcal M\) comparisons lead to opposite qualitative conclusions for the shadow [2604.22684].

Weak-lensing observables also depend on which parameters are held fixed. In the \((\alpha,\beta)\) formulation, the asymptotic deflection angle is
\[
\hat{\vartheta}_{\infty}(b)=\frac{4(M+\alpha)}{b}-\frac{3\pi(4\alpha\beta)}{4b^2}+\mathcal{O}(b^{-3}),
\]
so the leading term depends only on \(\mathcal M=M+\alpha\), while the first subleading term reduces the bending angle relative to Schwarzschild with the same asymptotic mass [2604.22684]. In the direct Hernquist-dressed Schwarzschild model, the weak-field deflection angle instead acquires positive halo corrections proportional to \(r_s^3\rho_s\) [2503.19938]. 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 [2509.11245]. In those images, increasing Hernquist parameters enlarges the photon sphere by \(\sim2\%\) to \(\sim30\%\) and decreases measured intensity in the spherical accretion models, with infalling accretion darker than static because of Doppler de-boosting [2509.11245].

A thin-disk study based on a Novikov–Thorne model in the direct Hernquist metric finds that increasing \(r_s\) or \(\rho_s\) results in cooler, dimmer disks with modified flux distributions and outward-shifted ISCOs [2601.03980]. 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 [2507.14305]. 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
\[
M=\frac{r_h(r_h-4\pi r_s^3\rho_s+r_s)}{2(r_h+r_s)}
\]
and
\[
T_H=\frac{2r_h r_s+r_h^2-4\pi r_s^4\rho_s+r_s^2}{4\pi r_h(r_h+r_s)^2},
\]
with a finite endpoint temperature
\[
T_H^M=\frac{1}{16\pi^2 r_s^3\rho_s}
\]
at the radius where the ADM mass vanishes [2503.19938]. 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 [2503.19938].

A related analysis writes
\[
T_H=\frac{1}{4\pi r_h}-\frac{\rho_s r_s^4}{r_h(r_h+r_s)^2}
\]
and defines a remnant radius
\[
r_{\text{rem}}=r_s\left(2r_s\sqrt{\pi\rho_s}-1\right)
\]
from the condition \(T_H(r_{\text{rem}})=0\), with a corresponding remnant mass obtained by substituting \(r_h=r_{\text{rem}}\) into the mass–radius relation [2507.14305]. 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
\[
T=\frac{1}{8\pi M+\displaystyle \frac{64\pi^2 M\rho_s r_s^3 (M+r_s)}{(2M+r_s)^2}},
\]
which is lower than the Schwarzschild value and decreases as halo parameters increase [2602.15420]. 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 [2602.15420].

The literature also contains genuine model-dependent extensions. The neutral limit \(g=0\) of a magnetically charged NED black hole immersed in a Hernquist halo reproduces a Schwarzschild–Hernquist metric of the form
\[
A_{\text{S-H}}(r)=1-\frac{2M}{r}-\frac{4\pi\alpha\beta^3}{r+\beta},
\]
and this framework exhibits parameter degeneracies in which halo effects can mimic Schwarzschild values of the horizon radius, shadow radius, or strong-lensing coefficients [2512.24753]. Another paper adds a cloud of strings and AdS curvature,
\[
f(r)=1-\alpha-\frac{2M}{r}-\frac{b}{r+r_s}+\frac{r^2}{\ell_p^2},
\]
and reports that strings enlarge the shadow whereas Hernquist dark matter shrinks it in that composite setting [2506.18457]. 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 [2604.22684, 2601.03980, 2509.04001, 2602.15420]. 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.

Source: https://www.emergentmind.com/topics/schwarzschild-hernquist-black-hole