Kiselev-Type Metric in Black Hole Physics

• Kiselev-type metric is defined as a family of static, spherically symmetric spacetimes that model black holes in anisotropic media, generalizing classical solutions.
• Its formulation uses a state parameter and normalization constant to incorporate effects of various surrounding fields such as dust, radiation, and quintessence on horizon and thermodynamic properties.
• Further studies reinterpret the metric through nonlinear electrodynamics, scalar fields, or modified gravity frameworks to capture the anisotropic stress-energy distribution.

The Kiselev-type metric is a family of static, spherically symmetric geometries used to model black holes immersed in an anisotropic surrounding medium parameterized by an equation-of-state variable. In the literature represented here, it appears both as a direct generalization of the Schwarzschild or Reissner–Nordström metric and as a broader effective template for spacetimes sourced by quintessence-like matter, dust, radiation, or related anisotropic sectors. The same structure has also been used in analogue gravity, modified-gravity constructions, regular black-hole models, wormholes, and compact-star studies (Santos et al., 25 Jun 2025, Majeed et al., 2015).

1. Canonical form and parameterizations

A standard Kiselev line element is written as

$$ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$

with

$$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\sigma}{r^{3\omega+1}}.$$

Here $M$ is the black-hole mass, $Q$ is the electric charge, $\sigma$ is a normalization parameter associated with the surrounding medium, and $\omega$ is the state parameter entering $p=\omega\rho$ in the effective description (Majeed et al., 2015, Morais et al., 2021). In uncharged cases the $Q^2/r^2$ term is omitted.

A second notation, used in analogue-gravity constructions, writes

$f(r)=1-\frac{D}{r}+\frac{C_\varpi}{r^{3\varpi+1}},$

where $D$ is a draining parameter and $$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\sigma}{r^{3\omega+1}}.$$0 is a fluid-dependent “charge” parameter (Santos et al., 25 Jun 2025). This reflects a different parametrization of the same functional motif: a Schwarzschild-like $$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\sigma}{r^{3\omega+1}}.$$1 term supplemented by a fluid term with exponent fixed by the state parameter.

Several special cases recur throughout the literature.

Surrounding medium	Parameter choice	Metric function $$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\sigma}{r^{3\omega+1}}.$$2
Radiation	$$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\sigma}{r^{3\omega+1}}.$$3	$$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\sigma}{r^{3\omega+1}}.$$4
Dust	$$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\sigma}{r^{3\omega+1}}.$$5	$$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\sigma}{r^{3\omega+1}}.$$6
Quintessence	$$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\sigma}{r^{3\omega+1}}.$$7, $$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\sigma}{r^{3\omega+1}}.$$8	$$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\sigma}{r^{3\omega+1}}.$$9

These cases are used to interpolate between standard vacuum black holes and “dirty” environments in which the asymptotic and horizon structures are modified by a surrounding field (Majeed et al., 2015). In thermodynamic studies, $M$0 is also identified with the cosmological-constant case and $M$1 with phantom energy (Morais et al., 2021).

2. Stress-energy content and source interpretation

Although the Kiselev family is often introduced through an equation-of-state parameter, the source is generically anisotropic rather than a perfect fluid. A general static, spherically symmetric stress tensor takes the form

$M$2

and, in the Kiselev construction, one commonly finds

$M$3

with the average pressure satisfying

$M$4

(Qu et al., 2023, Saadati et al., 2020). This distinction between directional pressures is central: the Kiselev geometry is not, in general, a perfect-fluid spacetime.

For the one-component Kiselev black hole, the explicit stresses are

$M$5

so that

$M$6

This directly controls the null energy condition (NEC): it is satisfied for $M$7, marginal for $M$8, and violated for $M$9 (Boonserm et al., 2019).

A persistent misconception is that the Kiselev metric generically represents a perfect fluid. Later analyses show that it can instead be decomposed into a perfect-fluid component plus either an electromagnetic component or a scalar-field component: $Q$0 The electromagnetic sector is selected when $Q$1, with

$Q$2

while the scalar sector is selected when $Q$3, with

$Q$4

(Boonserm et al., 2019). In the generalized $Q$5-component case, the same criterion is recast in terms of the density gradient $Q$6, leading to an “onion-like” radial structure in which the effective support can change with radius (Boonserm et al., 2019).

A further reinterpretation replaces the anisotropic fluid altogether by nonlinear electrodynamics. In power-Maxwell electrodynamics, the Kiselev geometry arises as an exact solution of Einstein’s equations coupled to a power-Maxwell Lagrangian, for both electric and magnetic ansätze, with the metric written as

$Q$7

In that framework the “quintessence” source is recast as a nonlinear electromagnetic sector rather than an ordinary fluid (Dariescu et al., 2022).

3. Horizons, causal structure, and geometric variants

Horizons are determined by the roots of $Q$8, and the number and character of those roots depend on the state parameter and normalization of the surrounding medium. For the widely studied quintessence case $Q$9,

$\sigma$0

and the horizons are

$\sigma$1

This yields the standard trichotomy: two horizons for $\sigma$2, a degenerate horizon at $\sigma$3, and no horizon beyond that threshold, producing a naked singularity (Jiao et al., 2016, Younas et al., 2015).

The horizon analysis extends directly to charged variants. For example, in the radiation case one has

$\sigma$4

while in the dust case

$\sigma$5

(Majeed et al., 2015, Sadeghi et al., 2018). These formulas make explicit that the ambient field can shift horizon radii and alter extremality conditions.

A less-studied regime, termed the reduced Kiselev black hole, takes $\sigma$6 and removes the standard mass term. Writing $\sigma$7 with $\sigma$8, the metric function becomes

$\sigma$9

Even with $\omega$0, a black-hole-type horizon appears at $\omega$1. The causal structure is Schwarzschild-like, $\omega$2 is a Killing horizon, and the thermodynamic quantities are

$\omega$3

(Qu et al., 2023).

Rotating generalizations have been constructed through Newman–Janis-type procedures. In $\omega$4 gravity, the rotating solution is written with

$\omega$5

where the exponent

$\omega$6

depends on both the state parameter $\omega$7 and the matter-geometry coupling $\omega$8. The solution reduces to Kerr for $\omega$9 and to Kerr–Newman for $p=\omega\rho$0, $p=\omega\rho$1 (Ghosh et al., 2023).

4. Accretion, lensing, chaos, and perturbative probes

The Kiselev family has been used extensively as a dynamical background. In spherical accretion without back-reaction, baryon-number conservation gives

$p=\omega\rho$2

while energy-momentum conservation yields a Bernoulli equation,

$p=\omega\rho$3

The sonic-point conditions are

$p=\omega\rho$4

and the quintessence parameter strongly affects the location of the critical point and the mass accretion rate (Jiao et al., 2016).

For null geodesics, the $p=\omega\rho$5 Kiselev black hole admits a detailed strong-lensing analysis. The effective potential is

$p=\omega\rho$6

the photon sphere is located at

$p=\omega\rho$7

and the exact bending angle can be written in terms of elliptic integrals. In that specific study, the bending angles satisfy the ordering

$p=\omega\rho$8

for comparable impact parameters (Younas et al., 2015).

Charged-particle dynamics exhibits additional structure. For the charged Kiselev black hole,

$p=\omega\rho$9

the Lyapunov exponent governing orbital instability is computed from the effective potential and a Jacobian-matrix treatment of perturbations. Near the horizon, the chaos bound is satisfied for fixed charge-to-mass ratio, whereas away from the horizon it can be violated when the black-hole charge $Q^2/r^2$0 and particle angular momentum $Q^2/r^2$1 are sufficiently large and the normalization factor $Q^2/r^2$2 is small (Gao et al., 2022).

Perturbation theory also enters through scalar waves. In analogue Kiselev acoustic black holes, the massless Klein–Gordon equation separates into radial equations of Heun type, quasibound states can be obtained by the Vieira–Bezerra–Kokkotas method, and quasinormal modes can be extracted with sixth-order WKB methods (Santos et al., 25 Jun 2025).

5. Thermodynamics, holography, and evaporation

Thermodynamic studies of Kiselev black holes emphasize horizon products, heat capacities, and deformations of the Hawking sector. For radiation and dust backgrounds, the products of horizon areas and entropies are mass-independent: $Q^2/r^2$3 for radiation, and

$Q^2/r^2$4

for dust. By contrast, for the Schwarzschild black hole surrounded by quintessence, the corresponding products depend on the mass and are therefore not universal (Majeed et al., 2015).

The same thermodynamic relations have been used in a CFT setting. For Kiselev black holes surrounded by radiation or dust, the universal entropy products support a Kiselev/CFT correspondence in which the left- and right-moving central charges are found to be equal: $Q^2/r^2$5 for the radiation case, with an analogous equality in the dust case (Sadeghi et al., 2018).

Beyond general relativity, Rainbow-gravity studies use

$Q^2/r^2$6

with $Q^2/r^2$7 and $Q^2/r^2$8. In that setting, cosmic-fluid effects impose a maximum allowed horizon radius, the Hawking temperature can vanish at a finite critical horizon where black-hole and cosmological horizons merge, and black-hole remnants do not generically appear except in the peculiar case $Q^2/r^2$9, $f(r)=1-\frac{D}{r}+\frac{C_\varpi}{r^{3\varpi+1}},$0 (Morais et al., 2021).

A distinct thermodynamic construction links the Kiselev metric to Rényi entropy. Requiring the Hawking temperature to equal the Rényi temperature yields

$f(r)=1-\frac{D}{r}+\frac{C_\varpi}{r^{3\varpi+1}},$1

which is of Kiselev form with $f(r)=1-\frac{D}{r}+\frac{C_\varpi}{r^{3\varpi+1}},$2 and $f(r)=1-\frac{D}{r}+\frac{C_\varpi}{r^{3\varpi+1}},$3. In the Schwarzschild-like case $f(r)=1-\frac{D}{r}+\frac{C_\varpi}{r^{3\varpi+1}},$4, the geometry admits a maximum mass

$f(r)=1-\frac{D}{r}+\frac{C_\varpi}{r^{3\varpi+1}},$5

at which the temperature vanishes and the horizons coincide; the third law is then interpreted as a cosmic censor preventing dynamical access to the naked-singularity regime (Czinner et al., 23 Apr 2025).

Evaporation studies introduce a dynamical state parameter $f(r)=1-\frac{D}{r}+\frac{C_\varpi}{r^{3\varpi+1}},$6 in

$f(r)=1-\frac{D}{r}+\frac{C_\varpi}{r^{3\varpi+1}},$7

Lowering $f(r)=1-\frac{D}{r}+\frac{C_\varpi}{r^{3\varpi+1}},$8 reduces the non-final-stage temperature and markedly prolongs the evaporation lifetime; as $f(r)=1-\frac{D}{r}+\frac{C_\varpi}{r^{3\varpi+1}},$9, the metric approaches Schwarzschild–de Sitter behavior and enters an ultra-slowly evaporating regime. That mechanism is explicitly distinguished from the ultra-long lifetimes found in PFDM and Horndeski black holes (Wu et al., 17 Jun 2026).

6. Analogue, regularized, and matter-coupled extensions

The Kiselev form is sufficiently flexible to reappear in several adjacent research programs. In analogue gravity, Gross–Pitaevskii theory for Bose–Einstein condensates yields an acoustic metric

$D$0

with a radial flow engineered so that

$D$1

This realizes analogue Kiselev acoustic black holes and provides quasinormal and quasibound spectra for scalar perturbations (Santos et al., 25 Jun 2025).

Regularity constructions replace the Kiselev singular core by de Sitter space and match the two regions across a thin shell using Barrabes–Israel junction conditions. The exterior is

$D$2

the interior is

$D$3

and stable stationary solutions arise for suitable parameter ranges, yielding nonsingular black-hole spacetimes with a de Sitter core (Saadati et al., 2020).

The same anisotropic template has been generalized to traversable wormholes. With the Morris–Thorne ansatz

$D$4

and Kiselev-inspired density

$D$5

integration gives the shape function

$D$6

The flare-out condition requires $D$7, while the NEC and WEC are violated in the physically traversable sector, so the configuration remains supported by exotic matter in the usual wormhole sense (Yuennan et al., 2024).

Compact-star applications replace the phenomenological anisotropic fluid by a chameleon scalar field. In that setting the Kiselev-type metric

$D$8

is coupled to modified TOV equations, and scalar gradients induce anisotropy through

$D$9

The scalar is screened in the high-density core and unscreened outside a critical radius $$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\sigma}{r^{3\omega+1}}.$$00, producing stable neutron-star configurations consistent with the conservative mass, radius, and tidal-deformability bounds quoted in the study (Keshav et al., 6 Aug 2025).

Taken together, these developments show that the Kiselev-type metric functions less as a single-source solution than as a reusable geometric ansatz for anisotropic environments. The precise physical interpretation depends on context—quintessence-like matter, nonlinear electrodynamics, effective laboratory media, thin-shell matching, or screened scalars—but the defining signature remains the same: a Schwarzschild-type potential deformed by a state-parameter-dependent power law.