Kiselev-type Quintessence Fluid

• Kiselev-type Quintessence Fluid is an anisotropic matter source that deforms the Schwarzschild metric by adding a power-law term to the lapse function.
• It exhibits a variable equation-of-state parameter (w) that distinguishes dark-energy-like behavior from radiation or dust, influencing horizon structure and geodesic behavior.
• The model has broad implications in thermodynamics, lensing, and quantum corrections, and invites reinterpretation through modified gravity and nonlinear electrodynamics.

Searching arXiv for recent and foundational papers on Kiselev-type quintessence fluid and related critiques/generalizations. Kiselev-type Quintessence Fluid (QF) is the conventional label for the anisotropic matter source used in Kiselev metrics, most commonly written in static, spherically symmetric form as

$$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_2^2, \qquad f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}},$$

or equivalently $f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$, depending on notation (Younas et al., 2015, Visser, 2019). In much of the literature, $w$ or $w_q$ is treated as a quintessence equation-of-state parameter and the range $-1<w<-1/3$ is singled out as dark-energy-like; however, later analyses make clear that the source is generically anisotropic, not a perfect fluid, and not standard cosmological quintessence in the scalar-field sense (Visser, 2019, Semiz, 2020).

1. Canonical metric form and parameterization

The defining geometric feature of the Kiselev construction is the replacement of the Schwarzschild lapse by a power-law matter term. In the notation of different papers this contribution appears as $-K/r^{1+3w}$, $-\sigma/r^{3w_q+1}$, $-2c\,r^{-3\omega-1}$, $-N_q/r^{3w_q+1}$, or $-\mathrm N/r^{3w+1}$, but the structural role is the same: a nonvacuum source deforms the metric directly rather than acting as a perturbative afterthought (Younas et al., 2015, Rodrigues et al., 2022).

For the standard “quintessence-like” sector, the interval

$f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$0

is the one repeatedly used. Within that range, the Kiselev term decays more slowly than Schwarzschild and, for special values, simplifies sharply. The case $f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$1 yields a Schwarzschild–de Sitter form,

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

while the frequently studied value $f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$3 produces the linear term

$f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$4

which is the analytically simplest nontrivial Kiselev background (Younas et al., 2015, Majeed et al., 2015).

The same parameterization is used outside the quintessence interval as well. In particular, several papers treat $f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$5 as radiation, $f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$6 as dust, and $f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$7 as quintessence, so the Kiselev form functions as a broader anisotropic-source ansatz rather than only a dark-energy model (Majeed et al., 2015, Bagchi et al., 8 Oct 2025).

$f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$8 or $f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$9	Kiselev term in $w$0	Interpretation used in the literature
$w$1	$w$2	radiation
$w$3	$w$4	dust
$w$5	$w$6	quintessence
$w$7	$w$8	cosmological constant

This parameter table also explains why $w$9 dominates explicit calculations: the background remains Schwarzschild-like in form, but the matter term is linear in $w_q$0, which simplifies horizon, lensing, and thermodynamic formulas (Younas et al., 2015, Majeed et al., 2015).

2. Stress-energy structure and the status of the “quintessence” interpretation

The technically correct source is anisotropic. In Visser’s formulation, the Einstein tensor of the Kiselev metric implies

$w_q$1

so that

$w_q$2

Hence $w_q$3 is an average-pressure parameter, not the pressure-to-density ratio of an isotropic perfect fluid (Visser, 2019).

A compact measure of the anisotropy is

$w_q$4

which is generally nonzero and constant. Only for $w_q$5 does the anisotropy vanish, in which case the solution reduces to Schwarzschild–(anti)-de Sitter and the source becomes a cosmological constant rather than generic Kiselev matter (Visser, 2019).

This is the basis of the now-standard criticism of the phrase “black hole surrounded by quintessence.” Semiz argues that the Kiselev spacetime should not be read as a generic black hole in a quintessence-dominated universe, both because the source is anisotropic and because the original derivation imposed the extra condition $w_q$6 after already fixing the radial coordinate, thereby selecting a non-generic subclass of static spherically symmetric spacetimes (Semiz, 2020). The same point reappears in later work, which treats “Kiselev-type quintessence fluid” as a conventional but physically imprecise label (Xia et al., 1 Dec 2025).

A more precise characterization is therefore: a Kiselev-type source is a static anisotropic matter distribution with

$w_q$7

and with the metric deformation controlled by a power law $w_q$8 (Qu et al., 2023, Visser, 2019).

3. Parameter regimes, horizon structure, and special cases

In the commonly analyzed Schwarzschild-like quintessence case $w_q$9,

$-1<w<-1/3$0

For $-1<w<-1/3$1, the metric can be factorized as

$-1<w<-1/3$2

with

$-1<w<-1/3$3

At $-1<w<-1/3$4 the horizon is degenerate, and for $-1<w<-1/3$5 the roots become imaginary and the curvature singularity at $-1<w<-1/3$6 is uncovered (Younas et al., 2015). A closely related thermodynamic treatment uses the same roots but attaches different horizon nomenclature to $-1<w<-1/3$7; this suggests that horizon naming conventions are not uniform across the literature, even when the algebraic structure is the same (Majeed et al., 2015).

The same background also shifts the photon sphere. In the $-1<w<-1/3$8 geometry the circular null orbits satisfy

$-1<w<-1/3$9

and the unstable photon orbit is identified with

$-K/r^{1+3w}$0

which reduces to $-K/r^{1+3w}$1 in the Schwarzschild limit (Younas et al., 2015). This shift is central to strong-deflection phenomenology.

A distinct and much less studied regime is the “reduced Kiselev black hole,” obtained by setting the Schwarzschild mass term to zero and moving to

$-K/r^{1+3w}$2

Then

$-K/r^{1+3w}$3

still possesses a Killing horizon at $-K/r^{1+3w}$4, despite the absence of an explicit Schwarzschild term, and the resulting causal structure is Schwarzschild-like (Qu et al., 2023). This shows that the anisotropic Kiselev source can by itself generate a black-hole spacetime in the reduced branch.

The same Kiselev power law also extends naturally to cylindrical and string-like topologies. For a charged AdS black string immersed in a Kiselev-type quintessence fluid and a Letelier string cloud, the metric function becomes

$-K/r^{1+3w}$5

with the $-K/r^{1+3w}$6 specialization again producing a linear $-K/r^{1+3w}$7 term (Barbosa et al., 4 Jun 2026).

4. Thermodynamics and horizon relations

Thermodynamic analyses of the Schwarzschild–Kiselev quintessence solution with $-K/r^{1+3w}$8 and $-K/r^{1+3w}$9 give

$-\sigma/r^{3w_q+1}$0

with the reality condition $-\sigma/r^{3w_q+1}$1 (Majeed et al., 2015). The associated horizon quantities are

$-\sigma/r^{3w_q+1}$2

$-\sigma/r^{3w_q+1}$3

A notable conclusion is that, for Schwarzschild black holes surrounded by quintessence, even the area and entropy products are mass dependent: $-\sigma/r^{3w_q+1}$4 This differs from the radiation and dust Kiselev cases studied in the same work, where the corresponding products are universal (Majeed et al., 2015).

The same paper verifies a first-law form

$-\sigma/r^{3w_q+1}$5

and gives the heat capacity

$-\sigma/r^{3w_q+1}$6

For the example $-\sigma/r^{3w_q+1}$7, $-\sigma/r^{3w_q+1}$8, the heat capacity is positive for $-\sigma/r^{3w_q+1}$9 and $-2c\,r^{-3\omega-1}$0, negative for $-2c\,r^{-3\omega-1}$1, and vanishes at $-2c\,r^{-3\omega-1}$2 and $-2c\,r^{-3\omega-1}$3, exhibiting the phase-transition pattern emphasized in that study (Majeed et al., 2015).

In regular-black-hole generalizations, the Kiselev parameter enters thermodynamics as an independent work term. For the Bardeen–Kiselev–(A)dS geometry

$-2c\,r^{-3\omega-1}$4

the Smarr formula contains the characteristic factor

$-2c\,r^{-3\omega-1}$5

and the corrected first law includes the conjugate $-2c\,r^{-3\omega-1}$6, so the quintessence normalization becomes a bona fide thermodynamic variable (Rodrigues et al., 2022). This suggests that Kiselev matter, once embedded in more elaborate models, is naturally incorporated into black-hole chemistry rather than merely altering the lapse function.

5. Geodesics, lensing, accretion, quasinormal spectra, and evaporation

Strong-field lensing is one of the most developed applications. In the $-2c\,r^{-3\omega-1}$7 Kiselev background, the exact bending angle can be written in elliptic-integral form, and the main qualitative result is that the Kiselev term enhances deflection in black-hole sectors relative to Schwarzschild. The ordering reported in the lensing study is

$-2c\,r^{-3\omega-1}$8

with the enhancement controlled by the normalization parameter $-2c\,r^{-3\omega-1}$9 (Younas et al., 2015). A homotopy-perturbation treatment reaches the same conclusion from the orbit equation

$-N_q/r^{3w_q+1}$0

and yields especially simple leading deflection formulas at $-N_q/r^{3w_q+1}$1 and $-N_q/r^{3w_q+1}$2 (Shchigolev et al., 2016).

The Kiselev term also modifies accretion and evaporation. In the Kazakov–Solodukhin–Kiselev geometry, the quintessence contribution $-N_q/r^{3w_q+1}$3 alters transonic flow, horizon structure, and proper density profiles, while the quantum correction regularizes the center (Nozari et al., 2020). In evaporation studies using

$-N_q/r^{3w_q+1}$4

more negative $-N_q/r^{3w_q+1}$5 lowers the non-final-stage Hawking temperature

$-N_q/r^{3w_q+1}$6

and markedly prolongs the lifetime, with the enhancement becoming substantial as $-N_q/r^{3w_q+1}$7 (Wu et al., 17 Jun 2026).

Optical and perturbative observables in AdS backgrounds show the same sensitivity. For a Schwarzschild–AdS black hole with cloud of strings and Kiselev-type quintessence, the photon-sphere condition contains the explicit QF correction

$-N_q/r^{3w_q+1}$8

and the scalar-field effective potential contains both the Kiselev term in $-N_q/r^{3w_q+1}$9 and its derivative contribution

$-\mathrm N/r^{3w+1}$0

In that setup, decreasing $-\mathrm N/r^{3w+1}$1 lowers $-\mathrm N/r^{3w+1}$2 and makes $-\mathrm N/r^{3w+1}$3 more negative, which the authors interpret as stronger confinement and faster damping of scalar perturbations (Ahmed et al., 10 Aug 2025).

Analogue-gravity constructions preserve the same metric logic. In the Gross–Pitaevskii realization of “analogue Kiselev acoustic black holes,” the effective acoustic metric takes

$-\mathrm N/r^{3w+1}$4

with $-\mathrm N/r^{3w+1}$5 reproducing the quintessence-like linear term. The exact quasibound spectrum in that sector is

$-\mathrm N/r^{3w+1}$6

showing that the Kiselev normalization appears directly in laboratory-accessible spectral quantities (Santos et al., 25 Jun 2025).

6. Modified-gravity generalizations, source reinterpretations, and current status

Several papers detach the Kiselev metric from its original GR-fluid reading. In Rastall gravity, the surrounding-field term acquires a modified radial power,

$-\mathrm N/r^{3w+1}$7

so the same intrinsic source parameter $-\mathrm N/r^{3w+1}$8 can look, at the metric level, like a GR Kiselev fluid with a different effective equation-of-state parameter (Heydarzade et al., 2017). In $-\mathrm N/r^{3w+1}$9 gravity, the Kiselev contribution to black-string geometry becomes

$f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$00

with $f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$01 and $f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$02, so matter–geometry coupling changes both amplitude and exponent of the effective source term (Santos et al., 24 Feb 2026).

A more radical reinterpretation replaces the fluid altogether. In power-Maxwell electrodynamics, the Kiselev metric

$f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$03

is shown to be an exact Einstein–nonlinear-electrodynamics solution. The required power-Maxwell exponent is

$f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$04

for the electric branch and

$f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$05

for the magnetic branch, with the Kiselev coefficient $f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$06 determined by the corresponding electromagnetic charge. This undermines any claim that the Kiselev geometry uniquely requires a quintessence-fluid source (Dariescu et al., 2022). Closely related Einstein–NLED constructions in AdS recover Kiselev-type terms as explicit matter contributions in regular black-hole solutions (Al-Badawi et al., 2 Mar 2025, Rodrigues et al., 2022).

The strongest viability criticism concerns AdS. For

$f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$07

the Seiberg–Witten brane action becomes negative somewhere outside the horizon whenever the source lies in the dark-energy regime $f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$08. The derived stability condition is

$f(r)=1-\frac{2M}{r}-\sigma r^{-(3w_q+1)}$09

which excludes the very Kiselev parameter range normally associated with quintessence. The conclusion is not that the classical metric fails to solve Einstein’s equations, but that the Kiselev–AdS black hole is nonperturbatively unstable and therefore not viable as a consistent AdS realization of anisotropic dark-energy matter (Xia et al., 1 Dec 2025).

The present status is therefore two-layered. Geometrically, Kiselev-type QF remains a useful exact ansatz for studying how a power-law anisotropic source reshapes horizons, null geodesics, thermodynamics, and perturbation spectra. Physically, later work has made three restrictions unavoidable: the source is generically anisotropic rather than perfect-fluid, the word “quintessence” is historically entrenched but technically misleading, and the metric admits alternative microscopic realizations—especially nonlinear electrodynamics and modified-gravity effective fluids—that weaken any literal identification with cosmological quintessence (Visser, 2019, Semiz, 2020, Dariescu et al., 2022).