Kiselev-Type Behavior in Black Hole Physics

• Kiselev-Type Behavior is a framework describing modified spacetime metrics near black hole horizons with contributions from anisotropic fluids and nonextensive thermodynamic features.
• It employs local metric deformations to enforce the equivalence of Hawking and Rényi temperatures, thereby supporting unique phase structures and energy conditions.
• The extension to rotating and charged black holes highlights a universal mechanism where anisotropic fluid coupling yields mass corrections and novel horizon dynamics.

Kiselev-type behavior refers to a suite of gravitational, thermodynamic, and matter interactions characterized by the geometry and physics of black holes (and related compact objects) in the presence of anisotropic fluids or matter fields, as captured by the “Kiselev solution” and its variants. Originally conceived to model black holes surrounded by “quintessence”-like dark energy, Kiselev-type behavior encompasses a spectrum of modifications to standard spacetime metrics, energy conditions, and thermodynamic relations—especially around horizons—arising from the inclusion of nontrivial matter content with radial–transverse pressure anisotropy, often extending to include electromagnetic and rotational degrees of freedom. Recently, it has also found crucial application in nonextensive thermodynamic frameworks (notably, Hawking-Rényi gravity), neutron star interiors with screened scalar fields, and in analog gravity models.

1. Local Kiselev-type Behavior Near Black Hole Horizons

A fundamental property of Kiselev-type behavior is that to enforce specific thermodynamic requirements—such as the equivalence between the thermodynamically defined Rényi temperature ($T_R$) and the geometrically defined Hawking temperature ($T_H$) at the event horizon—it is sufficient for the spacetime metric to exhibit a Kiselev-type modification only within an infinitesimal neighborhood of the horizon (Czinner et al., 30 Aug 2025). This is termed local Kiselev-type behavior (Editor's term), where the entire global structure of the metric may vary, but the near-horizon limit is governed by an effective metric deformation:

$M \to M[1 + \lambda h(r, a)]$

with %%%%2%%%%. The near-horizon deformation function,

$g(\lambda, r, a) = r[1 + \lambda \pi (r^2 + a^2)],$

ensures that the deformed metric function,

$$\Delta_g = r^2 + a^2 - 2 M r [1 + \lambda \pi (r^2 + a^2)],$$

precisely mimics a Kiselev black hole surrounded by an anisotropic fluid. Crucially, this local requirement means that the specifics of the deformation away from the horizon are arbitrary, provided the global Einstein equations are satisfied.

2. Rényi Nonextensive Thermodynamics and the HR Condition

Kiselev-type horizons have gained renewed relevance in the context of nonextensive black hole thermodynamics, particularly models using the Rényi entropy. For rotating black holes, the Rényi entropy is

$S_R = \frac{1}{\lambda} \ln(1 + \lambda S_{BH}),$

with $S_{BH} = \pi(r_+^2 + a^2)$. The Hawking-Rényi (HR) approach postulates that the first law,

$dM = T_R dS_R + \Omega_H dJ,$

should yield a temperature $T_R$ exactly matching the Hawking temperature $T_H$ derived from the surface gravity. This requirement—$T_R = T_H$ at the horizon—constrains the metric deformation to be locally of Kiselev type (Czinner et al., 30 Aug 2025). The constraint leads to a partial differential equation for $g(\lambda,r,a)$, with the unique solution (under minimal assumptions on boundary terms) corresponding to the Kiselev form above. Notably, in the charged (Kerr–Newman) case, the functional structure of this deformation persists, confirming the universality of this link between nonextensive entropy and local Kiselev behavior.

3. Anisotropic Fluid Coupling and Effective Stress–Energy

To support a Kiselev-type deformation, the spacetime must be sourced by a specially tailored anisotropic fluid: $T^{ab} = \mathrm{diag}(\rho, p_r, p_t, p_t).$ For example, in the static case the energy-momentum profile is

$\rho = -p_r = -2p_t = \frac{\lambda M}{2r}.$

Rotation modifies the anisotropic pressure terms but preserves the necessity for negative radial pressure and anisotropy between radial and transverse directions. This explicit matter content—qualitatively analogous to what is often interpreted as “quintessence” or dark energy—supplies the additional stress required to maintain the HR condition in the presence of the metric deformation (Czinner et al., 30 Aug 2025). The “hair” parameter arising from this coupling shifts the effective black hole mass, especially in rotating scenarios, where

$\bar{M} = M [1 + \pi \lambda a^2].$

4. Extension to Rotating and Electrically Charged Black Holes

The local Kiselev-type mechanism is robust under the inclusion of both rotation ($a \neq 0$) and electric charge ($Q \neq 0$). In the Kerr–Newman-HR framework, the deformed metric function adapts to

$\Delta_g = r^2 + a^2 + Q^2 - 2M g(\lambda, r, a)$

with the modified mass definition

$M = \frac{r_+^2 + a^2 + Q^2}{2g(r_+)}.$

Electric charge leads to the inclusion of an electromagnetic potential in the first law and at the horizon, but does not modify the basic local Kiselev-type form of the metric function. Instead, the anisotropic fluid is augmented by electric contributions, with the unique locally Kiselev-type deformation persisting as the solution enforcing the HR condition.

5. Thermodynamic and Phase Structure Implications

Imposing locally Kiselev-type behavior to achieve Hawking–Rényi equivalence produces several thermodynamic modifications:

• The effective black hole mass receives an additive shift proportional to both the rotation parameter $a$ and Rényi parameter $\lambda$.
• The identification of the anisotropic fluid with a specific equation of state parameter ($w = -2/3$ in some cases) suggests the possibility of new phase structures, analogous to the Hawking–Page transition, and new phenomenology in the stability analysis for the full class of black holes realizing the HR condition (Czinner et al., 30 Aug 2025).
• The universality of the near-horizon Kiselev structure indicates that, despite potentially diverse global physics, the local thermodynamic behavior remains controlled by the underlying matter–geometry coupling, with consequences for entropy–area relations, critical exponents in phase transitions, and the microphysical interpretation of entropy.

6. Universality, Model-Independence, and Physical Interpretation

A key outcome is that Kiselev-type behavior is a minimal and universal requirement for consistent nonextensive black hole thermodynamics in both rotating and charged scenarios. The ability to match $T_R$ and $T_H$ by constraining only a thin region near the horizon (locally Kiselev-type)—while leaving the global solution otherwise unconstrained—emphasizes the independence of horizon thermodynamics from the details of far-field geometry. It also points to the underlying physical role played by radial–transverse pressure anisotropy and the nature of the matter content supporting the spacetime: the existence of an “exotic” anisotropic fluid is not merely a curiosity, but an essential ingredient for extending classical thermodynamic equilibrium concepts to more general gravitational settings.

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Summary Table: Metric Deformation and Matter Structure

Feature	Functional Form Near Horizon	Physical Role
Metric deformation	$g(\lambda, r, a) = r[1 + \lambda \pi (r^2 + a^2)]$	Enforces $T_R = T_H$ in the HR model
Anisotropic fluid	$\rho = -p_r = -2p_t = \frac{\lambda M}{2r}$	Supports Kiselev-type geometry
Mass shift	$\bar{M} = M (1 + \pi \lambda a^2)$	Rotation-induced correction

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In essence, the imposition of Hawking–Rényi thermodynamics on black hole horizons—whether static or rotating, charged or uncharged—requires the metric to exhibit locally Kiselev-type behavior, sourced by an anisotropic fluid whose equation of state and energy–momentum distribution are precisely those originally found in Kiselev’s construction. This framework generalizes and clarifies the role of "exotic" matter in modified black hole thermodynamics, and underscores the universality of horizon physics in settings beyond traditional (extensive and isotropic) paradigms (Czinner et al., 30 Aug 2025).