Regular Black Holes from Anisotropic Source with Hydrodynamic Equation of State

Abstract: We study regular black hole solutions sourced by an anisotropic energy momentum tensor. It is well known that the geometry of the interior of a spherically symmetric regular black hole approaches the dS metric. Having decomposed the energy momentum tensor into its isotropic and anisotropic components, we assume a hydrodynamic equation of state, $P= P(ρ)$, for the pressure, and look for spherically symmetric, regular black hole solutions. We consider different forms of $P(ρ)$ which yield the previously known regular black hole solutions, as well as various new metrics. We show that the profile of $P(ρ)$ has a root and a maximum as it approaches $0^+$ at large distances. Consequently, the square of the sound speed of perturbations, $c_s<sup>2$, changes sign at the point where $P$ reaches its maximum, indicating a potential hydrodynamic instability. In addition, imposing the subluminal bound on $c_s$ puts strong constraints on the model parameters, excluding models in which the energy density has an exponential fall off. We establish a universal hierarchy among the relative positions at which the strong energy condition is violated, at which $P$ has its root, and at which $P$ attains its maximum.

• The paper introduces a framework that derives regular black hole metrics by directly solving Einstein-hydrodynamic field equations with an anisotropic equation of state.
• It systematically examines energy conditions, de Sitter core formation, and the emergence of a universal hierarchy among key structural radii.
• The study recovers known models like Bardeen and Hayward while proposing novel solutions using polynomial, logarithmic, and trigonometric equations of state.

Regular Black Holes From Anisotropic Hydrodynamic Sources: Structure, Stability, and Constraints

Introduction

The paper investigates the class of regular black hole (BH) solutions in general relativity (GR) realized through anisotropic matter sources with prescribed hydrodynamic equations of state $P = P(\rho)$. Unlike classical BHs exhibiting a curvature singularity at the core, regular BHs replace the interior with a de Sitter (dS) geometry and manifest finite curvature invariants everywhere. The approach presented here deviates from traditional methodologies that typically inverse-engineer metric functions from assumed energy densities $\rho(r)$; instead, the study prescribes the equation of state (EOS) $P = P(\rho)$ and solves the coupled Einstein-hydrodynamic field equations directly.

This work provides a systematic generalization, recovering known solutions such as the Bardeen, Hayward, and Dymnikova BHs, and produces novel families of regular BH metrics corresponding to new polynomial, logarithmic, and trigonometric EOS. Key phenomena—such as constraints arising from hydrodynamic and causality conditions, interplay with energy conditions, sound speed behavior, and mass function structures—are systematically analyzed.

Formalism for Regular Black Holes with Anisotropic Sources

The analysis considers static, spherically symmetric spacetimes with a metric of the form

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

and an anisotropic energy momentum tensor decomposed into isotropic and anisotropic components. The isotropic pressure $P$ is defined via a hydrodynamic EOS $P = P(\rho)$, with $\rho$ the energy density and with the strong assumption of no heat flow.

A critical result is the energy conservation equation for the system:

$\rho'(r) + \frac{2}{r}\left(\rho + P \right) = 0,$

which structurally mirrors the FLRW cosmological energy conservation law (under the identification of $r$ as timelike within the inner BH region). The mass function is constructed as

$M(r) = 4\pi \int_0^r dx\, x^2 \rho(x),$

and the metric function is directly related to $\rho(r)$0. Regularity is imposed via $\rho(r)$1 and finiteness of $\rho(r)$2 and $\rho(r)$3.

Energy Conditions, Spacetime Structure, and Constraints

Regular BH solutions require:

• Weak Energy Condition (WEC): $\rho(r)$4 and $\rho(r)$5 everywhere, ensuring monotonic decay of $\rho(r)$6.
• Finiteness: Both $\rho(r)$7 and $\rho(r)$8 remain finite for all $\rho(r)$9.
• Asymptotic Flatness: $P = P(\rho)$0 must fall off faster than $P = P(\rho)$1, guaranteeing finite ADM mass.

Near $P = P(\rho)$2, the imposed regularity and the energy conservation law yield $P = P(\rho)$3, enforcing a dS core. At spatial infinity, different decay laws for $P = P(\rho)$4 (power-law vs. exponential) lead to distinct EOS asymptotics for $P = P(\rho)$5.

A universal hierarchy emerges among the radii where (i) the strong energy condition (SEC) is violated $P = P(\rho)$6, (ii) $P = P(\rho)$7 vanishes $P = P(\rho)$8, and (iii) $P = P(\rho)$9 attains its maximum $$ds^2 = -A(r) dt^2 + A(r)^{-1} dr^2 + r^2 d\Omega^2,$$0:

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

SEC violation ($$ds^2 = -A(r) dt^2 + A(r)^{-1} dr^2 + r^2 d\Omega^2,$$2) is inevitable for regularity and signals the interior dS patch.

Sound Speed and Stability Considerations

A critical and universal feature is found in the behavior of the squared sound speed:

$$ds^2 = -A(r) dt^2 + A(r)^{-1} dr^2 + r^2 d\Omega^2,$$3

which changes sign at the point where $$ds^2 = -A(r) dt^2 + A(r)^{-1} dr^2 + r^2 d\Omega^2,$$4 is maximized, i.e., where $$ds^2 = -A(r) dt^2 + A(r)^{-1} dr^2 + r^2 d\Omega^2,$$5, corresponding to $$ds^2 = -A(r) dt^2 + A(r)^{-1} dr^2 + r^2 d\Omega^2,$$6. For $$ds^2 = -A(r) dt^2 + A(r)^{-1} dr^2 + r^2 d\Omega^2,$$7, $$ds^2 = -A(r) dt^2 + A(r)^{-1} dr^2 + r^2 d\Omega^2,$$8, suggesting hydrodynamic instability in the BH interior. This contrasts with previous quasinormal mode (QNM) analyses, which typically consider only the exterior region and find stability. The crossover of $$ds^2 = -A(r) dt^2 + A(r)^{-1} dr^2 + r^2 d\Omega^2,$$9 often lies inside the Cauchy horizon ($P$0) for a broad parameter range, so exterior stability coexists with potential interior instability. The relationship between this behavior and known inner-horizon mass inflation instabilities remains to be fully clarified.

Imposed subluminal bounds ($P$1) place severe restrictions on the allowable EOS parameter space, excluding, for example, models where $P$2 has exponential falloff.

Construction of Known and Novel Solutions

Polynomial EOS and Known Metrics

The polynomial form

$P$3

captures the Bardeen, Hayward, Fan-Wang, and Dymnikova I solutions for appropriate $P$4. The EOS dictates both the mass function and the horizon structure, thus encoding the behavior of the interior and exterior regions in a physically meaningful way. Extensions to arbitrary $P$5 generate new regular BH metrics, with mass functions expressible, in select cases, via elementary functions or Gauss hypergeometric functions.

Logarithmic and Trigonometric EOS

Exponential decay of $P$6 corresponds to nonlinear/logarithmic EOS:

$P$7

yielding, for instance, the Dymnikova II BH metric and connecting to Einasto halo profiles in the context of dark matter. However, such EOS generically violate subluminal sound speed bounds at large distances, imposing further exclusion on this otherwise physically motivated class.

Trigonometric EOS of the form

$P$8

lead to a qualitatively distinct set of new regular BH solutions, with manageable mass functions and explicit constraint relations for physicality and causality (e.g., $P$9 bounds from WEC and subluminality).

Implications and Theoretical Consequences

Universality and Pathology

A key universal result is that the hydrodynamic approach, with physically-motivated EOS satisfying WEC and regularity conditions, ensures a dS core, necessitates SEC violation, and induces an interior region where hydrodynamic stability breaks down ($P = P(\rho)$0). Excluding all exponential decays by the subluminal bound is stringent and potentially surprising, given prior interest in such models for astrophysical applications (e.g., dark matter halo-mimicking BHs).

Field-Theoretic Realizability and Anisotropy

Although the hydrodynamic method is agnostic to the microscopic matter source, it is compatible with known field-theoretic realizations. For example, non-linear electrodynamics (NED) Lagrangians are known to yield many regular BHs, albeit often not reducing to Maxwell in the IR. The choice of anisotropic fluid decomposition isolates the isotropic sector for consistent EOS definition, avoiding ambiguities present in alternative conventions.

Stability, QNM, and Extensions

The link between sound speed sign change and potential interior instabilities suggests new directions in the study of the interior QNM spectrum, especially since these regions are also subject to classical mass inflation and quantum fluctuation amplification. Parameter ranges where $P = P(\rho)$1 outside the event horizon support exterior stability, but regions where this is violated invite further mathematical and physical scrutiny.

Future work can:

• Seek field-theoretical or fundamental UV completions yielding the prescribed EOS, or classify which can be consistently realized.
• Extend the analysis to more general matter couplings, rotation, and dynamical evolutions.
• Investigate implications for quantum gravity modifications, entropy bounds, and BH evaporation.

Conclusion

This paper provides a comprehensive framework for constructing and classifying spherically symmetric regular black hole solutions sourced by anisotropic fluids with general hydrodynamic EOS. Known regular BH metrics are encompassed as special cases of polynomial or nonlinear EOS, while new families of solutions are derived. The analysis elucidates universal features—such as dS cores, mandatory SEC violation, the hierarchy of key structural radii, and the sign change of the sound speed—highlighting both theoretical constraints and possible physical pathologies. The results underline the importance of EOS selection in constructing physically meaningful regular BHs, clarify the limits imposed by causality and hydrodynamic stability, and open several avenues for further investigation into the microphysical origins, stability, and phenomenological implications of regular black holes (2606.20109).