Generalized Polytropic Regular Black Holes in Arbitrary Dimensions

Abstract: We investigate static, spherically symmetric regular black holes with anti-de Sitter (AdS) asymptotics in arbitrary spacetime dimensions. They are solutions of Einstein gravity, sourced by an anisotropic fluid whose radial pressure corresponds to vacuum energy, while the tangential pressure satisfies a generalized polytropic equation of state. By solving the Einstein field equations, we derive a generic class of asymptotically AdS black hole solutions and determine the conditions required for spacetime regularity. We then investigate the dynamical formation of these regular black holes within the thin-shell formalism, assuming a linear barotropic equation of state for the shell matter. Next, we study the thermodynamics of the regular AdS black holes in arbitrary dimensions by verifying the first law of black hole thermodynamics and the corresponding Smarr relation. We analyze the thermodynamic stability and phase structure of solutions in four, five, and six spacetime dimensions, demonstrating the existence of dimension-dependent phase transitions.

• The paper constructs an exact D-dimensional family of regular black holes sourced by anisotropic generalized polytropic matter, with finite curvature invariants and a physical parameter range of 1/(D−2) < ω₁ ≤ 1.
• The paper shows that thin-shell collapse can produce nonsingular bouncing trajectories when the shell equation of state is ζ = −(D−3)/(D−2), while emphasizing that this fine-tuned condition is not universal.
• The paper verifies corrected extended-phase-space first laws and Komar-derived Smarr relations, finding van der Waals-like transitions with critical ratios of 0.1891, 0.2787, and 0.4191 in four, five, and six dimensions, respectively.

Overview

The paper constructs a family of static, spherically symmetric regular black hole solutions of Einstein gravity with a cosmological constant, valid in arbitrary spacetime dimension $D$ (2608.19606). The matter source is an anisotropic fluid whose radial pressure equals the negative energy density (vacuum energy), while the tangential pressure obeys a generalized polytropic equation of state. The work proceeds along three axes: exact solution construction and regularity analysis; dynamical formation via thin-shell collapse with a linear barotropic shell equation of state; and extended-phase-space thermodynamics, including verification of the first law and Smarr relation and an analysis of phase transitions in four, five, and six dimensions.

Construction of the D-dimensional solutions

The authors adopt the barotropic equations of state

$$P_r = \omega\rho + \bar\omega\,\frac{\rho^n}{\rho_0^{n-1}}, \qquad P_t = \omega_1\rho + \omega_2\,\frac{\rho^n}{\rho_0^{n-1}},$$

with central density $\rho_0 > 0$. Solving the Einstein field equations for the metric ansatz $ds^2 = -f(r)dt^2 + dr^2/f(r) + r^2 d\Omega_{D-2}^2$ requires $\omega = -1$, $\bar\omega = 0$, so that $P_r = -\rho$, and conservation ($\nabla_\mu T^\mu{}_r = 0$, i.e., the generalized TOV equation) fixes $\omega_2 = -\omega_1 - 1$. The resulting density profile is

$$\rho(r) = \frac{\rho_0}{\left[C_1(\rho_0 r^{(\omega_1+1)(D-2)})^{n-1} + 1\right]^{1/(n-1)}},$$

and integration of the field equations yields a metric function expressed through a Gauss hypergeometric function. Setting the Schwarzschild-like integration constant $$P_r = \omega\rho + \bar\omega\,\frac{\rho^n}{\rho_0^{n-1}}, \qquad P_t = \omega_1\rho + \omega_2\,\frac{\rho^n}{\rho_0^{n-1}},$$0 removes the singular contribution, which is essential for regularity.

The total mass is finite and positive provided $$P_r = \omega\rho + \bar\omega\,\frac{\rho^n}{\rho_0^{n-1}}, \qquad P_t = \omega_1\rho + \omega_2\,\frac{\rho^n}{\rho_0^{n-1}},$$1, $$P_r = \omega\rho + \bar\omega\,\frac{\rho^n}{\rho_0^{n-1}}, \qquad P_t = \omega_1\rho + \omega_2\,\frac{\rho^n}{\rho_0^{n-1}},$$2, $$P_r = \omega\rho + \bar\omega\,\frac{\rho^n}{\rho_0^{n-1}}, \qquad P_t = \omega_1\rho + \omega_2\,\frac{\rho^n}{\rho_0^{n-1}},$$3, $$P_r = \omega\rho + \bar\omega\,\frac{\rho^n}{\rho_0^{n-1}}, \qquad P_t = \omega_1\rho + \omega_2\,\frac{\rho^n}{\rho_0^{n-1}},$$4. Combined with the dominant energy condition, this restricts the parameter to

$$P_r = \omega\rho + \bar\omega\,\frac{\rho^n}{\rho_0^{n-1}}, \qquad P_t = \omega_1\rho + \omega_2\,\frac{\rho^n}{\rho_0^{n-1}},$$5

which automatically enforces the null and weak energy conditions as well. The strong energy condition is violated near the center, where the fluid approaches $$P_r = \omega\rho + \bar\omega\,\frac{\rho^n}{\rho_0^{n-1}}, \qquad P_t = \omega_1\rho + \omega_2\,\frac{\rho^n}{\rho_0^{n-1}},$$6 — an unavoidable feature of any de Sitter-core regularization.

Near the origin the metric behaves as de Sitter with effective cosmological constant $$P_r = \omega\rho + \bar\omega\,\frac{\rho^n}{\rho_0^{n-1}}, \qquad P_t = \omega_1\rho + \omega_2\,\frac{\rho^n}{\rho_0^{n-1}},$$7, plus subleading corrections scaling as $$P_r = \omega\rho + \bar\omega\,\frac{\rho^n}{\rho_0^{n-1}}, \qquad P_t = \omega_1\rho + \omega_2\,\frac{\rho^n}{\rho_0^{n-1}},$$8 with $$P_r = \omega\rho + \bar\omega\,\frac{\rho^n}{\rho_0^{n-1}}, \qquad P_t = \omega_1\rho + \omega_2\,\frac{\rho^n}{\rho_0^{n-1}},$$9. All curvature invariants ($\rho_0 > 0$0, $\rho_0 > 0$1, $\rho_0 > 0$2) remain finite at $\rho_0 > 0$3, confirming singularity avoidance. A notable caveat stated explicitly by the authors: since $\rho_0 > 0$4 is generally non-integer, the metric is not necessarily $\rho_0 > 0$5 at the origin — regularity here means finiteness of curvature scalars, not analyticity. At large radius the solution approaches Schwarzschild-(A)dS form, recovering earlier three- and four-dimensional results as special cases.

Thin-shell collapse and dynamical formation

Using Israel junction conditions, the authors derive the general thin-shell equation $\rho_0 > 0$6 in arbitrary $\rho_0 > 0$7, for a shell obeying the linear barotropic EoS $\rho_0 > 0$8, giving $\rho_0 > 0$9. The asymptotic behavior of $ds^2 = -f(r)dt^2 + dr^2/f(r) + r^2 d\Omega_{D-2}^2$0 at both $ds^2 = -f(r)dt^2 + dr^2/f(r) + r^2 d\Omega_{D-2}^2$1 and $ds^2 = -f(r)dt^2 + dr^2/f(r) + r^2 d\Omega_{D-2}^2$2 is classified by the value of $ds^2 = -f(r)dt^2 + dr^2/f(r) + r^2 d\Omega_{D-2}^2$3.

A key result is that the effective potential remains finite at both boundaries only for the fine-tuned choice

$ds^2 = -f(r)dt^2 + dr^2/f(r) + r^2 d\Omega_{D-2}^2$4

for which $ds^2 = -f(r)dt^2 + dr^2/f(r) + r^2 d\Omega_{D-2}^2$5 and $ds^2 = -f(r)dt^2 + dr^2/f(r) + r^2 d\Omega_{D-2}^2$6: the collapsing shell carries positive density and negative pressure (tension). The authors are careful to note this condition follows from demanding boundedness of the potential at both ends and "should not be interpreted as a universal condition" for regular black hole formation. For $ds^2 = -f(r)dt^2 + dr^2/f(r) + r^2 d\Omega_{D-2}^2$7 the shell reaches the origin in finite proper time; at the critical value $ds^2 = -f(r)dt^2 + dr^2/f(r) + r^2 d\Omega_{D-2}^2$8 it approaches $ds^2 = -f(r)dt^2 + dr^2/f(r) + r^2 d\Omega_{D-2}^2$9 only asymptotically, as $\omega = -1$0.

Three explicit models illustrate the mechanism:

Dimension	$\omega = -1$1	$\omega = -1$2	Shell EoS $\omega = -1$3	Outcome
$\omega = -1$4	$\omega = -1$5	$\omega = -1$6	$\omega = -1$7	bounce at $\omega = -1$8
$\omega = -1$9	$\bar\omega = 0$0	$\bar\omega = 0$1	$\bar\omega = 0$2	bounce at $\bar\omega = 0$3
$\bar\omega = 0$4	$\bar\omega = 0$5	$\bar\omega = 0$6	$\bar\omega = 0$7	bounce at $\bar\omega = 0$8

In each case the shell crosses both horizons, reaches a finite minimum radius inside the inner horizon where $\bar\omega = 0$9, bounces, re-expands into a new asymptotic region, and can repeat the cycle — in contrast to the Schwarzschild(-Tangherlini) comparison case, where the shell terminates on the singularity. In all three cases the near-origin approach is exponential in proper time, i.e., the core is reached only in infinite proper time within this model. One caveat deserves emphasis: the four-dimensional example uses $P_r = -\rho$0, outside the dominant-energy-condition bound $P_r = -\rho$1; it was chosen because it reproduces the Hayward metric, so the cleanest dynamically formed example does not satisfy the full set of energy conditions imposed on the general solution.

Thermodynamics in extended phase space

Treating $P_r = -\rho$2 as thermodynamic pressure, the authors decompose the mass into geometric and matter contributions and verify, for the general $P_r = -\rho$3-dimensional family, a corrected first law

$P_r = -\rho$4

where $P_r = -\rho$5 is a correction factor arising from the implicit dependence of $P_r = -\rho$6 on $P_r = -\rho$7, and $P_r = -\rho$8 is the conjugate to the central density treated as an independent thermodynamic variable. The Smarr relation,

$P_r = -\rho$9

is derived from the Komar identity rather than Euler's theorem, with an explicit matter contribution $\nabla_\mu T^\mu{}_r = 0$0 evaluated analytically via hypergeometric asymptotics. The authors state plainly that the Smarr formula is not equivalent to the Euler identity in this setting and leave the detailed relation between them open.

Phase structure across dimensions

Canonical-ensemble stability and phase structure were analyzed for the three explicit models:

$\nabla_\mu T^\mu{}_r = 0$1	$\nabla_\mu T^\mu{}_r = 0$2	$\nabla_\mu T^\mu{}_r = 0$3	$\nabla_\mu T^\mu{}_r = 0$4	$\nabla_\mu T^\mu{}_r = 0$5	$\nabla_\mu T^\mu{}_r = 0$6	$\nabla_\mu T^\mu{}_r = 0$7
4	3/2	2	0.0084	1.5043	0.0669	0.1891
5	4/3	1	0.0047	2.9408	0.0498	0.2787
6	7/6	1/2	$\nabla_\mu T^\mu{}_r = 0$8	8.0899	0.01542	0.4191

All three cases exhibit van der Waals-like behavior: oscillating isotherms below $\nabla_\mu T^\mu{}_r = 0$9, swallowtail structures in the free energy signaling first-order small/large black hole transitions, and heat capacity divergences marking second-order critical points with stable/unstable/stable branches. Two quantitative findings stand out. First, the four-dimensional critical ratio 0.1891 lies close to but distinct from the van der Waals value 3/37 ≈ 0.1875, indicating that the matter source modifies the universal critical behavior. Second, since $\omega_2 = -\omega_1 - 1$0 and $\omega_2 = -\omega_1 - 1$1 vary alongside $\omega_2 = -\omega_1 - 1$2 in these examples, the monotonic growth of the ratio with dimension reflects the combined effect of dimensionality and matter parameters — the authors conclude the critical ratio is not universal within this class.

An additional dimensional distinction emerges from the free energy: in five and six dimensions it remains negative at all temperatures considered, so the black hole always dominates over thermal AdS and no Hawking–Page transition occurs; the phase structure is governed solely by the small/large black hole transition. In four dimensions the swallowtail can extend into positive free energy, allowing a Hawking–Page transition.

Limitations and open questions

Several restrictions qualify the results. The regularity proof establishes finiteness of curvature scalars but not $\omega_2 = -\omega_1 - 1$3 smoothness, since $\omega_2 = -\omega_1 - 1$4 is generically non-integer. The dominant-energy-condition bound $\omega_2 = -\omega_1 - 1$5 conflicts with the Hayward-matching choice used in the flagship four-dimensional example. The shell EoS $\omega_2 = -\omega_1 - 1$6 is a fine-tuned requirement tied to potential boundedness rather than a generic formation criterion, and the thin-shell treatment assumes a specific interior geometry (pure AdS-like core) and sign choices for the junction normals. On the thermodynamic side, the analysis is restricted to the canonical ensemble and to dimensions four through six; the discrepancy between the Smarr formula and the Euler identity remains unresolved. Whether the phase structure persists in other ensembles, or whether the critical ratio becomes universal when $\omega_2 = -\omega_1 - 1$7 and $\omega_2 = -\omega_1 - 1$8 are held fixed while varying $\omega_2 = -\omega_1 - 1$9, are questions the paper leaves unanswered.

Conclusion

This paper delivers a unified, exactly solvable framework for regular black holes in arbitrary dimensions sourced by physically motivated anisotropic polytropic matter, demonstrates their consistency with gravitational collapse through thin-shell dynamics exhibiting nonsingular bounces, and establishes their extended thermodynamics including corrected first laws and Komar-derived Smarr relations. The principal quantitative contributions are the parameter window $$\rho(r) = \frac{\rho_0}{\left[C_1(\rho_0 r^{(\omega_1+1)(D-2)})^{n-1} + 1\right]^{1/(n-1)}},$$0 for physical regularity, the shell EoS $$\rho(r) = \frac{\rho_0}{\left[C_1(\rho_0 r^{(\omega_1+1)(D-2)})^{n-1} + 1\right]^{1/(n-1)}},$$1 compatible with nonsingular collapse, and the demonstration that critical ratios (0.1891, 0.2787, 0.4191 in $$\rho(r) = \frac{\rho_0}{\left[C_1(\rho_0 r^{(\omega_1+1)(D-2)})^{n-1} + 1\right]^{1/(n-1)}},$$2) depend on both dimensionality and matter parameters, precluding universality. The framework provides a basis for extending regular black hole studies to modified gravity theories and to the still-open question of the Smarr–Euler relation.