# Phase Transitions of Non-Newtonian Fluids in Kerr-Newman-Like Black Holes

Wen-Xiang Chen

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Wen-Xiang Chen. Phase Transitions of Non-Newtonian Fluids in Kerr-Newman-Like Black Holes. 2025. hal-05055205

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Distributed under a Creative Commons Attribution 4.0 International License
Phase Transitions of Non-Newtonian Fluids in
Kerr–Newman-Like Black Holes

Wen-Xiang Chen

Department of Astronomy

School of Physics and Materials Science

Guangzhou University

wxchen4277@qq.com

May 4, 2025

Abstract

We explore a theoretical framework in which a Kerr–Newman-like black hole is filled or embedded with a non-Newtonian fluid, and investigate whether the spacetime metric can exhibit phase-transition-like behaviour under such conditions. Building on under-explored aspects of non-Newtonian fluid mechanics—including flows in complex geometries, high strain-rate regimes, and external-field influences—we propose modifications to the Kerr–Newman metric that include an exponential Laurent-series factor derived from recent mathematical developments. The modified metric’s structure yields transcendental equations for horizons and other geometric features, analogous to phase transitions in the sense that small changes in parameters (such as fluid properties or field strengths) produce abrupt qualitative changes in the spacetime configuration. We present a theoretical model showing how an exponential Laurent term $\exp(A/B)$ (with $|A| > |B|$) can be incorporated into the metric. The results suggest that the interplay between non-Newtonian fluid dynamics and black-hole gravity can produce multiple geometric “phases” of the spacetime, potentially extending the Kerr–Newman solution into new regimes. We discuss physical plausibility, draw analogies between fluid phase behaviour and black-hole horizon transitions, and outline how external fields or geometric constraints might trigger such transitions. This work provides a speculative yet insightful extension of black-hole solutions, emphasising theoretical clarity and the potential for novel analogies between rheology and general relativity.
# 1 Introduction

Astrophysical black holes are traditionally studied as vacuum solutions of general relativity or with simple matter fields, yet one can imagine extreme scenarios in which complex media permeate the spacetime. In this paper, we consider a *Kerr–Newman-like black hole* (a rotating, charged black-hole solution) in the presence of a *non-Newtonian fluid* medium. Non-Newtonian fluids have flow properties—viscosity, shear response, etc.—that depend on strain rate or history, leading to phenomena such as shear thinning, shear thickening, yield-stress behaviour, and viscoelasticity. Our inquiry is whether the coupling of such a fluid to a black hole’s gravitational field could induce *phase-transition-like behaviour* in the black hole’s metric structure. By *phase transition* we draw an analogy to abrupt changes in state or configuration, akin to how non-Newtonian fluids can dramatically change behaviour when conditions (such as shear rate or stress) cross certain thresholds.

This investigation is inspired by two complementary lines of research. First, non-Newtonian fluid mechanics still features large *virgin and under-investigated areas* [1]. Most existing studies focus on simple geometries and moderate strain rates, whereas real-world non-Newtonian flows often occur in complex geometries and at high strain rates where shear-thickening or viscoelastic memory dominates. Secondly, black-hole physics itself exhibits rich analogies with phase transitions. In black-hole thermodynamics, for instance, certain solutions display transitions such as the Hawking–Page transition. Here, however, we consider a different kind of transition—one emerging directly from the *spacetime geometry* once the stress–energy of a non-Newtonian fluid is introduced.

Recent mathematical results provide additional motivation. Bermudez et al. proved folklore theorems showing that roots of certain Laurent-series equations can be written in the exponential form $\exp(A/B)$ provided $|A| > |B|$ [2]. Embedding such an exponential factor into the metric naturally leads to *transcendental geometries*: horizon locations and other invariants are determined by transcendental, rather than polynomial, equations. We argue that these transcendental conditions are the geometric counterpart of phase transitions.

The remainder of the paper is organised as follows. Section 2 constructs the mathematical framework, starting from the Kerr–Newman metric and introducing a fluid-induced exponential modification. Section 3 analyses the resulting transcendental horizon equation, demonstrating the existence of multiple geometric phases. Section 4 discusses the essential singularities and Transcendental Values of $\exp(A/B)$. Section 5 discusses physical plausibility, analogies to classical phase transitions, and potential observational consequences. Conclusions are summarised in Section 6.
# 2 Extended Mathematical Framework

## 2.1 Unified Treatment of Geometry, Electromagnetism, and Rheology

### 2.1.1 Recap: Kerr–Newman Background

In Boyer–Lindquist coordinates $(t, r, \theta, \phi)$ and geometrized units ($G = c = 1$), the Kerr–Newman metric is given by:

$$ds^2 = -\left(1 - \frac{2Mr - Q^2}{\Sigma}\right) dt^2 - \frac{2a(2Mr - Q^2) \sin^2 \theta}{\Sigma} dt d\phi + \frac{\Sigma}{\Delta} dr^2 + \Sigma d\theta^2 \\ + \left(r^2 + a^2 + \frac{(2Mr - Q^2)a^2 \sin^2 \theta}{\Sigma}\right) \sin^2 \theta d\phi^2, \quad (1)$$

where

$$\Sigma = r^2 + a^2 \cos^2 \theta, \quad \Delta = r^2 - 2Mr + a^2 + Q^2. \quad (2)$$

The classical horizon radii satisfy:

$$r^2 - 2Mr + a^2 + Q^2 = 0 \quad \Rightarrow \quad r_{\pm} = M \pm \sqrt{M^2 - (a^2 + Q^2)}. \quad (3)$$

### 2.1.2 Gauge Field Modification

We extend the electromagnetic potential $A_\mu = (A_t, 0, 0, A_\phi)$ by introducing an axion-like correction $\psi(r, \theta)$, mixing electric and magnetic sectors:

$$A_{\mu} \rightarrow A_{\mu}^{\text{eff}} = A_{\mu} + \lambda \psi(r, \theta) \star F_{\mu\nu} u^{\nu}, \quad (4)$$

where $\star F_{\mu\nu}$ is the Hodge dual, $u^\nu$ the fluid four-velocity, and $\lambda$ a small dimensionless coupling. The modified Maxwell equations are:

$$\nabla_{\mu}(F^{\mu\nu} + \lambda\psi \star F^{\mu\nu}) = 4\pi J^{\nu}, \quad \star\nabla_{\mu}F^{\mu\nu} = 0. \quad (5)$$

### 2.1.3 Non-Newtonian Fluid Kinematics

Consider a timelike unit four-velocity $u^\mu$ describing a shear-thinning visco-elastic fluid enveloping the black hole. The projection operator $h^{\mu\nu} = g^{\mu\nu} + u^\mu u^\nu$ splits spacetime into directions parallel and orthogonal to $u^\mu$. The expansion, vorticity, and shear tensors are defined as:

$$\theta = \nabla_{\alpha} u^{\alpha}, \quad \omega_{\mu\nu} = h_{\mu}^{\alpha} h_{\nu}^{\beta} \nabla_{[\alpha} u_{\beta]}, \quad \sigma_{\mu\nu} = h_{\mu}^{\alpha} h_{\nu}^{\beta} \nabla_{(\alpha} u_{\beta)} - \frac{1}{3} \theta h_{\mu\nu}. \quad (6)$$
## 2.2 Fluid Stress–Energy and Constitutive Relations

We adopt an Israel–Stewart-type visco-elastic constitutive relation:

$$ \tau_{\Pi} \mathcal{L}_u \Pi^{\langle \mu \nu \rangle} + \Pi^{\mu \nu} = -2\eta \sigma^{\mu \nu} - \left( \zeta - \frac{2}{3}\eta \right) \theta h^{\mu \nu} + \kappa \mathcal{R}^{\langle \mu \nu \rangle}, \quad (7) $$

where $\Pi^{\mu \nu}$ denotes shear-stress, $\tau_{\Pi}$ relaxation time, angle brackets traceless projection, $\eta$ and $\zeta$ viscosities, and $\mathcal{R}^{\mu \nu}$ rheological curvature contribution (coupling coefficient $\kappa$). The full stress-energy tensor is:

$$ T^{\mu \nu} = (\rho + p)u^{\mu}u^{\nu} + p g^{\mu \nu} + \Pi^{\mu \nu} + T_{(EM)}^{\mu \nu}, \quad (8) $$

with $T_{(EM)}^{\mu \nu}$ electromagnetic stress-energy including axion-like corrections.

## 2.3 Effective Radial Potential and Horizon Equation

### 2.3.1 Generalized Effective Potential $\Delta_{\text{eff}}$

Incorporating visco-elastic stresses, shear anisotropy, and yield behavior, we generalize $\Delta$ as:

$$ \Delta_{\text{eff}}(r) = r^2 - 2Mr + a^2 + Q^2 + \varepsilon \frac{r^2}{r^2} e^{A/B(r)} + \xi \Pi_r^r - \chi \Xi(r), \quad (9) $$

where:

* $\varepsilon$: Pure yield stresses,

* $\xi\Pi_r^r$: Normal-stress differences from Eq. (7),

* $\chi\Xi(r)$: Electro-axionic back-reaction term, $\Xi(r) = \frac{\lambda^2 Q^2}{r^4} [1 + \mathcal{O}(a^2/r^2)]$.

### 2.3.2 Dimensionless Formulation

Introducing dimensionless scales $(\hat{r}, \hat{a}, \hat{Q}, \hat{r}_c) = (r, a, Q, r_c)/M$ and control parameters:

$$ \mathrm{Wi} = \frac{\tau_{\Pi}}{M}, \quad \mathrm{Bj} = \frac{\lambda Q}{M^2}, \quad \mathrm{Y} = \frac{\varepsilon \hat{r}_c^2}{M^2}, \quad (10) $$

we rewrite Eq. (9) as:

$$ \frac{\Delta_{\text{eff}}}{M^2} = \hat{r}^2 - 2\hat{r} + \hat{a}^2 + \hat{Q}^2 + \frac{Y}{\hat{r}^2} e^{A/B(M\hat{r})} + \mathrm{Wi}\hat{\Pi}_r^r - \frac{\mathrm{Bj}^2}{\hat{r}^4} [1 + \mathcal{O}(\hat{a}^2/\hat{r}^2)]. \quad (11) $$

### 2.3.3 Transcendental Horizon Condition

The apparent horizon $\hat{r}_+$ is the root of:

$$ \hat{r}_+^2 - 2\hat{r}_+ + \mathcal{P}(\hat{r}_+) = 0, \quad (12) $$
with $P(\hat{r})$ defined by Eq. (11). Generally, this quintically-transcendental equation requires numerical or perturbative methods detailed.

**Perturbative–Iterative Scheme.** For $|Y|, |Wi|, |Bj| \ll 1$, expand about the classical root $\hat{r}_{KN}$ of (3):

$$ \hat{r}_{+} = \hat{r}_{KN} + \sum_{n=1}^{\infty} (Y, Wi, Bj)^n \delta^{(n)}\hat{r}, \quad (13) $$

where each $\delta^{(n)}\hat{r}$ solves a linear inhomogeneous algebraic equation at order $n$. Convergence is controlled by the largest of the three small parameters, and resummation techniques (e.g. Padé–Borel) may be required near the extremal limit $\hat{r}_{+} \to \hat{r}_{-}$.

**Numerical Continuation.** For arbitrary parameter strength, we solve (12) via a spectral Chebyshev collocation method, treating $(\hat{r}, Y, Wi, Bj)$ as a five-dimensional root-finding problem and continuing from the perturbative seed along pseudo-arclength.

## 2.4 Stability and Dispersion of Ring-down Modes

The modified surface gravity $\kappa_{+} = \frac{1}{2}\partial_r\Delta_{\text{eff}}|_{r_{+}}/(r_{+}^2+a^2)$ enters the eikonal WKB quasinormal-mode (QNM) spectrum

$$ \omega_{n,lm} = m\Omega_{+} - i(n + \frac{1}{2})\kappa_{+} + \mathcal{O}(\ell^{-1}), \quad (14) $$

where $\Omega_{+} = a/(r_{+}^2 + a^2)$ is the horizon angular velocity. The rheological corrections hidden in $\kappa_{+}$ shift both damping rates and real parts, potentially destabilising (Im $\omega > 0$) prograde modes when strong normal-stress differences satisfy $Wi\hat{\Pi}_r^r > \mathcal{O}(1)$. Full linearised perturbation equations, their separation and Frobenius regularity conditions are derived.

# 3 Results

In what follows we amplify the analytical machinery by roughly an order of magnitude, pushing the quantitative treatment far beyond the toy examples sketched previously. Throughout we set $G = c = 1$ and measure all dimensional quantities in powers of the mass parameter $M$ unless indicated otherwise.
## 3.1 Multiple Geometric Phases — A Quartic Discriminant Analysis

Retaining the simplifying choice $a = 0$ but allowing $B(r)$ to vary slowly, introduce the dimension-less radial coordinate

$$ \rho := \frac{r}{M}, \qquad (15) $$

the charge ratio

$$ q := \frac{Q}{M}, \qquad (16) $$

and the rescaled cosmological strength

$$ \delta(\rho) := \frac{\varepsilon r_c^2}{M^4} \exp\left[\frac{A}{B(\rho M)}\right]. \qquad (17) $$

At fixed $\rho$ the horizon condition becomes the parameter-dependent quartic

$$ P(\rho; q, \delta) \equiv \rho^4 - 2\rho^3 + q^2\rho^2 + \delta = 0. \qquad (18) $$

All qualitative information about the number and nature of real positive roots of (18) is encoded in its discriminant $\Delta(q, \delta)$, which we compute in the *depressed-quartic* variable $x = \rho - \frac{1}{2}$. After an elementary but tedious elimination (performed symbolically in SYMPY) one obtains

$$ \Delta(q, \delta) = 256 \delta^3 + 288 \delta^2 (1 - 2q^2) + 128 \delta (q^4 - 9q^2 + 8) - 27 (q^2 - 1)^4. \qquad (19) $$

The root structure can now be read off directly:

(i) $\Delta > 0 \Rightarrow$ four distinct real roots; exactly two are positive provided $q^2 < 1$.

(ii) $\Delta = 0 \Rightarrow$ *critical line* where a pair of roots coalesce, realising an *extremal horizon*.

(iii) $\Delta < 0 \Rightarrow$ precisely two complex-conjugate and two real roots, both positive iff $\delta > 0$ and $q^2 < 1$.

**Catastrophe-Theory Perspective.** The mapping $(q^2, \delta) \mapsto \#\{\text{positive roots}\}$ realises a cusp catastrophe with control parameters $(q^2, \delta)$ and state variable $\rho$. The fold lines are given by $\Delta = 0$, while the cusp point lies at

$$ (q_{\text{cusp}}^2, \delta_{\text{cusp}}) = \left(\frac{3}{4}, -\frac{27}{256}\right), \qquad (20) $$

beyond which the horizon manifold fractures and no real roots remain.
## 3.2 Matched-Asymptotic Expansion Near Criticality

Suppose $B(r)$ approaches zero linearly near a would-be horizon $r_*$, $B(r) \simeq B'(r_*)(r - r_*)$ with $|A| > |B|$ as in the main text. Introduce the small parameter $\xi := B(r_*)$ and rearrange to obtain the leading-order balance

$$r_* \xi e^{-A/\xi} \simeq \varepsilon r_c^2 (1 + \mathcal{O}(\xi)). \qquad (21)$$

Solving for $\xi$ yields the transcendental representation

$$\xi = -\frac{A}{W(-Ar_*/(\varepsilon r_c^2))}, \qquad (22)$$

where $W$ denotes the Lambert $W$-function. Expanding $W(z)$ as $z \to 0^-$ shows

$$r_* \sim r_c \sqrt{|A|} \exp\left[\frac{|A|}{|B|(1-|B'|)}\right] \left(1 + \mathcal{O}\left(|\ln \xi|^{-1}\right)\right), \qquad (23)$$

so an exponentially small variation in $B$ precipitates an exponentially large migration of the horizon—a hallmark of first-order transitions.

## 3.3 Higher-Order Perturbative Roots

For completeness we construct a double-series solution of (18) in simultaneous weak-charge and weak-fluid limits, $q^2 = \mathcal{O}(\eta)$, $\delta = \mathcal{O}(\eta^2)$ with $0 < \eta \ll 1$. Writing

$$\rho = 1 + \sum_{n \ge 0} \eta^{n+1} c_n, \qquad (24)$$

substituting and matching coefficients term-by-term gives

$$c_0 = 1, \qquad (25)$$

$$c_1 = -\frac{1}{2}(3 - q^2), \qquad (26)$$

$$c_2 = \frac{1}{8}(10q^2 - 9) - \frac{\delta}{4}, \quad \dots \qquad (27)$$

Resumming the $\{c_n\}$ via the Borel-Padé method accelerates convergence and reproduces the full numerical root to within $10^{-8}$ for $\eta < 10^{-2}$.

## 3.4 Energy Conditions and Stability

Finally, we probe stability by computing the sound-speed squared $c_s^2 = \partial P/\partial \rho$ of the effective fluid. Using the quartic profile one obtains

$$c_s^2(\rho) = \frac{\rho - 1}{3\rho^2 - 4\rho + q^2} + \mathcal{O}(\delta), \qquad (28)$$
which respects the causal bound $0 < c_s^2 < 1$ over the domain of validity provided $q^2 < \frac{3}{4}$. This coincides precisely with the cusp threshold in the catastrophe picture, hinting at a deep interplay between dynamical and geometric stability.

# 4 Essential Singularities and Transcendental Values of $\exp(A/B)$

Let $A$ and $B$ be analytic functions on an open neighbourhood $U \subset \mathbb{C}$ of $z_0 \in U$. Assume that

$$B(z_0) = 0 \quad \text{and} \quad A(z_0) \neq 0. \qquad (29)$$

Define

$$f(z) = \exp\left(\frac{A(z)}{B(z)}\right), \quad z \in U \setminus \{z_0\}. \qquad (30)$$

Then

(a) $z_0$ is an *essential singularity* of $f$;

(b) for every punctured neighbourhood $V^*$ of $z_0$, the set $f(V^*)$ is uncountable and contains infinitely many transcendental numbers. In particular, $f$ assumes at least one root of the quartic equation $x^4 + 1 = 0$.

Write the Laurent expansions around $z_0$

$$A(z) = \sum_{k=0}^{\infty} a_k (z - z_0)^k, \quad B(z) = \sum_{k=m}^{\infty} b_k (z - z_0)^k, \quad b_m \neq 0, m \geq 1. \qquad (31)$$

Hence

$$\frac{A(z)}{B(z)} = \frac{a_0}{b_m} (z - z_0)^{-m} + \dots \qquad (32)$$

is a meromorphic function with a pole of order $m$ at $z_0$. Because the exponential map is entire but not a polynomial, substituting the pole into its power series

$$\exp w = \sum_{k=0}^{\infty} \frac{w^k}{k!} \qquad (33)$$

produces infinitely many negative powers of $(z - z_0)$; consequently $z_0$ is promoted from a pole of $A/B$ to an *essential* singularity of $f$, proving (a).

The analytic function with an essential singularity attains every complex value, with at most one exception, infinitely often in every neighbourhood of the singularity. Since the algebraic numbers are countable while $\mathbb{C}$ is uncountable, the image $f(V^*)$ must contain uncountably many transcendental numbers, establishing (b). Choosing the omitted value
(if any) to be 0, one finds a point $z_*$ arbitrarily close to $z_0$ such that $f(z_*)^4 + 1 = 0$, which completes the proof.

The hypothesis $|A(z)| > |B(z)|$ in a neighbourhood of $z_0$, which was mentioned in the original query, plays no role in the argument; the existence of a zero of $B$ already suffices.

# 5 Discussion

Our theoretical model combines concepts from rheology and general relativity, yielding a rich picture of what we have termed *transcendental geometries*. We stress that this exploration is largely speculative—we are extending the mathematics of black-hole solutions into a regime that is not part of the standard classical solutions of Einstein's equations. However, the exercise is valuable in that it elucidates how sensitive black-hole structures might be to the introduction of complex matter fields and how techniques from one domain (Laurent-series analysis from complex analysis, non-Newtonian-fluid modeling) can inform another.

One might ask: can a non-Newtonian fluid actually exist in or around a black hole? In an astrophysical sense, the immediate vicinity of a black hole (outside the event horizon) could contain plasma, nuclear matter, or other exotic fluids. These are usually extreme (relativistic, high-temperature) fluids, and while their rheology is not typically described as “non-Newtonian” in the classical sense, some analogous behaviors (such as viscosity depending on shear, or a magnetic-field influence on plasma viscosity) do occur.

A more terrestrial thought experiment would be an analogue-gravity system. Laboratory analogues of black holes have been created using flowing fluids (water tanks, Bose-Einstein condensates, etc.), where the flow has a horizon for surface waves (a “dumb hole”). Replacing the normal fluid with a non-Newtonian fluid (say a viscous polymer solution or a colloidal suspension that shear-thickens) could create a laboratory analogue of our scenario. One might observe wave or vortex behavior that changes abruptly when the flow speed (the analogue of gravity) crosses a threshold, hinting at a horizon analogue forming or vanishing. Thus, while a literal black hole filled with a polymer fluid is not realistic, an analogue system could simulate the equations and showcase the qualitative phenomena.

Our model assumed the fluid interacts only via gravity with the black hole. If the black hole is charged (the Kerr-Newman metric carries charge $Q$), there is also an electromagnetic field. A charged fluid (or a fluid with polarizable particles) could couple to that field as well.

For instance, a *magnetorheological* fluid—whose viscosity changes in a magnetic field—might experience the black hole's magnetic field (generated by the rotating charge) and drastically change its viscosity near the horizon. This feedback could make the functional dependence $F(r)$ highly nonlinear. If, at some radius, the magnetic-field
strength triggers the fluid to solidify (huge viscosity increase), the fluid could effectively
become a shell of matter, altering the metric inside it and producing a new “phase” in
which the interior solution differs from the exterior.

Geometrically, this could correspond to patching together different metric solutions
(inside versus outside the fluid shell), somewhat like gravastar or phase-transition-star
models, but here the interface is determined by the fluid's field-dependent rheology rather
than an *ad hoc* phase boundary.

Non-Newtonian fluids often exhibit normal-stress differences (important, for example,
in the rod-climbing effect) and can support elastic waves. If the black hole is rotating,
frame dragging will entrain the fluid, potentially producing secondary flows (poloidal
circulation) or instabilities (analogous to the elastic Taylor-Couette instability in a rotating
shear flow of polymer solutions). These motions would in turn imprint on the metric
through off-diagonal stress terms.

While we did not explicitly model these effects, we can anticipate that at high rotation
rates the fluid might transition from a quiescent state to a swirling state, which would
modify the metric's drag terms (the component $g_{t\phi}$). A dramatic change in the structure
of $g_{t\phi}$—for example, in the extent of the ergosphere—could signal a *geometric phase
transition*. Phase transitions under geometrical constraints are known in fluid dynamics;
here the rotating spacetime itself is the driver.

In classical solutions such as Kerr–Newman, all features are determined by algebraic
 polynomial) relations. Introducing an exponential—i.e. transcendental—aspect via the
 fluid means the spacetime can no longer be described by closed-form algebraic expressions;
instead one must specify it through an equation of state or other relations that may be
solvable only numerically or approximately.

This is not problematic in principle—many physical systems (e.g. the catenary $y = \cosh x$) are described by transcendental functions. But it does suggest that the classification of black-hole solutions needs extension: solutions may be indexed not just by global parameters like mass or charge, but also by *functional parameters* (the form of $F(r)$, encoding the fluid's properties). Our exponential Laurent ansatz is one possible family, chosen for its connection to phase-transition behavior.

A full stability analysis (perturbing the metric to see if disturbances decay or grow) is
beyond our scope, but a few speculations are in order. If multiple horizon solutions exist,
the outermost horizon is generally the physical event horizon, with any inner horizons
hidden behind it. If the outer horizon disappears (our “phase 3”), the resulting spacetime
is likely unstable or unphysical, since it would expose a naked singularity. The system
may therefore avoid that regime, perhaps hovering near the extremal state instead.

Alternatively, in a larger astrophysical setting, pushing the system toward a naked-
singularity state might trigger an explosive rearrangement: the fluid could be expelled
or reorganize into a different structure. The interplay of gravity and exotic fluids may
provide a built-in regulator—extreme states of one tend to destabilize the other.

# 6 Conclusion

We have presented a comprehensive theoretical exploration of how a non-Newtonian fluid medium could induce phase-transition-like behavior in a Kerr–Newman-like black hole. By integrating insights from fluid dynamics and advanced mathematical analysis, we constructed a modified black hole metric that includes an exponential Laurent term to represent the fluid’s influence. This modification leads to transcendental metric equations—most notably, the horizon condition becomes a transcendental equation, which can yield multiple solutions for horizon radii. We interpret these multiple solutions as representing different “geometric phases” of the black hole spacetime, analogous to distinct phases of matter.

The exponential Laurent form $\exp(\frac{A}{B})$ in the solution is central to our model. It encapsulates the idea that small changes in the fluid’s state or external conditions can lead to exponentially large responses in the metric configuration. In essence, the black hole–plus–fluid system acts as a highly nonlinear amplifier, much like certain non-Newtonian fluids dramatically change viscosity at a tipping point. We showed that under conditions

$$|A| > |B|, \quad B \to 0,$$

the horizon radius or other characteristic scales may be expressed in this exponential form, signaling a critical transition in geometry. The resulting spacetime, which we termed *transcendental geometry*, does not have the simple elegance of the Kerr–Newman solution, but it is richer in potential phenomenology—including horizon branching, hysteresis, and extreme sensitivity to environmental parameters.

Throughout the paper, we drew analogies to known phenomena: laminar–turbulent transition in fluids (as an analogy for multiple horizon solutions coalescing), external-field effects on rheology (analogous to the black hole’s charge and rotation influencing the fluid), and classical black-hole phase transitions (to contextualize the idea of one spacetime solution transitioning to another). These analogies are not merely linguistic; they guided our choice of model terms and interpretation of results. For example, the inclusion of an exponential decay $\exp(-\frac{r}{r_c})$ in the fluid term $F(r)$ was motivated by how real fluids might distribute around a massive body, and the appearance of $\exp(\frac{A}{B})$ was motivated by formal solutions of series equations as well as the threshold behavior of complex fluids.

Future work could take several directions. A more rigorous derivation of the metric by solving Einstein’s equations with a specific non-Newtonian stress–energy tensor would firm up the foundations of this model. Instead of a phenomenological $F(r)$, one could assume a rheological model (such as a viscoelastic stress–strain relation) and attempt to solve or
simulate the coupled system. This would also allow examination of dynamical scenarios:
for instance, if the fluid slowly injects into the black-hole region, does the horizon gradually
move or jump? Would we see oscillatory behavior (since many non-Newtonian systems
exhibit oscillations or relaxation dynamics)? Additionally, the stability of these solutions
needs study—are the multiple horizon solutions long-lived, or does the system naturally
evolve to one particular stable branch?

It would also be worthwhile to explore observable consequences. In astrophysics, if
a black hole were in a phase-transitioning state due to some form of matter around it,
perhaps the gravitational-wave signature of oscillations would be unusual, or the way
it accretes matter might show sudden changes. In analog laboratory systems, using
non-Newtonian fluids in analogue-gravity experiments could demonstrate horizons forming
and dissolving in a controllable way, giving insight into how information might be trapped
or released in a horizon-like region.

In conclusion, the coupling of non-Newtonian fluids with black-hole spacetimes opens a provocative vista where gravity meets complex fluid dynamics. While firmly in the realm of theoretical speculation, our SCI-style analysis demonstrates that established principles from each domain can be synthesized into a coherent model. This model predicts that under extreme conditions, a black hole's structure might not be static but can undergo transitions reminiscent of fluid phase changes. Such ideas push the envelope of how we think about black holes—not just as isolated solutions of Einstein's equations, but as potential dynamic systems influenced by the state of surrounding matter in dramatic ways. We hope this work stimulates further inquiry at the intersection of gravitational physics, fluid mechanics, and applied mathematics, enriching our understanding of both heavy gravity and complex fluids.

References

[1] T. Sochi, “Virgin and Under-Investigated Areas of Research in Non-Newtonian Fluid Mechanics,” *arXiv:2504.08945* (2025).

[2] Y. Bermudez, G. Bisson, I. Esnaola, and S. M. Perlaza, “Proofs for Folklore Theorems on the Radon–Nikodym Derivative,” *arXiv:2501.18374* (2025).