Imaginary Time and Black Holes

                                                    

by Giacinto Plescia

Abstract

This article explores the speculative framework of Giacinto Plescia concerning the nature of black holes, imaginary time, and quantum cosmology, in dialogue with Stephen Hawking’s pioneering ideas. Plescia reinterprets black hole singularities not as entropic collapses, but as topological attractors that emit anti-entropic, morphogenetic energy through a virtual hypospace structured by imaginary numbers. By integrating topological models (e.g., trivarieties, chiasmic strings) and catastrophe theory (from René Thom), Plescia envisions black holes as ontological thresholds between being and non-being, where the vacuum is not void but creatively structured. The article further compares his models with the recent work of Capozziello and De Bianchi on regularized black holes and atemporality, demonstrating convergences and divergences in their approaches to quantum gravity, entropy, and the geometry of spacetime. By proposing new forms of catastrophe (such as the metahedron and tetra-farfalla-cusp), Plescia opens a morphogenetic perspective on the cosmos that integrates physics, philosophy, and topology.

Introduction

The relationship between black holes, entropy, and the topology of spacetime continues to provoke deep philosophical and physical inquiry. In this translated essay, Italian theorist Giacinto Plescia offers a radical reformulation of the black hole paradigm inspired by the work of Stephen Hawking, yet distinctly rooted in a topological and morphogenetic ontology.

Drawing from catastrophe theory and imaginary mathematics, Plescia introduces a unique model in which black holes are not endpoints of cosmic collapse but dynamic loci of creative emergence—entities that, rather than consume matter, emit anti-entropic energy through morphogenetic superstrings embedded in a fluctuating hypospace. Central to this vision is the application of imaginary time, not merely as a mathematical tool, but as a generative operator of ontological transformation.

This paper situates Plescia’s contribution in the broader context of theoretical physics and metaphysics, examining his alignment with and divergence from more recent proposals—particularly those of Salvatore Capozziello and Silvia De Bianchi, who argue for a regularized, atemporal model of black holes within relativistic cosmology. The comparative section highlights how Plescia’s symbolic and ontogenetic approach anticipates aspects of their models while also offering a richer integration with topology and aesthetic logic.

By extending René Thom’s theory of catastrophes into new morphological structures (e.g., the metahedron, the farfalla-cusp, the tetra-farfalla-cusp), Plescia articulates a cosmic theory that not only describes physical phenomena but also resonates with metaphysical depth. In so doing, he proposes a vision of the universe as a dynamic field of singularities and transitions—an unfolding morphology of the possible where physics, logic, and ontology converge.

Imaginary Time and Morphogenetic Black Holes: A Topological Dialogue between Plescia and Hawking

1. Introduction: A Scientific Event in Florence

In the summer of 1995, during the 14th International Conference on General Relativity and Gravitation held in Florence, S. Hawking overturned the prevailing view of black holes: no longer singularities of imploding cosmic space, but rather singularities—akin to Heidegger’s Geworfenheit—that emit continuous and anti-entropic energy into the universe. This paradigm shift constitutes a genuine scientific event.

2. What Paradigm Emerged?

A spatial model of virtual black holes situated in a hyperspatial-temporal framework.

Hawking’s concept of imaginary time in black holes unveiled, within the depths of relativistic Einsteinian spatiality, an underlying “hypospace” that is neither a void nor a nothingness, but a super-entity of fluctuating topology.

No one—perhaps for centuries—may be able to answer such a question definitively. Yet, by closely analyzing Hawking’s topological model, some revelations regarding fundamental physical phenomena may emerge.

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3. Indeterminacy and Event Horizons

According to Hawking’s mathematical model, and in line with Heisenberg’s Uncertainty Principle, it will never be possible—even with the most advanced future technē—to determine with absolute precision the status of elementary particles at the boundaries of the event horizon inside a black hole. The stable components of a black hole can be imagined as unstable if the system is static, and thus impermeable to any quantum tunneling phenomenon—or as unstable and ek-static, thus vibrating with singular, strange, or virtual emissions.

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4. The Superstring of Virtual Particles

Among the myriad possible, probable, imaginary, or virtual particles—whether homologous, coherent, symmetric, asymmetric, or supersymmetric—it follows that, within the hypospace underlying a black hole, there must exist at least one cosmic superstring of virtual particles, or super-photonic waves, or gravitons, capable of crossing the event horizon from one space-time domain to another in Figure 1,

By symmetry, the hypospatial chiasmus of this cosmic superstring could allow for unstable and virtual quantum jets: if symmetric, generating an implosive gravimagnetic field; if asymmetric, producing an explosive and ek-static fission field. Thus, from the void, the nothingness, or the cosmic nihil, emerges matter or virtual antimatter—spacetime singularities and chronotopical manifestations of quantum relativity in Figure 2,

Is this truly the case?

Once again, no one may answer for centuries—but a deeper analysis of Hawking’s topological model may shed light on fundamental physical phenomena.

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5. Hypospatial Chiasmus, Planck’s Constant, and Imaginary Time

The hypospatial chiasmus of a black hole can be envisioned as stable and static, or unstable and ek-static, or structurally stable and ek-static.

In Figure 3,

the grave-quantum curvatures of spacetime surrounding a black hole plunge into virtual hypospatial singularities, producing a positive, circular, and symmetric curvature precisely corresponding to a one-to-one geometrical structure: an infinitesimal quantum superstring with dimensions approaching the Planck scale (10⁻³⁵).

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6. Hawking’s Imaginary Numbers and Imaginary Time

In Figure 4, 

the gravitational surface of the universe undulates negatively in accordance with the rhythm of the imaginary numbers coined by Hawking, ultimately revealing—within the hypospace beneath the black hole—the morphogenetic superstrings of the grave-quantum field: if symmetric, it implodes; if characterized by asymmetric spins, it virtually projects new energy into the universe, enough to generate future or even past big bangs.

By virtue of super-symmetry, the hypospatial string plunges into hyper-chronotopia, converging toward proximate or distant symmetries, even light-years away.

In figure five, a hypospatial chiasmus is eventuated—a virtual morphogenesis of the black hole and of other multiverses that are singular, strange, or imaginary.

If science does not deceive us, and Hawking’s reflections are indeed dense with relevance and salience, then we are facing a vision of the kosmos that is both astonishing and paradigmatic—capable of relegating all prior theories to mere entertaining particularities.

Yet this vision is so significant that it opens up new models, useful for unfolding the imagined events of Hawking and for unveiling previously inconceivable salient features.

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7. A Cosmic Metabolic Model and an Imaginary Hyposace

The hysteresis of the virtual chiasmus depicted in figure five can be numerically characterized through the topological cusp of Figure 6. 

There, a cosmic metabolic model is revealed—eventuating from the virtual void or nihil—that forms a chiasmus as an imaginary string and, more broadly, an imaginary virtual hypospace.

This chronotopic morphogenesis serves to stabilize a grave-quantum field that is ek-static or dense with quantum gravity.

In that supersymmetric singularity, the two black holes illustrated may remain eternally intangible, static, or function as supergravity fields of peripheral chronotopias—yet they generate a communicating, fluctuating hypospatial field that emits matter and antimatter, virtual and strange particles, galaxies, and universes.

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8. Topological Varieties: Bivarieties and Trivarieties

To confer rigor and aesthetic elegance to such a model of virtual hypospatial singularity, one may inscribe the paradigm—described via imaginary numbers—within topological varieties, or more precisely, trivarieties.

Figure 7 

illustrates the double hypospatial chiasmus of the virtual black hole, represented as a bivariety into which extreme and inferior polarities plunge, especially when the black hole’s walls reveal themselves to be unstable, indeterminate, and ek-static.

In Figure 8, 

the virtual bivariety envisioned by Hawking descends into the hypospace of a topological torus, traversing a cosmic string that is itself constituted by a topological bivariety.

In the supersymmetry imagined by Hawking, the double toroidal bivariety is revealed as a virtual singularity of the topological chiasmus presented in Figure 9.

9. A Fractal Composition and Hawking’s Virtual Hypospace

What appears to our perception is nothing but a fractal composition of the trivariety—within which black holes may be distributed in absolute freedom throughout the universal chronotopia, without any stable temporal or spatial coessentiality. This configuration makes black holes appear as unique and unmistakably distinct singularities in the universe, while in reality, they are intricately embedded within the grave-quantum field through Hawking’s virtual hypospace.

If this paradigm is significant on the macrocosmic scale, it is equally potent on the microcosmic level. It is not difficult to imagine strings within the Planck-scale microregion that are supersymmetric relative to Hawking’s hypospatial configuration.

We should therefore reflect upon the virtual chronotopia generated by Hawking’s black hole, as well as the topological trivariety model illustrated in Figure 10.

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10. The Hypercubic Model and Geometric Representations

Should black holes eventually eventuate within adjacent cosmic superstrings that generate the grave-quantum field described by Hawking, the resulting topological model would achieve an unparalleled completeness and rigor—though here, only a fleeting glimpse can be offered. And yet, could it not also be intuited?

More generally, with a numerically, geometrically, or algebraically attuned sensibility, one might imagine the black hole immersed in a hypercube or a cuspidal cube—within which the difference and harmonious proportion between the areas of the black hole and the hypospaces of the grave-quantum superstring are disclosed.

In Figure 11, 

a black hole descends into a hypercube alongside its well-known supersymmetry. However, since the implosion of the chronotopia relativistically curves the surrounding spacetime, the hypercube metabolizes into a topological cuspidal cube, wherein the supersymmetric singularity of the black hole unfolds elliptically, in co-essence with the hypospatial chiasmic superstring.

In Figure 12-a, 

the black hole curves spacetime with negative curvature, while Figure 12-b 

reveals the model’s completeness—highlighting the difference and harmony with the chiasmic singularity of the grave-quantum hypospatial superstring.

This section reveals a general logic—whether geometric, mathematical, or algebraic—of difference, since it is possible to discern the numerical proportion between the black hole and the hypospatial superstring within the framework of grave-quantum chronotopia.

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11. The Topological Model and Imaginary Time

In Figure 13/a-b

the trivariety is traversed—through the empty absence of the chiasmic string—by singular events generated by virtual, indeterminate, and strange black holes.

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12. Conclusion: Imaginary Time in Hawking’s Black Holes

Within hyperbolic topology, the events illustrated in Figure 13/b 

emerge from within the hypospatial string of the trivariety. They appear as static singularities—stable, yet born from the unstable salience and the ek-staticity of the nihilistic void.

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13. A Morphogenetic Interpretation of the Cosmos

Where Hawking envisioned vacuum fluctuations, Plescia discerns lines of creation; where science speaks of probability, he sketches possible forms.

His is a quantum metaphysics in which thought and the cosmos are not separate entities but rather co-generate one another:

“Being reveals itself as a strange attractor: it is never what appears, but what pulses in the invisible.”

Plescia’s philosophical vision leads to an aesthetics of topological difference, where beauty arises from the coherence between structure and emergent potential.

The black hole, in his interpretation, is not merely an astrophysical object, but a metaphor of the invisible and the unknown: a trace of Being, a locus of the possible. It represents otherness and the event that escapes the grasp of traditional categories of thought.

The event horizon—that point of no return beyond which nothing can escape—becomes, for Plescia, a symbol of the threshold between the known and the unknowable, the visible and the invisible.

Unlike deterministic interpretations, reality—especially in the region of black holes—is subject to sudden transitions, to “events” that cannot be predicted but only described topologically through a morphogenetic reading of the cosmos.

To describe the universe as a morphogenetic system that emerges from nothing—from nihil—Plescia applies a nonlinear model that captures the abrupt transitions of the universe from null states to active singularities.

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14. Mathematical and Topological Formalizations of Plescia’s Theory

a. Spacetime Curvature and Imaginary Numbers

Plescia employs imaginary numbers to model negative spacetime curvature, inspired by Hawking’s application of complex time in cosmology. This curvature can be represented via a modified Kerr-Newman metric:

$$ds^2 = -\left(1 - \frac{2mr - Q^2}{\rho^2}\right) dt^2 + \left(\frac{\rho^2}{\Delta}\right) dr^2 + \rho^2 d\theta^2 + \left(r^2 + a^2 + \frac{2mr - Q^2}{\rho^2} a^2 \sin^2 \theta\right) \sin^2 \theta d\phi^2 - \frac{4mr a \sin^2 \theta}{\rho^2} dt d\phi$$

with:

$$\rho^2 = r^2 + a^2 \cos^2 \theta + i\epsilon,\quad \Delta = r^2 - 2mr + a^2 + Q^2 + i\delta$$

where $i\epsilon$ and $i\delta$ represent the imaginary contributions related to the hypospatial topology.

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b. Morphogenetic Superstrings in Hypospace

Plescia models morphogenetic superstrings using the Polyakov action for bosonic strings, with an imaginary component added to express quantum genesis:

$$S = -\frac{T}{2} \int d^2\sigma \sqrt{-h} h^{ab} \partial_a X^\mu \partial_b X_\mu + i\alpha \int d^2\sigma \sqrt{-h} h^{ab} \partial_a X^\nu \partial_b X_\nu$$

where:

• $T$ = string tension

• $h^{ab}$ = worldsheet metric

• $\alpha$ = imaginary coefficient for morphogenesis

• $X^\mu$ = spacetime coordinates

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c. Anti-Entropic Black Hole Emission

Plescia reinterprets Hawking’s radiation equation by introducing an imaginary component corresponding to anti-entropic creative energy:

$$T_H = \frac{\hbar c^3}{8\pi G M k_B} + i\beta$$

where $\beta$ is the imaginary, anti-entropic term representing energy generation instead of dissipation.

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d. Modified Einstein Field Equations

To include hypospatial influences, the Einstein equations are extended with a topological imaginary term:

$$R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} + i\gamma g_{\mu \nu} = \frac{8\pi G}{c^4} T_{\mu \nu}$$

Here, $\gamma$ modulates the morphogenetic contribution of virtual topologies to spacetime curvature.

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15. Application of René Thom’s Catastrophe Theory

Elementary Catastrophes

• Fold Catastrophe:

$$V(x) = \frac{1}{3}x^3 + \alpha x$$

• Cusp Catastrophe:

$$V(x, y) = \frac{1}{4}x^4 + \frac{1}{2} \alpha x^2 + \frac{1}{2} y^2$$

These are applied to gravitational curvature and string bifurcations.

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Plescia’s Extensions: New Catastrophe Models

a. Metahedron

A synthesis of multiple catastrophe layers, encoding memory of past topologies:

$$V(x, y, z) = x^4 + y^4 + z^4 + \alpha x^2 + \beta y^2 + \gamma z^2 + \delta xy + \epsilon yz + \zeta xz$$

This model captures morphogenetic feedback loops and recursive topology.

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b. FarfallaCuspide (Butterfly-Cusp)

A hybrid catastrophe embodying spatial desire and the emergence of alterity:

$$V(x, y, z, w) = x^5 + y^5 + z^4 + \alpha x^3 + \beta y^3 + \gamma z^2 + \delta w^2 + \epsilon x^2 + \zeta y^2 + \eta z + \theta w$$

This potential represents event-rich spatial invention.

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c. TetraFarfallaCuspide

Describes dialogical morphogenesis between social intelligence and media-driven space:

$$V(x, y, z, w, u) = x^6 + y^6 + z^5 + w^4 + u^4 + \alpha x^4 + \beta y^4 + \gamma z^3 + \delta w^2 + \epsilon u^2 + \zeta x^2 + \eta y^2 + \theta z + \iota w + \kappa u$$

It formalizes the emergent geometry of collective and individual spatiality.

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Other Morphogenetic Topologies

• Necklace Model (discrete interconnected states):

$$V = \sum_{i=1}^N \left( a_i x_i^2 + b_i y_i^2 + c_i z_i^2 + d_i x_i y_i + e_i y_i z_i + f_i z_i x_i \right)$$

• Diadem Model (circular multi-state organization):

$$V = \sum_{i=1}^N \left( g_i x_i^3 + h_i y_i^3 + j_i z_i^2 + k_i w_i^2 + l_i x_i y_i z_i + m_i w_i \right)$$

• Umbilical Sphere (layered morphogenetic shell):

$$V = \sum_{i=1}^N \left( n_i x_i^4 + o_i y_i^4 + p_i z_i^4 + q_i x_i^2 + r_i y_i^2 + s_i z_i^2 \right)$$

These expressions symbolize nested singularities, transitions, and structural self-organization within quantum-cosmological processes.

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16. Comparative Framework: Plescia, Capozziello, and De Bianchi

Shared Concepts and Diverging Interpretations

a. Interaction with the Event Horizon

• Plescia introduces the notion that virtual particles, photonic superwaves, or gravitons—organized in morphogenetic superstrings—can cross the event horizon of black holes from one space-time to another. This suggests a hidden hypospatial complexity in the black hole’s structure.

• Capozziello and De Bianchi, by contrast, propose that due to atemporality, crossing the event horizon is impossible—yet they acknowledge the structural complexity at the boundaries of space-time.

b. Spacetime Curvature

• Plescia describes negative curvature using imaginary numbers and emphasizes the role of hypospace and quantum topology.

• Capozziello and De Bianchi advocate for a regularized spacetime geometry that avoids singularities by maintaining finite curvature—a purely geometric-relativistic approach.

c. Alternative Models of Black Holes

• Plescia develops the concepts of hypospatial chiasmus and morphogenetic superstrings, offering a spatial model of virtual black holes within temporal hyperspace.

• Capozziello and De Bianchi formulate the theory of atemporality, in which the approach to the event horizon requires infinite time—thus eliminating the singularity by temporal redefinition.

d. Energy and Entropy

• Plescia suggests that black holes emit anti-entropic energy, generating matter and antimatter from unstable singularities—redefining their role from entropic to creative entities.

• Capozziello and De Bianchi do not explicitly discuss anti-entropy but aim to stabilize curvature through geometrical regularization, thereby managing energy behavior without divergence.

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17. Philosophical and Epistemological Perspectives

Plescia’s Vision: From Nihil to Form

Plescia incorporates a morphogenetic-symbolic logic in which form emerges from the nihil via fluctuations of the grave-quantum field. He emphasizes the topology of Being and the role of imaginary numbers in shaping reality. His theory bridges physics with aesthetics and metaphysics, viewing black holes as morphogenetic attractors.

Capozziello and De Bianchi’s Framework

Their theory of regular black holes is based on geometric consistency and a timeless reformulation of relativistic collapse. They pursue a description of spacetime without singularities, rooted in relativistic geometry and supported by theoretical constraints from quantum gravity and loop cosmology.

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18. Chronology and Original Contributions

It is notable that Plescia’s speculative framework, developed as early as 1995, anticipated themes that Capozziello and De Bianchi would later articulate with different tools and goals. While his approach is more ontological, symbolic, and topological, theirs remains analytic, geometric, and relativistic.

Despite these differences, both theories:

• Challenge the classical singularity-based model of black holes

• Seek to integrate relativity with quantum insights

• Envision a more structured and generative spacetime

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19. Comparative Summary Table

Concept	Plescia	Capozziello & De Bianchi
Event Horizon	Crossed by virtual strings and particles (hypospace)	Inaccessible due to atemporality
Spacetime Curvature	Negative curvature via imaginary numbers	Regularized curvature to avoid singularities
Black Hole Model	Hypospatial chiasmus + morphogenetic strings	Atemporal evolution + regular black holes
Entropy	Anti-entropic energy emission from singularity	Not explicitly addressed; curvature stabilization emphasized
Temporal Framework	Imaginary time + hyper-chronotopia	Atemporal relativistic formalism
Mathematical Language	Topological catastrophes, morphogenetic equations, imaginary numbers	Geometrical regularization, relativistic invariants
Epistemological Approach	Symbolic, ontological, metaphysical integration	Analytical, physicalist, relativistic

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20. Concluding Considerations

While stemming from different disciplinary origins—Plescia from topological philosophy and nonlinear morphogenesis, and Capozziello–De Bianchi from quantum gravity and relativistic cosmology—their approaches converge in offering models that transcend the limits of classical black hole theory.

Plescia’s emphasis on imaginary time, anti-entropic emission, and symbolic morphogenesis may offer valuable insights into the epistemological implications of quantum gravity. At the same time, Capozziello and De Bianchi contribute to the formal stability of such models within standard theoretical physics.

Together, they form a dialogical constellation of approaches to spacetime, entropy, and singularity—models that open new ontological and cosmological possibilities.