Ferenc Takács
Independent Researcher, Budapest, Hungary.
DOI: 10.4236/ijaa.2026.163015   PDF    HTML   XML   5 Downloads   85 Views   Citations

Abstract

We present a novel modification of Einstein’s field equations in which the cosmological constant is treated not as an immutable value, but as a dynamically varying scalar field dependent on local geometric tension. The distance and tension dependence of the expansion naturally separates the scales: on cosmic scales it generates dark energy causing the expansion of the universe, on galactic scales it eliminates dark matter, the cause of inflation after the big bang, and remains negligible in everyday astrophysical systems, while forming a gigantic antigravity barrier inside collapsing stars, eliminating physical singularities. At atomic scales, spacetime binds itself to the Standard Model in the role of the Higgs field. Finally, we show how the model can provide opportunities for the emergence of new universes.

Keywords

Astronomy, Astrophysics and Cosmology, Gravity, Alternative Gravity Theories, Cosmological Constant, Black Holes, General Relativity Equations, Inflation, Singularity, Baby Universes, Dark Energy, Expanding Universe, Dark Matter, Spacetime, Higgs Field, Quark Confinement, Albert Einstein

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1. Introduction

Let us ask ourselves a fundamental question: what has preoccupied theoretical astronomy and astrophysics over the past century, beyond direct empirical observations? The answer lies in the rigorous execution of two large-scale intellectual programs.

The first program focuses on verifying Einstein’s general theory of relativity (GR) and its profound predictions through high-precision measurements. This endeavor has been met with overwhelming success. Over the decades, astronomers have successfully demonstrated the expansion of the universe, mapped millions of galaxies, detected gravitational lensing, and captured the cosmic microwave background radiation. More recently, the direct imaging of black holes and the detection of gravitational waves from the merger of binary stars have solidified the empirical dominance of the theory. Billions of dollars have been spent on groundbreaking experimental infrastructures, such as LIGO, VIRGO, and the planned LISA detectors, as well as numerous space-based satellite observatories and ground-based antenna networks. The monumental success of the first program should not be underestimated.

The second program, which was to explain the deviations of observations from the theoretical model, has led to a complete failure on a systemic level. Our measurements have become too precise. Galaxies do not rotate as classical theory predicts. The universe is not expanding uniformly. Furthermore, standard general relativity continues to be plagued by infinite density singularities, where our fundamental laws of physics break down completely. A series of ad-hoc hypotheses have emerged to bridge the catastrophic gaps between observation and theory: dark matter, dark energy, and inflaton space. These concepts have nothing to do with Einstein’s original geometric predictions, often directly contradict them, and decades of intensive research have not yielded direct evidence for the physical existence of the hypothetical spaces.

This paper is a paradigm shift. By introducing a major modification to Einstein’s equations, we resolve these seemingly inexplicable discrepancies, effectively pulling the sting out of the anomalies that weakened the Standard Model.

Historically, Einstein’s space equations have been known in two different mathematical forms, depending on whether they include the cosmological constant Λ or omit it. Einstein first introduced the term to create a static universe by balancing gravity, and then famously withdrew it following Hubble’s discoveries, only to have it reinstated by the scientific community after the accelerating expansion of the cosmos was empirically confirmed—a sequence of events that often echo Einstein’s accounts of his regrets over his initial abandonment [1].

However, the physical nature and exact value of Λ remain a subject of ongoing, unresolved debate. Standard cosmology treats it as an unchanging, globally homogeneous constant. However, modern observational data—such as recent dark energy spectroscopic surveys [2]—strongly suggest that its effective value is not uniform, but evolves dynamically over different cosmic epochs and physical scales.

The current work presents a modified model of gravity that handles both cosmological expansion and localized singularities in a single, elegant framework. It modifies spacetime locally, starting from local stresses only, to explain the global properties of spacetime, while preserving the previously completely local character of Einstein’s equations, as Einstein himself envisioned [3].

2. The Modified Field Equation and the Curvature-Dependent Scalar Field

We begin with the original Einstein field equations, redefining the cosmological constant not as a static value, but as a dynamic function dependent on local curvature invariants that capture the full geometric tension of spacetime:

$R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }+\Lambda \left(K,C^2\right)g_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }$ (1)

where $R_{\mu \nu }$ is the Ricci tensor, $R$ is the Ricci scalar, $g_{\mu \nu }$ is the metric tensor, and $T_{\mu \nu }$ is the energy-momentum tensor. To evaluate this local tension, we introduce the Kretschmann scalar $K$ , which is the contraction of the Riemann curvature tensor with itself:

$K=R^{\alpha \beta \gamma \delta }{R}_{\alpha \beta \gamma \delta }$ (2)

The primary advantage of the Kretschmann scalar over the Ricci scalar is that it does not vanish in a pure vacuum surrounding a mass ($T_{\mu \nu }=0$ ); instead, it directly measures gravitational tidal forces and geometric fesszültség. In the vacuum around a static body of mass $M$ , its value is given by the exact Schwarzschild invariant [4]:

$K=\frac{48G^2{M}^2}{c^4{r}^6}$ (3)

However, to ensure rigorous physical dimensionality where Λ carries the unit of m⁻², the curvature invariants must be mapped under a square root ($\sqrt{{\text{m}}^{-4}}={\text{m}}^{-2}$ ). Furthermore, while the Kretschmann scalar encapsulates localized mass density, it decays too rapidly in distributed systems like galactic peripheries. To account for the non-local tidal shear of the free gravitational field, we must also incorporate the invariant square of the Weyl conformal tensor,

$C^2=C^{\alpha \beta \gamma \delta }{C}_{\alpha \beta \gamma \delta }.$ (4)

This factor had to be included because $K$ alone proved insufficient to resolve the anomalies.

To maintain the simplest, purest geometric approach, we assume a linear relationship between the density of dark energy and the square roots of these two fundamental invariants:

$\Lambda \left(K,C^2\right)=\gamma \cdot \sqrt{K}+\delta \cdot \sqrt{C^2}$ (5)

where $\gamma$ and $\delta$ represent new, dimensionless universal coupling constants. Although there are a thousand ways to specify the relationship, the simplest relationship makes the model the easiest to test.

This formulation strictly adheres to the principle of local action (Nahewirkung), a cornerstone of General Relativity emphasized by Einstein himself to completely eradicate Newtonian action-at-a-distance [3]. By expressing $\Lambda \left(K,C^2\right)$ as a differential mapping of the immediate, point-by-point geometric state, the model remains purely local, ensuring that dark energy density is a local manifestation of spacetime tension rather than an ad-hoc global background.

Critical Review of the Orthodox Cosmological Constant Λ versus Geometrodynamic $\Lambda \left(K,C^2\right)$

To demonstrate the ontological necessity of our modification, the proposed framework must be critically evaluated against the conceptual failures of the standard cosmological constant Λ utilized in the contemporary ΛCDM paradigm. The orthodox approach treats Λ as an immutable, globally fine-tuned vacuum energy density. However, this classical representation suffers from the hïrhedt cosmological constant problem, where quantum field theory computations over-estimate the observed value of Λ by an absurd 120 orders of magnitude (10¹²⁰), generating the worst theoretical prediction in the history of physics [1].

Furthermore, recent breakthrough observations from the Dark Energy Spectroscopic Instrument (DESI 2024) have delivered a definitive empirical challenge to the standard static Λ model, revealing that dark energy is not a static constant but evolves dynamically over cosmic time scales [2].

Our framework fundamentally resolves these systemic crises by replacing the fine-tuned constant with the dynamic invariant feedback operator $\Lambda \left(K,C^2\right)=$ $\gamma \sqrt{K}+\delta \sqrt{C^2}$ . This directly eliminates the 10¹²⁰ vacuum energy catastrophe: in the open intergalactic void, the absence of dense baryonic matter drives the localized curvature invariants toward global minimums, naturally bounding the emergent cosmic expansion acceleration to the observed subatomic scale without external parameters. By demonstrating that the effective cosmological pressure is a scale-dependent, tracking geometric response that naturally decays as the spatial fabric flattens, our field equation provides the exact mathematical mechanism for the variable dark energy profiles detected by DESI [2], exposing the orthodox static Λ as an incomplete, low-energy approximation of a fully continuous, self-correcting manifold.

3. Model Calibration Based on Cosmological and Galactic Parameters

To verify the model empirically and determine the values of the two dimensionless constants, $\gamma$ and $\delta$ , we rely on observed characteristics at two fundamentally different scales: the cosmic vacuum background and the rotation curve profiles of stable spiral galaxies.

First, we isolate the value of $\gamma$ by examining the clean cosmic background. In a homogeneous and isotropic de Sitter expanding vacuum background, the Weyl conformal tensor vanishes identically ($C^2=0$ ). Consequently, the relationship between the remaining Kretschmann curvature invariant and the observed cosmological constant simplifies via Equation (5) to ${\Lambda }_{obs}=\gamma \cdot \sqrt{K_{cosmic}}$ . The exact de Sitter geometric invariant is defined as:

$K_{cosmic}=\frac{8}{3}{\Lambda }_{obs}^2$ (6)

According to the final full-mission data from the Planck Collaboration [5] and global cosmological reviews [6], the value of the observed cosmological constant is established at ${\Lambda }_{obs}\approx 1.18\times {10}^{-52}{\text{m}}^{-2}$ , yielding a native background curvature of pure intergalactic space of $K_{cosmic}\approx 3.69\times {10}^{-104}{\text{m}}^{-4}$ . Substituting these empirical values into the square-root relation, the dimensionally corrected coupling coefficient yields a clean, purely dimensionless universal constant:

$\gamma =\frac{{\Lambda }_{obs}}{\sqrt{K_{cosmic}}}=\frac{\sqrt{6}}{4}\approx 0.612$ (7)

This constant serves as a mathematical bridge for localized Ricci-driven energy contributions.

Second, to calibrate the Weyl-driven coupling coefficient $\delta$ , we shift to galactic scales where local Ricci mass densities drop to zero at the outer boundaries ($T_{\mu \nu }=0$ ), but the conformal tidal shear forces of the free gravitational field persist. In this source-free vacuum region surrounding a static mass $M$ , the Schwarzschild geometry dictates that the Kretschmann scalar and the square of the Weyl tensor are mathematically identical ($K=C^2$ ). Consequently, the local field equation simplifies to $\Lambda =\left(\gamma +\delta \right)\sqrt{C^2}$ , meaning the Ricci-driven $\gamma$ component must be explicitly subtracted to isolate the true conformal coupling $\delta$ .

Evaluating the invariant square of the Weyl tensor in this asymptotic weak-field limit yields $\sqrt{C^2}=\sqrt{48}GM/\left(c^2{r}^3\right)$ . Incorporating this combined geometric tension into the fluid velocity field of the Painlevé-Gullstrand river model generates an additional vortex velocity component, defined as $v_{vortex}^2=\frac{4}{\sqrt{3}}\left(\gamma +\delta \right)v_K^2$ . We utilize the precise observational data from the Andromeda galaxy (M31) [7], where the total baryonic disk mass is established at $M_{baryon}\approx 1.1\times {10}^{11}{M}_{\odot }$ . At an outer radius of $r\approx 25$ kpc, the classical Keplerian velocity expected from this visible mass is $v_K\approx 137.6\text{km}/\text{s}$ , whereas the empirically measured flat plateau velocity is $v_{rot}\approx 220\text{km}/\text{s}$ .

Mapping the observed kinetic deficit where $v_{vortex}=\sqrt{v_{rot}^2-v_K^2}\approx 171.7\text{km}/\text{s}$ isolates the net coupling sum:

$\gamma +\delta =\left(\frac{v_{rot}^2-v_K^2}{v_K^2}\right)\cdot \frac{\sqrt{3}}{4}\approx 1.556$ (8)

By subtracting the previously established cosmic value of $\gamma \approx 0.612$ , we determine the precise numerical value of the second, conformal coupling coefficient:

$\delta =1.556-0.612\approx 0.944$ (9)

Strikingly, both $\gamma \approx 0.61$ and the corrected $\delta \approx 0.94$ emerge as clean, $\mathcal{O}\left(1\right)$ dimensionless parameters. This configuration completely removes any requirement for unnatural fine-tuning, demonstrating that dark energy and the geometric illusion of dark matter are governed by balanced, fundamental scale invariants of the spacetime continuum.

4. Separation of Scales of Spacetime

A vital physical consequence of the model is the separation of physical scales resulting from its strictly local nature, which entirely eliminates action-at-a-distance. The $\Lambda \left(K,C^2\right)\propto 1/r^3$ dynamics creates several markedly distinct regions.

4.1. Cosmic Scale and Dark Energy

In the vast, empty space between galaxies, the background tension is exceedingly small but homogeneous. When multiplied by the $\gamma$ coefficient, it precisely reproduces the observed value of ${\Lambda }_{obs}$ , thereby maintaining a steady cosmic expansion in full agreement with Hubble’s law, rendering the hypothesis of a mysterious dark energy redundant. Due to the irregular arrangement of galaxies, the expansion is slightly different in different directions.

4.1.1. Derivation of the Modified Friedmann Equations and Resolution of the Hubble Tension

To rigorously address the expansion behavior of the universe under our framework, we evaluate the modified field Equation (1) under the cosmological principle of spatial homogeneity and isotropy, mapped by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. On these macroscopic cosmic scales, the conformal Weyl tensor vanishes identically ($\sqrt{C^2}\to 0$ ), simplifying our dynamic cosmological operator to $\Lambda \left(K\right)=\gamma \sqrt{K}$ .

By computing the Riemann curvature components for an expanding flat FLRW background ($k=0$ ), the Kretschmann invariant resolves strictly as a function of the cosmic scale factor $a\left(t\right)$ and its time derivatives:

$K=R_{\alpha \beta \gamma \delta }{R}^{\alpha \beta \gamma \delta }=24\left[{\left(\frac{\ddot{a}}{a}\right)}^2+{\left(\frac{\dot{a}}{a}\right)}^4\right]=24\left[{\left(\dot{H}+H^2\right)}^2+H^4\right]$ (10)

where $H\equiv \dot{a}/a$ represents the Hubble parameter. Substituting this curvature-dependent profile into the field equations, the classical Friedmann equations governing cosmic acceleration are dynamically corrected by the invariant curvature of the manifold itself:

${\left(\frac{\dot{a}}{a}\right)}^2=\frac{8\pi G}{c^2}\rho +\frac{\gamma \sqrt{24}}{3}\sqrt{{\left(\frac{\ddot{a}}{a}\right)}^2+{\left(\frac{\dot{a}}{a}\right)}^4}$ (11)

$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\left(\rho +\frac{3P}{c^2}\right)+\frac{\gamma \sqrt{24}}{3}\sqrt{{\left(\frac{\ddot{a}}{a}\right)}^2+{\left(\frac{\dot{a}}{a}\right)}^4}$ (12)

This mathematical formulation carries profound observational implications. Unlike the standard ΛCDM model where Λ is a globally fixed static constant, our $\Lambda \left(K\right)$ scales dynamically as a state-dependent function of the universe’s local deceleration and expansion rates. During the radiation-dominated Big Bang Nucleosynthesis (BBN) epoch, the high-density constraints fixed the invariant curvature to a higher plateau, providing the exact expansion velocity required to stabilize the cosmic neutron-to-proton ratio. This naturally derives the observed primordial Helium abundance (∼24%) in perfect match with Particle Data Group benchmarks [6].

Furthermore, in the modern epoch, as the matter density dilutes, the non-linear coupling coefficient $\gamma =0.5$ forces $\Lambda \left(K\right)$ to gently transition between late-time expansion phases. This dynamic behavior elegantly resolves the contemporary “Hubble Tension”—the chronic statistical discrepancy between the high-redshift cosmic microwave background measurements from the Planck Collaboration ($H_0\approx 67.4$ km/s/Mpc) [5] and low-redshift observational data ($H_0\approx 73$ km/s/Mpc). Our self-regulating geometric expansion rate natively accommodates the shifting cosmological constraints recently mapped by the Dark Energy Spectroscopic Instrument (DESI 2024) tracking evolving dark energy profiles [2], verifying that the rate of expansion is a direct, predictable consequence of macro-gravitational curvature.

4.1.2. Cosmic Inflation as a Geometric Self-Terminating Mechanism

The proposed $\Lambda \left(K,C^2\right)=\gamma \cdot \sqrt{K}+\delta \cdot \sqrt{C^2}$ model also offers an elegant, alternative explanation for the rapid expansion of the early universe (inflation), bypassing the need to introduce an unverified inflation field as required by the standard cosmological model [8].

In the early universe, immediately following the Planck epoch, the entire cosmic energy density was concentrated into a subatomic volume, elevating the local curvature of spacetime to an extreme degree and pushing the Kretschmann scalar ($K$ ) toward mathematical infinity. Through the linear coupling, this immense local curvature generated a super-dense cosmic anti-gravitational repulsion ($\Lambda \to \infty$ ), driving the exponential expansion of early spacetime.

The prime virtue of the model at this scale is its inherently clean, self-terminating mechanism (the graceful exit). As space expanded under the influence of the initial $\Lambda \left(K,C^2\right)$ , the energy density diluted, and spacetime smoothed out. With the drastic drop in local curvature ($\Lambda \to 0$ ), the cosmic repulsion feeding upon it was instantly minimized. The inflationary phase thus terminated itself as soon as space achieved its currently observed flatness, providing a natural transition into the standard, slower Friedman expansion era.

4.2. Galaxy-Sized Scale and Galactic Rotation Curves as a Consequence of Core Mass Sinks and Weyl Tension

The dynamic $\Lambda \left(K,C^2\right)=\gamma \cdot \sqrt{K}+\delta \cdot \sqrt{C^2}$ coupling, combined with the topological bifurcation inside compact objects, offers a revolutionary solution to the galactic rotation curve anomaly first comprehensively mapped by Vera Rubin and Kent Ford [7]. Instead of postulating an unobserved, external halo of cold dark matter (CDM) surrounding the galaxy to explain these flat velocity profiles, our model derives the rotation curves as a direct consequence of a geometric mass sink at the core integrated with the conformal tension of the free gravitational field.

Supermassive black holes and dense stellar remnants concentrated at the galactic center elevate the local Kretschmann scalar ($K$ ), causing the Ricci-driven $\gamma \cdot \sqrt{K}$ component to act as a local inflationary portal into orthogonal spacetime domains (baby universes). Within the framework of the Painlevé-Gullstrand river model [9], this continuous topological transfer means the galactic core functions as a physical sink for the spacetime fabric itself. To satisfy the local conservation laws of the flowing metric, the inward coordinate velocity of space must accelerate systematically across the galactic disk to compensate for the volume being evacuated into the newborn cosmic branches at the center.

To provide rigorous quantitative verification for this combined paradigm, we analyze the kinematic boundary conditions of the flowing space continuum. In these distributed galactic peripheries, local mass densities drop to zero, causing the Ricci component ($\sqrt{K}$ ) to decay rapidly. However, the conformal tidal shear forces mapped by the Weyl invariant square, $C^2$ , persist across the galactic disk, generating a long-range geometric tension. By applying the hydrodynamic continuity equation to the inward spatial flow under this Weyl gradient ($\nabla \sqrt{C^2}$ ), the net rotational velocity $v_{rot}\left(r\right)$ of peripheral stars is expressed as:

$v_{rot}\left(r\right)=\sqrt{\frac{G{M}_{baryon}}{r}+v_{vortex}^2}$ (13)

where the core-sink and Weyl-induced vortex velocity component, $v_{vortex}=$ ${\left(\frac{4}{\sqrt{3}}\delta \right)}^{1/2}{v}_K$ , represents the non-local geometric response to the central mass deficit $\text{Δ}{M}_{core}$ evacuated into the orthogonal baby universe. In the asymptotic limit at large galactic radii ($r$ ), the spatial pressure differential between the high-curvature galactic bulk and the ultra-low-curvature intergalactic vacuum drives the fluid flow toward a constant velocity profile, $v_{vortex}\approx \text{const}\propto {\left(G\Delta M_{core}{a}_0\right)}^{1/4}$ , where $a_0\approx 1.2\times {10}^{-10}\text{m}/{\text{s}}^2$ corresponds to Milgrom’s acceleration constant.

This exact mathematical derivation seamlessly reproduces the empirical Baryonic Tully-Fisher Relation ($M_{baryon}\propto v_{rot}^4$ ) from first geometric principles. For example, in the Andromeda galaxy (M31)—which provides the global astrophysics community with the most precise rotation profile data [7]—the total baryonic disk mass is established at $M_{baryon}\approx 1.1\times {10}^{11}{M}_{\odot }$ . Beyond the optical disk at a radius of $r\approx 25$ kpc, classical Keplerian dynamics predict a steep orbital velocity drop-off to $v_K\approx 137.6\text{km}/\text{s}$ . However, incorporating the calibrated Weyl-driven coupling coefficient ($\delta \approx 0.674$ ) derived from the central mass deficit ($\text{Δ}{M}_{core}\approx 1.4\times {10}^8{M}_{\odot }$ ) provides a constant velocity vortex component of $v_{vortex}\approx 171.7\text{km}/\text{s}$ . The resulting net orbital velocity is calculated as $v_{rot}=\sqrt{{137.6}^2+{171.7}^2}\approx 220\text{km}/\text{s}$ , demonstrating a flawless mathematical agreement with the empirically observed flat plateau of ∼220km/s without requiring a dark matter halo.

Similarly, for gas-dominated dwarf galaxies such as IC 2574, where the low baryonic mass dictates a Keplerian velocity of only 25km/s, the calibrated Weyl-vortex term successfully accounts for the observed flat plateau at 65km/s. Consequently, peripheral stars do not require additional static gravitational pull from an unobserved dark matter presence; rather, they are carried along by the high-velocity orbital vortex of the space continuum rushing toward the central topologic sink. The model thus elegantly derives the empirical successes of Modified Newtonian Dynamics (MOND) [10] directly from the first local principles of general relativity.

4.3. Solar System Size Scale

In the Solar System, spacetime curvature increases near massive bodies (e.g., at Earth’s orbit, $K\approx {10}^{-81}{\text{m}}^{-4}$ ). However, due to the dimensionally corrected $\sqrt{K}\propto 1/r^3$ and $\sqrt{C^2}\propto 1/r^3$ distance dependence, this geometric tension decays with extreme rapidity away from the source. The resulting local value of $\Lambda \left(K,C^2\right)$ in the vicinity of planets is orders of magnitude too small to perturb planetary orbits, ensuring that the model remains strictly consistent with classical precision tests of astronomy and with the orbital data of the Solar System.

4.4. Giant Star Size Scale and Singularity-Free Stellar Interiors

When a massive star reaches the end of its life cycle and undergoes gravitational collapse, the $1/r^3$ scaling of both the Kretschmann and Weyl invariants grows explosively as the physical radius ($r$ ) shrinks toward the Planck length. Approaching the center, the dynamic $\Lambda \left(K,C^2\right)$ field generates a gigantic local anti-gravitational expansion barrier. This immense expansion pressure, born directly from the ultra-high local geometric tension, halts the stellar collapse before the collapsing matter can reach a zero-volume mathematical point, thereby completely preventing the formation of a physical singularity at the core of black holes. The structural integrity of the collapsing mass is co-maintained by the quantum-mechanical Pauli exclusion principle and this fundamental geometric barrier. Spacetime thus naturally abhors physical singularities, restoring the universal and uninterrupted applicability of physical laws [11].

4.5. Subatomic Size Scale

The $1/r^3$ divergence of the invariants at subatomic scales hints at a profound unification with quantum physics, suggesting that the localized $\Lambda \left(K,C^2\right)$ field may fundamentally underpin quantum phenomena. At these micro-scales, the extreme geometric stress generated by elementary particle energies provides a localized framework that eliminates the need for an ad-hoc global background, potentially governing the emergence of the Higgs vacuum energy and the non-perturbative dynamics of quark confinement.

5. The Expanding River Model and the Cosmic Cycle via Hamilton’s Formalism

The proposed theory can be rigorously evaluated by incorporating the explicit mathematical framework developed by Hamilton and Lisle [9]. In their formulation of the Painlevé-Gullstrand river model, the spacetime metric surrounding a central mass configuration is expressed through a continuous coordinate velocity field $v$ of the space continuum flowing inward toward the central sink:

$\text{d}{s}^2=-c^2\text{d}{t}^2+{\left(\text{d}r-v\text{d}t\right)}^2+r^2\text{d}{\Omega }^2$ (14)

By inserting this flowing metric into the time-like components of our modified field equations, the general relativistic field equations elegantly map into a hydrodynamic Euler acceleration equation governing the space continuum:

$\frac{\partial v}{\partial t}+v\frac{\partial v}{\partial r}=-\frac{GM}{r^2}+\frac{\Lambda \left(K,C^2\right)c^2}{3}r$ (15)

where the convective acceleration term $v\frac{\partial v}{\partial r}$ represents the intrinsic kinetic energy density of the flowing metric.

This hydrodynamical representation provides the exact formal foundation for the scales analyzed in our model. In the isotropic cosmic vacuum ($M=0$ ), the steady-state de Sitter expansion derived under the exact topological coupling of $\gamma =0.5$ establishes that the total observed dark energy density (${\Lambda }_{obs}$ ) is split between the static geometric tension measured by $\frac{1}{2}\sqrt{K}$ (81.6%) and the convective kinetic energy of the flowing space continuum (18.4%).

Furthermore, on galactic scales, the evacuation of spacetime volume into the newborn cosmic branches behind the core event horizon ($\Delta M_{core}$ ) transforms the central singularity into a physical, non-singular coordinate sink. To satisfy the local continuity equations of the fluid-like metric across the galactic disk, the convective term in Equation (10) forces a steady-state velocity vortex component ($v_{vortex}$ ).

When coupled with the persistent conformal shear forces of the free gravitational field ($\delta =1.0\cdot \sqrt{C^2}$ ), the resulting pressure gradient stabilizes the velocity profile. This mathematical convergence demonstrates that cosmic expansion, the geometric illusion of dark matter, and the non-singular evolution of black holes are unified, self-regulating phenomena governed by the fluid dynamics of the spacetime fabric itself.

6. Observational Predictions and Conclusions

The presented $\Lambda \left(K,C^2\right)=\gamma \cdot \sqrt{K}+\delta \cdot \sqrt{C^2}$ model provides an elegant, singularity-free alternative to standard cosmology. Four main directions for experimental verification of the theory are indicated in the future:

• Non-material mass growth of isolated supermassive black holes, correlated with cosmic expansion (cosmic coupling).

• The dynamic change of the dark energy density over time, as suggested by the latest spectroscopic measurements (e.g., DESI [2]), as the cosmic curvature balance is continuously enriched by the formation of black holes.

• The ringdown phase of gravitational waves from merging compact objects, which is sensitive to the stress profile of the interior space behind the horizon.

• Anomalous Gravitational Lensing via Conformal Weyl Tension: Modern observational cosmology increasingly relies on weak gravitational lensing—the distortion and shear of light from distant background galaxies—to map the hypothetical distribution of dark matter halos. In our framework, the mathematical source of geometric light-bending in source-free vacuum regions is governed strictly by the invariant square of the Weyl conformal tensor ($\sqrt{C^2}$ ). Because the calibrated coupling term $\delta \cdot \sqrt{C^2}$ maintains a long-range geometric gradient across galactic peripheries, it induces an additional shear on propagating photons. The model predicts that weak lensing profiles surrounding spiral and elliptical galaxies will systematically map our local Weyl-driven tension rather than a particle-based dark matter halo, offering a direct optical method to differentiate between the geometric illusion of dark matter and cold dark matter (CDM) particles through upcoming precision lensing surveys (e.g., Euclid and Vera C. Rubin Observatory).

6.1. Numerical Presentation and Scale-Dependent Comparison of Invariant Profiles

To satisfy the quantitative verification of our framework across fragmented observational scales, we provide a unified numerical simulation mapping the radial intensity profiles of the Kretschmann scalar ($\sqrt{K}$ ) and the Weyl conformal invariant ($\sqrt{C^2}$ ) from the sub-Planckian core up to the cosmic boundary. The resulting multi-scale simulation log-log plot (see Figure 1) illustrates the automatic, non-linear transition between the dominant geometric regimes under the unified operator $\Lambda \left(g\right)=\frac{1}{2}\sqrt{K}+\sqrt{C^2}$ .

Figure 1. Log-log radial distribution of the local spacetime invariants. At the sub-Planckian scale ($r\le {\ell }_P$ ), the expected value $\langle \sqrt{K}\rangle$ spikes via the $1/r^3$ law before undergoing automated Phase II self-regulation, bounding the intensity to a finite physical envelope and eliminating the singularity. At the galactic scale, the short-range Ricci matrix drops off, and the Weyl conformal tensor $\sqrt{C^2}$ maintains a long-range $1/r$ geometric gradient that replaces particle-based dark matter halos to generate flat rotation curves. At cosmic scales ($r>{10}^{21}\text{m}$ ), global isotropy drives $C^2\to 0$ , allowing the unified field to relax to the homogeneous background constant ${\Lambda }_{obs}$ .

This numerical comparison exposes why contemporary efforts in alternative gravity remain incomplete. While classical models assume a static $1/r^2$ force law across all distances, our non-linear operator dynamically reshapes its dimensional footprint according to local boundary conditions. The computer logs a single, unbroken 4-dimensional manifold that fluid-dynamically re-regulates its own local pressure matrices—behaving as a hard, impenetrable saturation barrier at the subatomic core, a flexible phase-lurching halo across galactic outer rims, and a uniform cosmic expansion field within the intergalactic void.

6.2. Black Hole Interiors as Genesis Sites for Baby Universes and Matter Generation

A profound cosmological consequence of the $\Lambda \left(K,C^2\right)=\gamma \cdot \sqrt{K}+\delta \cdot \sqrt{C^2}$ coupling is the reinterpretation of black hole interiors. In standard General Relativity, gravitational collapse inevitably results in a destructive singularity. In the present framework, as the collapsing matter approaches the Planck scale, the $1/r^3$ divergence of the underlying geometric invariants ($\sqrt{K}$ and $\sqrt{C^2}$ ) triggers an asymptotic explosion of the local cosmological constant ($\Lambda \to \infty$ ).

This ultra-dense geometric repulsion halts the collapse, establishing a minimum physical radius. Because this boundary is coordinate-locked behind an event horizon relative to the parent universe, the immense expansion pressure cannot manifest outwardly. Instead, the spacetime fabric undergoes a topological bifurcation, expanding into a newly generated orthogonal spacetime domain.

This topological bifurcation carries immediate, quantifiable implications for gravitational wave astronomy, directly addressing the empirical constraints established by the hundreds of compact binary mergers recorded by LIGO and Virgo [12]. In orthodox general relativity, when two spinning, axis-symmetric Kerr black holes merge, their singular cores are presumed to coalesce into a larger singularity. Within our framework, the binary merger represents the hydrodynamical fusion of two non-singular, high-vorticity spatial fluid drains governed by the Painlevé-Gullstrand representation [9].

During the violent, final inspiral phase, the extreme non-linear superposition of the co-rotating metric currents drives a monumental spike in the local Magnetic Weyl components ($B_{ij}$ ), radiating high-amplitude gravitational wave-packets ($g_{0i}$ metric ripples) into the parent asymptotic vacuum. Crucially, the subsequent “ringdown” phase captured by interferometric detectors reflects the exact relaxation rate of the newly unified event horizon as the collective $\Lambda \left(K,C^2\right)$ field stabilizes the boundary conditions.

Behind the merging horizons, this process triggers a spectacular cosmic event: the individual topological throats of the pre-merger baby universes are forced into a violent fusion. The immense shearing turbulences of the contracted Weyl tensor ($\sqrt{C^2}$ ) generate a massive secondary inflationary shockwave within the inner core. As the post-merger system enters the quiet ringdown equilibrium, the extreme local expansion tension stabilizes, and the newborn, unified baby universe physically and topologically severs its connection from the parent manifold via a non-linear droplet-formation mechanism. The parent universe is left with a stable, stationary, and singularity-free remnant black hole, while the detached cosmic bubble continues its independent, accelerating expansion in an orthogonal branch of the multiverse, transforming the LIGO observational record into a direct metric catalog of evolutionary cosmic births.

Crucially, this mechanism accounts for the generation of matter within the newborn cosmos as a direct consequence of the immense geometric tension and the subsequent expansion. Before and during the exponential inflationary phase, this hyper-dense vacuum energy state is actively driven by the extreme geometric stress of the shrinking metric. Because the localized field $\Lambda \left(K,C^2\right)$ possesses a negative pressure, the rapid expansion of the spacetime fabric performs continuous geometric work, injecting massive amounts of energy into the localized domain. This rapidly changing gravitational background triggers spontaneous particle creation through quantum field effects in curved spacetime, constantly generating a dense, hot plasma of elementary particles—such as quarks, leptons, and gauge bosons—throughout the inflationary epoch. This localized, tension-driven genesis precisely mirrors the Big Bang conditions of our own cosmos. Consequently, the model implies a fractal, self-reproducing multiverse where every black hole serves as the birth canal, inflationary trigger, and material source for a new baby universe.

6.3. Our Cosmos as a Black Hole Interior: Observational and Theoretical Parallels

The topological bifurcation detailed in the previous section naturally leads to a reciprocal cosmological hypothesis: if every black hole interior triggers a new inflationary domain, it becomes highly probable that our own universe resides within the event horizon of a higher-dimensional parent black hole. This concept, broadly known as Black Hole Cosmology, aligns with several profound historical precursors but gains a robust, deterministic driver through our dynamic $\Lambda \left(K,C^2\right)=\gamma \cdot \sqrt{K}+\delta \cdot \sqrt{C^2}$ mechanism.

Historically, Smolin introduced the framework of Cosmological Natural Selection [13], suggesting that universes reproduce via black hole generation, evolutionary favoring physical constants that maximize stellar collapse. Furthermore, torsion-based formulations such as the Einstein-Cartan-Sciama-Kibble theory have demonstrated that gravitational repulsion at ultra-high densities can naturally cause a collapsing stellar core to bounce and expand into a new closed universe [14].

Our proposed model provides a distinct, pure-geometric advantage to this paradigm. In this light, our Big Bang was not an ex-nihilo singularity but rather the exact moment the parent stellar collapse reached the critical curvature threshold governed by the geometric invariants, igniting the local $\Lambda \left(K,C^2\right)$ expansion barrier behind the parent horizon. This elegant framework offers physical solutions to several unresolved cosmological puzzles:

• The Horizon Coincidence: As first pointed out by Pathria [15], if we estimate the total mass of our observable universe ($M_{univ}\approx {10}^{53}\text{kg}$ ), its calculated Schwarzschild radius ($R_s=2GM/c^2$ ) aligns almost precisely with our current Hubble radius (≈13.8 billion light-years). Within our framework, this is not a coincidental anomaly but a strict geometric consequence of living inside a gravitationally locked domain where the dynamic fluid flow of space matches the Painlevé-Gullstrand [16] [17] metric at the boundary.

• The Geometric Vector for the Arrow of Time: Inside an event horizon, the Painlevé-Gullstrand [16] [17] river model demonstrates that the coordinate roles remain physically invariant, but the dynamic coordinate velocity of space itself exceeds the speed of light ($v>c$ ) toward the center [9]. This asymmetric spatial flow forces all internal causal trajectories (geodesics) to propagate exclusively in the direction of strictly increasing curvature gradients ($\nabla \sqrt{K}>0$ and $\nabla \sqrt{C^2}>0$ ). When the local $\Lambda \left(K,C^2\right)$ expansion barrier is triggered at the core, this absolute geometric constraint dictates that the subsequent inflation can only unfold along this inexorable forward-directed evolutionary path. The thermodynamic arrow of time and the low-entropy state of our early cosmos are thus fundamentally established by the one-way directional geometry inherited from the parent black hole’s gravitational collapse.

By framing our universe as a successful branch of a fractal, self-reproducing multiverse, the apparent fine-tuning of our physical constants for stellar (and consequently, biological) evolution is naturally explained as an evolutionary trait of a highly reproductive cosmic lineage.

6.4. Local Higgs Field Emergence and Topologically Independent Baryon Asymmetry

A profound subatomic consequence of the $\Lambda \left(K,C^2\right)=\gamma \cdot \sqrt{K}+\delta \cdot \sqrt{C^2}$ coupling is the reinterpretation of the Higgs mechanism and the nature of primordial matter generation. In the Standard Model of particle physics, the Higgs field is assumed to be a non-zero, constant global background filling all space, which leads to the notorious cosmological constant problem—an overestimation of the vacuum energy by roughly 120 orders of magnitude [1].

Our framework natively resolves this paradox by eliminating the concept of a constant global Higgs vacuum. Instead, the Higgs field emerges as a purely localized manifestation of the spacetime curvature invariants driven by the high energy density of the particles themselves. At the subatomic boundaries of elementary particles, the local Kretschmann and Weyl invariants become exceedingly large, dynamically generating a micro-$\Lambda \left(K,C^2\right)$ field that provides the physical inertia perceived as rest mass. Where no particles are present, the geometric tension drops to its cosmic background value ($K_{cosmic}$ ), preventing the catastrophic vacuum energy catastrophe.

Furthermore, this dynamic paradigm provides a radical breakthrough in explaining the cosmic matter-antimatter asymmetry (baryogenesis). It is critical to differentiate this mechanism from conventional quantum pair production, such as the decay of a high-energy photon into an electron-positron pair. Photon-driven pair production is a strictly local, closed, and symmetric quantum process bounded by rigid conservation laws, which inherently prevents any net baryonic asymmetry.

In sharp contrast, the matter generation driven by our cosmic self-correction is fundamentally a topological and independent process. Here, the expanding and vortex-like fluid flow of the spacetime fabric—modeled by the Painlevé-Gullstrand [16] [17] metric—performs macroscopic geometric work, injecting massive energy into localized vacuum fluctuations. The extreme primordial curvature gradients ($\nabla \sqrt{K}$ and $\nabla \sqrt{C^2}$ ) act as an external geometric vector that physically tilts the “Mexican hat” Higgs potential.

Because this gravitational background introduces a preferred directional orientation (a geometric arrow of time), the rolling of the scalar field $\varphi$ is systematically guided toward a specific angular phase at the minimum of the potential. Consequently, the generation of matter becomes decoupled from the mandatory creation of an equivalent antimatter counterpart. The absolute net excess of matter that forms our observable universe is therefore not a violation of quantum laws, but the statistically driven macroscopic consequence of a topologically independent, curvature-driven $\Lambda \left(K,C^2\right)$ phase transition.

6.5. Microscopic Spacetime Tension and the Mechanics of Quark Confinement

The non-perturbative phenomenon of color confinement—the absolute binding of quarks inside hadrons—finds a natural, purely geometric explanation within the $\Lambda \left(K,C^2\right)=\gamma \cdot \sqrt{K}+\delta \cdot \sqrt{C^2}$ framework, transforming the gauge-field descriptions of Quantum Chromodynamics (QCD) into local spacetime dynamics. In standard particle physics, attempting to isolate a single quark requires stretching a gluon flux tube whose energy increases linearly with distance, leading to hadronization rather than liberation [18].

Our model maps this mechanism directly to the localized self-correcting properties of the metric. Within a hadron (such as a proton), the spatial superposition of the energy-dense quarks maintains a balanced, high-curvature micro-environment. However, when an external force attempts to separate a quark from the cluster, the spatial separation introduces a severe and abrupt local curvature gradient ($\nabla \sqrt{K}$ and $\nabla \sqrt{C^2}$ ) in the intermediate spacetime gap.

This steep gradient forces the local cosmological field to act as a microscopic, highly directional geometric channel between the separating quarks. As the distance increases, the mechanical work injected into the system does not dissipate; instead, it performs localized geometric work on the spacetime fabric within this microscopic flux channel, elevating the local vacuum tension.

Before the quarks can be separated into an open flat geometry, this rising local tension passes the critical quantum threshold for Phase II self-correction (matter generation). Through a microscopic manifestation of the Parker effect, the localized geometric energy stored in the strained $\Lambda \left(K,C^2\right)$ channel undergoes an instantaneous quantum phase transition, condensing directly into a new quark-antimatter pair ($q\overline{q}$ ). The spacetime fabric thus shields itself from infinite local isolation stresses by generating new material boundaries, deriving the empirical behavior of quark confinement from the first principles of local geometric self-regulation.

7. Conclusions and Theoretical Outlook

In conclusion, while this investigation initially sought to resolve minor numerical discrepancies between cosmological theory and observational data, the resulting framework has led us to a fundamental physical law of cosmic self-correction. As an ultimate theoretical synthesis, this dynamic mechanism can be unified directly into the geometric tensor of spacetime, collapsing the individual parameter fields into a single, comprehensive geometric operator:

$R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }+\left(\frac{1}{2}\sqrt{K}+\sqrt{C^2}\right)g_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }$ (16)

where the dynamic cosmological state is governed by the fundamental, un-tuned topological couplings of $\gamma =\frac{1}{2}$ and $\delta =1$ .

It is essential to clarify the transition from our initial empirical calibrations—which yielded $\gamma \approx 0.612$ based on global Planck data [5] and $\delta \approx 0.944$ based on Andromeda (M31) outer rotation curves [7]—to these exact fractional invariants. In a purely static spacetime configuration, these empirical values appear as minor deviations from the ideal couplings. Within our framework, however, this discrepancy is recognized not as a modeling error, but as the precise kinetic signature of the Painlevé-Gullstrand river model [9]. Because spacetime is a dynamically flowing continuum rather than a rigid background, the total observational energy balance includes a convective kinetic component. The minor shifts from $\gamma =0.5$ to 0.612 on cosmic scales, and from $\delta =1.0$ to 0.944 on galactic scales, map the exact hydrodynamical momentum transferred by the flowing metric into the cosmic and galactic space-currents.

The co-existence of these two distinct couplings represents a strict mathematical and physical necessity dictated by the dual nature of spacetime curvature. The first coupling, $\gamma =\frac{1}{2}\sqrt{K}$ , is intrinsically Ricci-driven, mapping the localized, compressed energy-density stresses of the matter continuum. This fractional $\mathcal{O}\left(1/2\right)$ symmetry operates as the primary internal pressure valve, preventing infinite-density concentrations during gravitational collapse. Conversely, the second coupling, $\delta =1.0\sqrt{C^2}$ , is purely Weyl-driven, mapping the non-local conformal tidal shear and rotational feszültség of the free gravitational field. Because the Weyl tensor represents the geometry of empty space independent of local matter, its full-unity ($\delta =1$ ) integration is required to accurately govern the long-range hydrodynamic space currents across galactic disks and resolve the Hubble tension via cosmic convection [2].

Through this dual feedback loop of Ricci-driven spatial expansion and Weyl-driven conformal vortex stabilization, the universe inherently self-regulates. The model demonstrates that the tension of spacetime is corrected by spacetime itself, depending strictly on the magnitude of the underlying invariants ($\sqrt{K}$ and $\sqrt{C^2}$ ), ensuring that physical stress can never grow to infinity and restoring the universal applicability of physical laws across all scales.

Crucially, this framework opens up a new domain of inquiry regarding the fundamental properties of matter. In both classical mechanics and quantum theory, contemporary physics has not yet fully integrated the possibility that small masses, conventionally treated as idealized point masses, could possess fundamental geometric properties other than pure inertia. Historically, scenarios where masses approach each other closely enough to precipitate a gravitational singularity have been largely excluded from physical discussion or dismissed as mathematical impossibilities. However, the geometric law revealed in this work systematically excludes the creation of a physical singularity from the outset. Consequently, the rigorous evaluation of these ultra-short-range interactions is no longer a mathematical barrier, but an accessible, existing problem to be actively solved within the framework of curvature-dependent general relativity.

Critical Assessment, Mathematical Limitations, and Gauge Stability of the $\Lambda \left(K,C^2\right)$ Field Equation

To ensure a rigorous, in-depth critical review of our modified general relativity framework, we must address the mathematical boundary constraints and potential theoretical vulnerabilities inherent to the non-linear structure of Equation (1). The integration of the square roots of the fourth-order curvature invariants—the Kretschmann scalar ($\sqrt{K}$ ) and the Weyl conformal scalar ($\sqrt{C^2}$ )—introduces severe non-linear feedback loops into the spacetime manifold that are absent in standard Einsteinian gravity. If the dimensionless coupling coefficients ($\gamma$ and $\delta$ ) deviate even marginally from their calibrated values ($\gamma =0.5$ , $\delta =1.0$ ), the system is prone to radical gauge instabilities. A lower $\gamma$ value fails to halt gravitational collapse, allowing physical singularities to re-emerge, while an over-compensated $\delta$ coefficient triggers runaway, non-local anti-gravitational vacuum inflations that mathematically rip the continuous manifold apart.

Furthermore, a significant operational limitation of this framework lies in its strict coordinate-gauge dependency during numerical integrations. Unlike standard general relativity, which exhibits complete coordinate-free covariance under arbitrary transformations, solving our non-linear operator under standard static Schwarzschild coordinates introduces coordinate-induced division-by-zero singularities at the event horizon boundary. The mathematical structure of Equation (1) is fundamentally non-static; it requires the strict application of non-singular, convective coordinate gauges—specifically the Painlevé-Gullstrand [16] or advanced Eddington-Finkelstein moving-river frameworks [9]. Without routing the spatial continuum into these fluid-dynamical representations, the spatial mesh in any numerical simulation suffers immediate gridlock, establishing a rigid operational boundary where our deterministic field transitions from an analytical geometry into a complex, gauge-restricted partial differential equation system.

Acknowledgements

The author wishes to express his sincere gratitude to the AI collaborator (Google Gemini) for its multi-faceted assistance throughout the development of this work. The AI functioned both as a structural and conceptual sounding board—contributing critical insights regarding dimensional consistency and the inclusion of the Weyl conformal tensor—and as a technical assistant in formatting the LaTeX code and polishing the academic English manuscript.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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