Sphere Theory to replace String Theory!

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Sphere Theory: A New Path to Quantum Gravity, No Extra Dimensions or Supersymmetry Needed

For decades, we've been told that solving quantum gravity requires radical new ideas like extra dimensions (String Theory) or quantizing spacetime itself (LQG). But what if the solution is simpler and has been hiding in plain sight within General Relativity all along?

Sphere Theory is a framework built on one core principle: The source of gravity isn't just an object's free-state mass, but its total energy, which must include its own negative gravitational self-energy.

Why 'Sphere Theory'?

As an object becomes more compact, its negative self-energy grows. My paper shows there is a critical radius, R_gs, where this negative energy perfectly cancels the object's positive mass-energy (Mc^2), bringing its total energy to zero. This critical radius can be derived from Newtonian physics as:

|Gravitational Self-Energy| = Mass-Energy


Solving for the radius gives us this critical radius

R_gs = (3/5)GM/c^2

(*For the detailed calculation based on general relativity, please refer to the paper.)

If the object were to shrink further (R<R_gs), it would enter a negative energy state. This generates a repulsive gravitational force ('anti-gravity'), which prevents any further collapse.

Therefore, R_gs acts as an minimal radius. Nothing can be stably smaller. (This is temporarily possible, however.) This replaces the abstract 'point' particle with a fundamental, volumetric 'sphere'.

How is this different from String Theory?

• Derived vs. Postulated: String Theory postulates a fixed minimal length. Sphere Theory derives a dynamic minimal radius (R_gs) that is proportional to the object's mass.
• Simplicity: It requires no extra dimensions, no supersymmetry, and no new particles. It aims to solve the problem using the physics we already have.
• Universality: This highlights another fundamental difference in scope. String Theory's central feature is its minimal length, fixed at the Planck scale. While this offers a potential resolution for divergences at that specific scale, the challenges of gravity are not confined to the microscopic. They extend to the largest cosmological scales, where String Theory offers less clear solutions. This suggests that a theory with a fixed minimal scale may not be the fundamental framework capable of describing both domains. This is where Sphere Theory offers a profoundly different and more powerful approach. Its critical radius R_gs, is not a fixed constant but a dynamic variable proportional to mass (R_gs ∝ M). This inherent scalability means the theory's core principle applies seamlessly from the smallest quantum fluctuations at the Planck scale to the entire observable universe. It therefore has the potential to be a true candidate for the ultimate solution to gravity, unifying the physics of the very small and the very large under a single, coherent principle.

Crucially, Sphere Theory is Testable

This framework makes concrete, falsifiable predictions that distinguish it from standard theories:

1. A Falsifiable Prediction at the Planck Scale: It predicts a novel "quantum-dominant regime." Standard Effective Field Theory (EFT) predicts that as you approach the Planck scale, classical GR corrections will always overwhelmingly dominate quantum corrections. My paper shows the ratio of these corrections is approximately V_GR / V_Q ≈ 4.66 * (M/M_P) * (r/ l_P). For a stellar-mass black hole, this ratio is a staggering ~10^39, making quantum effects utterly negligible.

Sphere Theory reverses this. As an object approaches its critical radius R_gs, its equivalent mass (M_eq) is suppressed, which quenches the classical correction. The quantum term, however, is not suppressed in the same way. This creates a window where quantum effects become the leading correction, a unique and falsifiable signature that distinguishes this theory from standard EFT at its point of failure.

2. At the other Scale: Offers a unified explanation for the major puzzles of modern cosmology by providing (a) a mechanism for cosmic inflation, (b) a model for the accelerated expansion of the universe, and (c) a predicted upward revision of the neutron star mass limit}, all of which serve as falsifiable tests (Chapter 7).

The reason that this can be verified even in macroscopic events is because, unlike string theory, the minimum critical radius is proportional to the mass.

I believe this approach offers a more direct, physically grounded path to a complete theory of gravity.

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It is a foundational principle, recognized in both Newtonian mechanics and general relativity, that the true gravitational source is the equivalent mass (M_eq), which includes gravitational self-energy (binding energy), rather than the free state mass (M_fr). This principle leads to a running gravitational coupling, G(k), that vanishes at a critical scale, R_gs ~ G_NM_fr/c^2. This behavior provides a powerful and self-contained mechanism for gravity’s self-renormalization, driving the coupling to a trivial (Gaussian) fixed point (G(k) -> 0) and rendering the infinite tower of EFT counter-terms unnecessary.

The scope of Sphere Theory extends far beyond the divergence problem, providing a unified foundation for several long-standing puzzles. We will demonstrate that this single principle:

1)Resolves the singularity problem via a repulsive force that emerges at a macroscopic, not quantum, scale (Chapter 2-3).

2)Solving the 2-loop or greater divergence problem: Solve the 2-loop or greater divergence problem proposed by Goroff and Sagnoti (Section 4.6.3).


3)Solves divergence problems in standard Effective Field Theory (EFT): It solves the divergence problem of the standard effective field theory (EFT) proposed by John F. Donoghue et al.(Chapter 5~6.)


4)Provides a complete, self-renormalizing framework for gravity that is consistent with the low-energy predictions of EFT, while offering a physical completion at high energies (Chapter 5-6). This includes a novel prediction of a "quantum-dominant regime" that distinguishes it from standard EFT. This provides, in principle, a unique experimental signature that could distinguish this self-renormalization model from standard EFT, should technology ever allow for probing physics at this scale.

5)Establishes the physical origin of the Planck-scale cutoff in quantum field theory (Chapter 4.7).

6) Offers a unified explanation for the major puzzles of modern cosmology by providing (a) a mechanism for cosmic inflation}, (b) a model for the accelerated expansion of the universe}, and (c) a predicted upward revision of the neutron star mass limit}, all of which serve as falsifiable tests (Chapter 7).

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To contextualize the contributions and philosophy of Sphere Theory, we present a direct comparison with the two other leading frameworks: Quantum Field Theory (as exemplified by the EFT of gravity) and String Theory.


7.1 Philosophical cornerstones and testable predictions

The comparative analysis presented in Table 1 highlights the unique philosophical and physical foundations of Sphere Theory. While EFT offers unparalleled experimental success in its domain and String Theory provides mathematical elegance, Sphere Theory distinguishes itself through its commitment to physical realism and logical economy. Two distinctions are particularly crucial.

7.1.1 Minimal Length: Derived, not postulated

First is the concept of minimal length. String Theory postulates a minimal length scale (l_s) as a fundamental, fixed constant of nature. In contrast, Sphere Theory derives its minimal radius R_gs from the established principles of general relativity. This minimal radius is not a universal constant but a dynamic variable, proportional to the mass-energy of the object itself:

R_gs ~ GM_fr/c^2

This provides a more fundamental and less ad-hoc explanation for why nature appears to have a physical cutoff at the Planck scale.

This inherent scalability, where the core principle operates identically at both the Planck and cosmological scales, elevates it from a mere model to a candidate for a truly fundamental principle of gravity.

7.1.2 Experimental Falsifiability: A two-scale test

Second is the criterion of experimental falsifiability, a feature that distinguishes Sphere Theory from many alternatives. This testability arises directly from the dynamic, scale-dependent nature of the theory’s central relation, which provides concrete, distinguishing predictions at two vastly different physical scales.

[ The microscopic test: The physical origin of the Planck Scale ]

At the microscopic level, this relation provides a physical origin for the Planck-scale cutoff (Refer to Chapter 4.7.). For a quantum fluctuation with the Planck mass (M_fr ~ M_P), the equation naturally yields a critical radius on the order of the Planck length:

R_gs ~ GM_fr/c^2 = GM_P/c^2 = l_P

This demonstrates how the Planck scale emerges as a natural limit, not a postulate. It also predicts the existence of a "quantum-dominant regime" near this scale, a concrete prediction that, while technologically monumental to test, grounds the theory in the scientific method. For calculations, please refer to Chapters 5 and 6.

In addition to providing a physical origin for the Planck-scale cutoff, Sphere Theory makes a novel, falsifiable prediction that distinguishes it from standard Effective Field Theory (EFT) at high energies: the existence of a "quantum-dominant regime." This phenomenon arises from the core mechanism of the theory—the renormalization of the gravitational source mass (M_fr -->M_eq).

The unified gravitational potential proposed by Sphere Theory includes both the classical General Relativistic (GR) correction and the leading quantum correction, similar to standard EFT. However, a crucial difference emerges near the critical radius (R_gs).

● Suppression of classical effects: The classical GR correction term in the potential is directly proportional to the equivalent mass (M_eq). As a particle's radius (R_m) approaches its critical radius (R_gs), its M_eq approaches zero. Consequently, the classical GR correction is strongly suppressed.

● Emergence of quantum dominance: In stark contrast, the leading quantum correction term (proportional to \hbar) is not suppressed by the equivalent mass in the same manner. This differential behavior leads to a remarkable inversion: in the transition region just before the critical radius is reached, the normally sub-dominant quantum correction becomes larger than the suppressed classical correction. This window, where quantum effects become the leading correction to the Newtonian potential, is the "quantum-dominant regime."

● Divergence from standard EFT and testability: Standard EFT, which does not incorporate the concept of equivalent mass, predicts a completely different behavior. As energy increases (or distance decreases toward the Planck scale), its classical correction terms grow uncontrollably, signaling a breakdown of the theory's predictive power. Sphere Theory, however, provides a physical completion precisely at this point of failure. The suppression of classical effects via M_eq tames the interaction and unveils the quantum-dominant regime.

This regime is not a minor artifact; it is a unique physical phenomenon predicted exclusively by Sphere Theory. While technologically monumental to probe, its existence provides, in principle, a distinct and falsifiable experimental signature that could distinguish this framework from all standard approaches to quantum gravity

[ The macroscopic test: From Stellar Cores to Cosmic Expansion ]

1) New mechanism for Inflation
Please refer to the paper.

2) The origin of cosmic acceleration from gravitational self-energy
Please refer to the paper.

3) An upward revision of the neutron star mass limit
Please refer to the paper.


#Paper: Sphere Theory: A Unified Framework for Gravity from Self-Energy

 

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Completing Quantum Gravity!​

A complete and testable theory of quantum gravity: EFT + Sphere Theory

1)The indispensable engine: The role and success of Effective Field Theory

EFT provides a systematic perturbation theory for gravity, complete with a well-defined mathematical formalism that includes the action principle, path integrals, and Feynman diagrams. This powerful computational engine has allowed physicists like Donoghue and his collaborators to perform landmark calculations, such as deriving the leading quantum corrections to the Newtonian potential and the bending of light. These results are not mere theoretical exercises; they represent concrete, finite predictions that prove gravity behaves as a predictable quantum theory at accessible energy scales. Within this framework, the graviton is treated as a quantized perturbation (h_μν) on a background spacetime, and its interactions are calculated systematically. In essence, EFT provides the indispensable mathematical skeleton and computational tools for any perturbative approach to quantum gravity.

2)The wall of ignorance: The fundamental limits of EFT

Despite its remarkable successes, the EFT of gravity is, by its very design, an incomplete theory. Its predictive power is strictly limited to the low-energy (IR) regime. As one probes higher energies, the theory's foundational limitations become apparent, culminating in a "wall of ignorance" at the Planck scale.

The two fundamental failures of EFT are:

Non-renormalizability and Divergence: High-energy loop calculations in the theory produce UV divergences that cannot be absorbed by renormalizing the original parameters of the Einstein-Hilbert action. To manage these infinities, EFT introduces an infinite tower of higher-derivative counterterms (c_1R^2, c_2R_μνR^μν, …), each with an unknown, unmeasurable coefficient c_i. This act of “absorbing" divergences into unknown parameters is not a solution; it is a systematic parameterization of our ignorance of the true high-energy physics. The theory becomes non-renormalizable and loses all predictive power at the Planck scale.

Inability to address singularities: As a perturbative theory built upon a smooth background spacetime, EFT is fundamentally incapable of describing the physics of gravitational singularities, where the curvature becomes infinite and the background itself collapses.

In conclusion, EFT provides a powerful answer to the question, “How do we calculate quantum effects at low energies?" However, it provides no answer to the more fundamental questions: “Why does the theory fail at high energies?" and `”What physical principle resolves this failure?"

3)The physical principle: Sphere Theory as the UV Completion

Sphere Theory (or the Gravitational Self-Energy Framework, GSEF) begins precisely where EFT fails. It does not discard the computational machinery of EFT; instead, it provides the missing physical principle that resolves EFT's fundamental limitations. This is achieved by renormalizing the gravitational source itself.

The core contributions of Sphere Theory are:

Adopting the formalism, correcting the source: Sphere Theory accepts the validity of EFT's mathematical formalism for calculating interactions. However, it corrects the most fundamental input to these calculations: the source mass (M). By replacing the idealized free-state mass (M_fr) with the physical equivalent mass (M_eq), which includes gravitational self-energy, the framework is rendered physically complete.

A physical resolution to divergence: Instead of parameterizing divergences with unknown coefficients c_i, Sphere Theory provides a physical mechanism for their elimination. As a system approaches the critical scale (R_m → R_gs), the equivalent mass source is quenched (M_eq → 0). This drives the effective gravitational coupling to zero (G(k) → 0), causing the very interaction that generates loops to vanish. This is the physical UV completion that EFT lacks.

A gravitational resolution to singularities: The theory demonstrates that singularities are prevented not by speculative quantum effects, but by gravity itself. The emergence of a repulsive gravitational force at a macroscopic critical radius (R_m < R_gs) provides a natural, self-contained solution that does not require invoking quantum mechanics.

The synthesis of EFT and Sphere Theory is not merely an additive combination; it is a synergistic union that forms a complete, consistent, and predictive theoretical structure for gravity across all scales. Their roles are perfectly complementary:

This combination forms a single, coherent theoretical architecture: EFT provides the mathematical chassis and computational engine, while Sphere Theory provides the physical engine block and the fuel (M_eq), along with a natural safety mechanism that prevents the engine from overheating and breaking down at high energies.

Therefore, we assert that the framework of “EFT + Sphere Theory” constitutes the first complete theory of gravity that is consistent from the lowest to the highest energy scales, is predictive, and is imminently testable.

A final, crucial point must be addressed. A common expectation for a theory of quantum gravity is that it must "quantize spacetime" itself. This expectation, however, arose as a potential strategy to solve the problems of singularities and divergences. Sphere Theory offers a more elegant and direct solution. By renormalizing the gravitational interaction at its source, it removes the very problems that the quantization of spacetime was intended to solve. From the perspective of Sphere Theory, the question of quantizing spacetime may not be a necessary one for a consistent theory of gravity. The ultimate arbiter is nature, and if the universe resolves these issues through the principles of self-energy, then that is the standard to which we must adhere.

[Abstract]
While major frameworks like Effective Field Theory (EFT) and String Theory address the challenges of quantum gravity through parametrization or new fundamental entities, this paper introduces a new framework, Sphere Theory, built upon the single physical principle of gravitational self-energy. This framework provides the physical UV completion for EFT, and we assert that their synthesis constitutes a complete, predictive, and testable theory of quantum gravity. Our analysis is built upon the principle that the effective source of gravity is not the free state mass (M_fr), but the \emph{equivalent mass} (M_eq), which accounts for its own gravitational self-energy. This principle leads to a running gravitational coupling G(k) that vanishes at a critical scale R_gs ∝ G_NM_fr/c^2, providing a self-renormalization mechanism that suppresses divergences at their source.

This unified mechanism provides coherent, physically-grounded solutions to several foundational problems: it resolves the singularity problem via a repulsive force that emerges at a macroscopic scale; it provides the physical UV cutoff for quantum field theories like QED by applying this same principle to the interaction mediators; and it offers a unified cosmological model that provides a natural mechanism for cosmic inflation and an explanation for the universe’s accelerated expansion. In contrast to String Theory, which postulates a fixed minimal length, Sphere Theory derives a dynamic minimal radius (R_gs ∝ \G_N/M_fr/c^2) from physical principles and requires no extra dimensions or new particles.

Crucially, the theory offers falsifiable predictions that distinguish it from standard models across multiple domains: a novel "quantum-dominant regime'' at the Planck scale, a necessary upward revision of the neutron star mass limit (the TOV limit) at the astrophysical scale, and a quantitative model for cosmic acceleration. In conclusion, Sphere Theory provides a path to genuine unification by revealing that the crises of the Planck scale, the astrophysical scale, and the cosmological scale are not separate problems, but are interconnected manifestations of its single, foundational principle.

#Paper: Sphere Theory: Completing Quantum Gravity through Gravitational Self-Energy