Sphere Theory: Beyond String Theory, Completing Quantum Gravity!

[icarus2]

Sphere Theory: Beyond String Theory, Completing Quantum Gravity!​

For decades, we have been working to perfect the theory of quantum gravity, exploring radical new ideas such as extra dimensions (string theory) or the quantization of spacetime itself (loop quantum gravity). Moreover, significant unresolved problems related to gravity—such as the divergence problem, the singularity problem, the cause or driving mechanism of inflation, and the problem of cosmic accelerated expansion—span from the smallest to the largest scales.

This strongly suggests that we may be missing something crucial in our understanding of gravity.

Although these four representative gravity-related problems (Divergence, Singularity, Inflation, and Dark Energy) appear to exist at different scales and in different contexts, they could, in fact, be manifestations of a single underlying issue related to gravity.
That issue is the necessity of antigravity or repulsive forces. If antigravity exists in the context of gravity, all four of these problems could be resolved. If this antigravity is scale-dependent, it could address issues across different scales.

I believe the physical concept that mainstream physics is overlooking is the gravitational self-energy or binding energy inherent to an object. The effective source of gravity is not the free-state mass (M_fr) but the equivalent mass (M_eq) corresponding to the total energy of the object. And this equivalent mass includes the gravitational self-energy (negative binding energy) that has a negative value. Since gravitational self-energy is negative energy, it satisfies the anti-gravity requirement. Also, since it is scale-dependent, it can solve the gravity problem from the smallest scale to the largest scale.
By accounting for this gravitational self-energy, we can resolve the four aforementioned problems and complete a theory of quantum gravity.

1. Why "Sphere Theory"?​

The concept of gravitational self-energy(U_gs) is the total of gravitational potential energy possessed by a certain object M itself. Since a certain object M itself is a binding state of infinitesimal mass dMs, it involves the existence of gravitational potential energy among these dMs and is the value of adding up these. M = ΣdM. The gravitational self-energy is equal to the minus sign of the gravitational binding energy. Only the sign is different because it defines the gravitational binding energy as the energy that must be supplied to the system to bring the bound object into a free state.

*To understand the basic principle, we can look at the problem in Newtonian mechanics, and for the actual calculation, we can use the binding energy formula of general relativity to find the value.

U_gs=-(3/5)(GM^2)/R

In gravitationally bound systems, changes in configuration (e.g., orbital reduction) lead to a decrease intotal energy and equivalent mass due to energy radiation, as seen in celestial mechanics. Although potential energy changes to kinetic energy, in order to achieve a stable bonded state, a part of the kinetic energy must be released to the outside of thesystem. As a result, this leads to a decrease in the equivalent mass of the system.

In the case of a spherical uniform distribution, the total energy of the system, including gravitational potential energy (binding energy), is

E_T = Σm_ic^2 + Σ-(Gm_im_j/r_ij) = Mc^2 - (3/5)(GM^2/R)

In the general case, the value of total gravitational potential energy (gravitational self-energy) is small enough to be negligible, compared to mass energy Mc^2.

However, as R gets smaller, the absolute value of U_gs increases. For this reason, we can see that U_gs is likely to offset the mass energy at a certain radius. The mass defect effect due to binding energy has already been demonstrated in particle physics.

Thus, looking for the size in which gravitational self-energy becomes equal to rest mass energy,


At the critical radius R_gs, the negative gravitational self-energy cancels out the positive mass energy, so the total energy becomes zero, and therefore the gravity becomes zero.

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

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


M_eq=M_fr - M_gs = M_fr - |U_gs|/c^2

M_fr is the free-state mass, -M_gs is the equivalent mass of gravitational self-energy (U_gs). G_N is Newton's gravitational constant, G(k) is running gravitational coupling.

G(k)=G_N(M_eq/M_fr)) = G_N(1 - M_gs/M_fr) = G_N(1- |U_gs|/M_frc^2)

The integration of the gravitational binding function is not analytical. Using the first-term approximation, we obtain the value R_{gs-GR-1st} ~ 1.16G_NM_fr/c^2 ~ 0.58R_S. If we calculate the integral itself numerically and apply the virial theorem to it, we obtain the value R_{gp-GR-vir} ~ 1.02G_NM_fr ~ 0.51R_S. Since the process in which actual celestial bodies contract gravitationally to become black holes is very complex, these values may be slightly different.

The important thing here is not the exact value, but the fact that there exists a actual critical radius R_gs where the negative gravitational self-energy offsets the positive mass energy. In addition, these R_gs are estimated to be GM/c^2 ~ 2GM/c^2.

R_gs ~ GM/c^2

What this critical radius R_gs means is that,
If the object were to shrink further (R<R_gs), it would enter a negative energy state. This generates a repulsive gravitational force or effect ('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'.

Where QFT can be viewed as a “Point Theory” and String Theory as a “String Theory”, "Sphere Theory" is built upon the physical principle that all fundamental entities are not mathematical idealizations but physical objects possessing a three-dimensional volume.

This framework, which can also be more descriptively referred to as the Gravitational Self-Energy Framework (GSEF), does not postulate new entities but rather rigorously applies a core tenet of general relativity: that all energy, including an object’s own negative self-energy, acts as a gravitational source.



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

 

[icarus2]

2. How is this different from String Theory?​

1)Minimal Length: Derived, not 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.

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/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.

At the microscopic level, this relation provides a physical origin for the Planck-scale cutoff (Refer to section 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(M=M_P) ~ GM_P/c^2 ~ l_P

For a Planck-mass entity, the critical scale where the gravitational interaction dynamically vanishes emerges naturally at the Planck scale itself.

If R_m < R_gs, then G(k)<0, signifying that the system enters a state of negative equivalent mass and experiences repulsive gravity. This repulsive force provides a dynamic stabilization mechanism. While the system can temporarily enter this state, the repulsive effect between negative mass components causes the distribution to expand. This expansion increases R_m, driving the system back towards the stable equilibrium point where G(k)=0. Thus, the Planck scale (R_gs ~ l_P) serves as a dynamic physical boundary, enforced by the interplay of gravitational self-energy and repulsive gravity.

2)Simplicity: It requires no extra dimensions, no supersymmetry, and no new particles. Use existing physics to solve the problem.

3)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 ∝ GM/c^2). This inherent scalability means the theory's core principle applies seamlessly from the quantum fluctuations at the Planck scale to the 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.

3. What problems does Sphere Theory solve?​

It is a foundational principle, recognized in both Newtonian mechanics and general relativity, that the effective 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 demonstrate that this single principle:

1) Resolves the singularity problem via a repulsive force that emerges at a macroscopic, not quantum, scale (Section 2, 3, 4.5).

2) Solves the non-renormalizability of gravity, as exemplified by the 2-loop divergence of Goroff and Sagnotti, by demonstrating that the interaction is dynamically turned off at a critical scale (Section 4.6.3).

3) Resolves the unitarity crisis in high-energy graviton scattering by showing that the scattering amplitude naturally vanishes as the physical source of the interaction is quenched (Section 4.9).

4) Provides the physical origin of the UV cutoff for quantum field theories, demonstrating that the gravitational self-energy of force mediators (e.g., photons, gravitons) dynamically suppresses their propagation at the Planck scale (Section 4.7~ 4.9).

5) Resolves the divergence and the Landau pole problem in QED, transforming it into a potentially fundamental theory by providing a physical cutoff mechanism rooted in the gravitational self-energy of the photon (Section 4.8).

6) Provides a UV completion for EFT. It resolves divergence problems arising in (1)the gravitational potential between two masses and (2)the bending of light. This approach also renders the infinite tower of unknown EFT coefficients (c_i) unnecessary and makes a novel, falsifiable prediction of a "quantum-dominant regime” (Section 5).

7) Offers a unified explanation for major cosmological puzzles by providing (1)a mechanism for cosmic inflation, (2)a model for the universe's accelerated expansion}, and (3)a predicted upward revision of the neutron star mass limit} (TOV limit) (Section 7).

8) Finally, it culminates in a declaration that the synthesis of EFT and Sphere Theory constitutes a self-consistent and testable framework for quantum gravity, providing a consistent description from the lowest to the highest energy scales (Section 8).

4. How can Sphere Theory be tested?​

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.

This demonstrates how the Planck scale cutoff 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 Sections 5 and 6.

2) At the Macroscopic Scale: Offers a unified explanation for the major puzzles of modern cosmology by providing (1) a mechanism for cosmic inflation, (2) a model for the accelerated expansion of the universe, and (3) a predicted upward revision of the neutron star mass limit (TOV limit), all of which serve as falsifiable tests (Section 7).

The reason this model can be tested for macroscopic events is that, unlike string theory, the critical radius is proportional to mass or energy.

 

[icarus2]

5. Unified framework for 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. 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.

2)-1) The two fundamental failures of EFT are

Non-renormalizability and Divergence: ~~~

Inability to address singularities: ~~~

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.

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

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:

 

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*Table 2. EFT and Sphere Theory's complementary roles in a unified quantum gravity.

Problem / Role	Effective Field Theory (EFT)	Sphere Theory (GSEF)
Calculation Engine	Provides the mathematical formalism.	Adopts and utilizes the formalism.
Low-Energy Physics	Delivers confirmed predictions.	Agrees with and preserves all predictions.
Gravitational Divergence (High-Energy)	Fails (Non-renormalizable, Divergence).	Resolves (Dual mechanism: Source quenching (M_eq --> 0) & Mediator suppression (E_{total} --> 0)).
Unitarity Crisis (High-Energy Scattering)	Fails (Violates unitarity).	Resolves (Scattering amplitude vanishes as the source is quenched).
QFT Divergences (e.g., QED, Landau Pole)	Fails (Incomplete, requires ad-hoc regularization).	Resolves (Provides a physical UV cutoff via mediator self-energy).
Singularity Problem	Fails (Inapplicable).	Resolves (Gravitational repulsion at a macroscopic scale).
New Predictions	Limited.	Provides (Quantum-dominant regime, TOV limit, Dark energy, etc.).

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.


[ A Common Origin for Two Gravitational Crises ]

It is telling that modern physics' two most significant challenges lie at the extremes of scale, and both are fundamentally problems of gravity. The non-renormalizability of gravity at the microscopic level and the unexplained cosmic acceleration at the macroscopic level point to a common, missing ingredient in our understanding of gravitation.

Sphere Theory asserts that this missing element is the negative gravitational self-energy inherent to the object itself. Because the critical radius, R_gs, derived from this overlooked self-energy is proportional to mass (R_gs ∝ G_NM_fr/c^2), it applies to both extremes of scale, and because its nature is that of negative energy, it can produce a repulsive effect. This repulsive effect can halt the collapse that leads to divergences at the quantum level and can drive the expansion that appears as dark energy at the cosmic level.

Therefore, Sphere Theory offers a potential path to a genuine unification, suggesting that the solutions to the crises of the very small and the very large are not separate problems, but are two manifestations of a single, deeper principle of gravity.


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

 

[exchemist]

More botshit, by the look of it.

 

[Pinball1970]

More botshit, by the look of it.

Botshit? I'm guessing batshit, bullshit or botfly shit which I have never encountered.
Certainly more interesting that the OP.

 

[exchemist]

Botshit? I'm guessing batshit, bullshit or botfly shit which I have never encountered.
Certainly more interesting that the OP.

Botshit is a thing: https://hbr.org/2024/07/the-risks-of-botshit

 

[DaveC426913]

Botshit is a thing: https://hbr.org/2024/07/the-risks-of-botshit

Bah! Subscription wall

 

[Pinball1970]

Bah! Subscription wall

Just to be clear, botshit is excrement from Bots? Like Ai making up stuff?

 

[DaveC426913]

Just to be clear, botshit is excrement from Bots? Like Ai making up stuff?

Dunno. Not my term.

I wanted to read the Ex's article

 

[exchemist]

Dunno. Not my term.

I wanted to read the Ex's article

Oh I just quoted it to show the term is real. Here’s another:https://www.sciencedirect.com/science/article/abs/pii/S0007681324000272
You can read the abstract at least.

As I understand it, botshit is false information from LLMs that becomes incorporated in the body of available knowledge on-line, thereby contaminating it, including it potentially being picked up by other LLMs.

 

[Pinball1970]

potentially being picked up by other LLMs.

Nasty, so it could spread like a virus? They need to invent a bot who's sole purpose is clearing out garbage info from platforms, like a Botdoc?
OR just specific things, antibotty?

 

[exchemist]

Nasty, so it could spread like a virus? They need to invent a bot who's sole purpose is clearing out garbage info from platforms, like a Botdoc?
OR just specific things, antibotty?

This seems to be already a real problem, with chatbots ingesting crap created by other chatbots and disseminating it further, corrupting the knowledge base - with botshit.

The solution is not easy, since these so-called AI bots are mere "stochastic parrots" (© Prof. Emily Bender), with no ability to think critically about what they paraphrase.

 

[DaveC426913]

Yes. Chatbots are poisoning their own food. There is a lot of speculation that they will poison themselves to death. (i.e. become useless echo chambers of disinformation)