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.

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 the case of a spherical uniform distribution, the total energy of the system, including gravitational potential energy, is

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.)

The integration of the gravitational binding function is not analytical. Using the first-term approximation, we obtain the value R_gs-GR ~ 1.16G_NM_fr ~ 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. It is also estimated that this R_gs exists between the radius of the black hole 0.5R_S and 1R_S.

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 ('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.

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 ∝ GM/c^2). 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.

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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 (Section2-3).

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

3) Solving divergence of the gravitational potential between two masses in standard EFT: It solves the divergence problem of the standard effective field theory (EFT) proposed by John F. Donoghue et al.(Section 5~6.)

4) Provides a UV completion for EFT by resolving the predictive obstacles in key quantum calculations. We resolve the divergence problems arising in (1)the gravitational potential between two masses, (2)the bending of light by applying the principle of source renormalization. This approach shows that the infinite tower of unknown EFT coefficients (c_i) is rendered unnecessary because the interaction source is dynamically quenched (M_eq} --> 0) in the UV limit, making the question of their values moot. This framework also makes a novel prediction of a "quantum-dominant regime” that distinguishes it from standard EFT (Section 5).

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

6) 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).

7) Declaration of completion of quantum gravity: Finally, it culminates in a declaration that the synthesis of EFT and Sphere Theory constitutes a complete and testable framework for quantum gravity. We will argue that this unified model, built on the physical principle of self-energy, provides the first consistent theory of gravity from the lowest to the highest energy scales (Section 8).

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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.

2) At the other 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.

7. A new framework for gravity: Sphere Theory

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:

7.1 Philosophical cornerstones and testable predictions

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:

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: quantum-dominant regime ]

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:

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 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

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2) The origin of cosmic acceleration from gravitational self-energy

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3) An upward revision of the neutron star mass limit

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[ 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.

8. A unified framework for Quantum Gravity

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:



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 our theories must adhere.

#Paper:

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

Edited June 23Jun 23 by icarus2

8 hours ago, icarus2 said:

*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 the case of a spherical uniform distribution, the total energy of the system, including gravitational potential energy, is

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

According to general relativity, this is incorrect. Birkhoff's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. This means that the spacetime outside of any spherically symmetric distribution of energy-momentum must be given by the Schwarzschild metric. It may be considered the general relativistic version of the shell theorem in Newtonian gravity. The significance of Birkhoff's theorem to what you wrote is that if any spherically symmetric object expands or contracts or arbitrarily pulsates, then the external gravitation does not change. In other words, the Newtonian gravitational binding energy, which depends on the radius of the spherical source of gravitation, does not produce a change in the total mass of the spherical source of gravitation as the radius changes.

I should remark that if the radius decreases, the change in gravitational binding energy would ultimately result in a corresponding change in thermal energy, which would eventually radiate away from the spherical object, resulting in a decrease in total mass corresponding to the change in gravitational binding energy. I suppose it is this change in total mass due to gravitational binding energy that the OP is referring to rather than energy-conserving transformation of gravitational binding energy to thermal or other energy during gravitational collapse.

2 hours ago, KJW said:

I should remark that if the radius decreases, the change in gravitational binding energy would ultimately result in a corresponding change in thermal energy, which would eventually radiate away from the spherical object, resulting in a decrease in total mass corresponding to the change in gravitational binding energy. I suppose it is this change in total mass due to gravitational binding energy that the OP is referring to rather than energy-conserving transformation of gravitational binding energy to thermal or other energy during gravitational collapse.

It is not possible to explain every detail in the summary, and the point you raised is, in fact, a matter of common knowledge.

It is widely understood that, during gravitational contraction or orbital transitions, energy corresponding to the difference in binding energy must be released to the outside of the system in order to achieve a stable bound state. Therefore, I did not consider it necessary to elaborate on this in the summary.

However, this point is briefly mentioned in the main body of the paper.
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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 [8].
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Even if we apply the kinetic energy and virial theorem, the radius only decreases as negative energy cancelsout positive energy, but the core claim that “there is a point where the negative gravitational potential energycancels out the positive mass energy.” remains unchanged. 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

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If you cannot accept the negative energy state, please refer to the following explanation.If, R_m = R_{gs−NM} , the total energy of the system, including the gravitational potential energy, is

From the point of view of mass defect, r = R_{gs−NM} (or R_{gs−NM−vir}) is the point where the total energy ofthe system is zero. For the system to compress more than this point, there must be an positive energy release from the system. However, since the total energy of the system is zero, there is no positive energy that thesystem can release. Therefore, the system cannot be more compressed than r = R_{gs−NM} (or R_{gs−NM−vir}). Soblack hole doesn’t have singularity

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Applying the core principle of Sphere Theory, which posits that total energy includes self-energy, even if some particle comes into the radius of black hole, it is not a fact that it contracts itself infinitely to the point R = 0. From the point R_{gs−NM} (or R_{gs−NM−vir}), gravity is 0, and when it enters into the area of R_{gs−NM} (or R_{gs−NM−vir}), total energy within R_{gs−NM} (or R_{gs−NM−vir} ) region corresponds to negative values enablinganti-gravity to exist. This R_{gs−NM} (or R_{gs−NM−vir} ) region comes to exert repulsive effects of gravity on the particles outside of it, therefore it interrupting the formation of singularity at the near the area R = 0. However, it still can perform the function as black hole because the emitted energy will exist in a region larger than r > R_{gs−NM} (or R_{gs−NM−vir} ). Since the emitted energy cannot escape the black hole, it is distributed inthe region R_{gs−NM} (or R_{gs−NM−vir}) < r < RS_ . Since the total energy of the entire range (0 ≤ r < R_S ) insidethe black hole is positive, it functions as a black hole.

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2 hours ago, icarus2 said:

It is not possible to explain every detail in the summary, and the point you raised is, in fact, a matter of common knowledge.

It is widely understood that, during gravitational contraction or orbital transitions, energy corresponding to the difference in binding energy must be released to the outside of the system in order to achieve a stable bound state. Therefore, I did not consider it necessary to elaborate on this in the summary.

Did you not see my post before the one you quoted? The post you quoted was connected to the one before it. In that post, I was saying that in general relativity, when a spherically symmetric distribution of matter expands or contracts or arbitrarily pulsates, the external gravitation does not change, implying that the total mass does not change. The second post was merely to clarify an aspect I overlooked in the first post.

4 hours ago, KJW said:

Did you not see my post before the one you quoted? The post you quoted was connected to the one before it. In that post, I was saying that in general relativity, when a spherically symmetric distribution of matter expands or contracts or arbitrarily pulsates, the external gravitation does not change, implying that the total mass does not change. The second post was merely to clarify an aspect I overlooked in the first post.

Your argument rests on a fundamental misapplication of Birkhoff's theorem. You are incorrectly extending it to dynamic, energy-radiating processes as if the total mass of the system were immutable.

1)Birkhoff's Theorem Describes a State, Not a Process: The theorem states that the external spacetime of a spherically symmetric mass is the Schwarzschild metric for a given total mass, M. It does not state that M must remain constant if the system radiates energy.

2)Binding Energy and Mass Defect are Real: Any system that becomes more tightly bound radiates energy and, as a consequence of E = mc², loses mass. This is a well-established fact, observable from the atomic scale to the cosmological:

• Hydrogen Atom: A hydrogen atom's mass is less than the sum of a free proton and a free electron because ~13.6 eV of binding energy is released.

• The Earth: Earth's mass is little less than the mass of its constituent dust and gas in a dispersed state, as gravitational potential energy was radiated away as heat during its formation.

Gravitational Contraction is a Dynamic Process. When a star contracts, it radiates energy (as photons, neutrinos, etc.) to become more gravitationally bound. This process is inherently non-static. The initial mass (M) decreases to a new, final mass (M') because energy has left the system.

Birkhoff's theorem applies perfectly to the initial state (with mass M) and to the final, more stable state (with mass M'). However, it does not forbid the transition from M to M'. Your claim that the total mass remains unchanged contradicts the law of conservation of energy.

Please consider the mass reduction of a hydrogen atom or Earth due to binding energy.

Edited June 24Jun 24 by icarus2

2 hours ago, icarus2 said:

Gravitational Contraction is a Dynamic Process. When a star contracts, it radiates energy (as photons, neutrinos, etc.) to become more gravitationally bound. This process is inherently non-static. The initial mass (M) decreases to a new, final mass (M') because energy has left the system.

If you have a black hole, how does the energy leave the system as the self-energy cancels the rest energy?

How does a massless (i.e. no energy) collection emit any radiation? It can’t be at 0K. It must have thermal energy, and as such, mass.

16 hours ago, swansont said:

If you have a black hole, how does the energy leave the system as the self-energy cancels the rest energy?

15 hours ago, swansont said:

How does a massless (i.e. no energy) collection emit any radiation? It can’t be at 0K. It must have thermal energy, and as such, mass.

A Model for the Interior State of a Black Hole

The process of releasing binding energy during gravitational contraction is inherently complex. However, for a general analysis, the core principle is that for a system to settle into a more stable, tightly-bound state, it must radiate away an amount of energy equivalent to the change in its gravitational potential energy. This results in a corresponding decrease in the system's total equivalent mass.

If we consider a rotating mass distribution, the virial theorem suggests that approximately half of the change in binding energy contributes to this reduction in the system's equivalent mass. While differential rotation adds further complexity, the exact numerical factors are not critical to the core argument, so we will omit them for this analysis. For now, let's consider the non-rotating case, keeping in mind that the inclusion of rotation would likely result in a smaller critical radius (R_gs) than the non-rotating case (perhaps by a factor within 0.5).

The release of binding energy external to a black hole is a well-understood process, so we can omit that discussion. Instead, let's focus on the problem of gravitational contraction and energy release inside a black hole.

Simplifying the Problem: The Low-Density Black Hole Analogy

The interior problem becomes exceedingly complex when considering ultra-high-density black holes, as it seems to require a full theory of General Relativity and quantum mechanics. However, it's well-known that the concept of a black hole is not exclusive to ultra-dense objects. To simplify the analysis, it is useful to first consider a sufficiently large black hole and then extrapolate the findings to higher-density cases.

For example, a supermassive black hole of about 4 billion solar masses has an average density comparable to that of Earth's air (~1.225 kg/m³). Conceptually, we can therefore picture this as a "black hole made of air."

Simplifying the Problem: The Low-Density Black Hole Analog

The interior problem becomes exceedingly complex when considering ultra-high-density black holes, as it seems to require a full theory of General Relativity and quantum mechanics. However, it's well-known that the concept of a black hole is not exclusive to ultra-dense objects. To simplify the analysis, it is useful to first consider a sufficiently large black hole and then extrapolate the findings to higher-density cases.

For example, a supermassive black hole of about 4 billion solar masses has an average density comparable to that of Earth's air (~1.225 kg/m³). Conceptually, we can therefore picture this as a "black hole made of air."

Contraction Inside the "Air Black Hole"

Now, let's analyze the gravitational contraction of this "air" inside the black hole. The critical radius (R_gs) at which the negative gravitational self-energy precisely cancels the positive rest mass energy can be estimated:

• In a Newtonian approximation, this radius is R_gs ≈ 0.3R_S, where RS is the Schwarzschild radius.

• In General Relativity, accounting for the virial theorem and rotation, the radius is approximately R_gs ≈ 0.51R_S.

While the values differ, the fundamental logic is the same. For the sake of clarity, I will proceed using the Newtonian result R_gs = 0.3R_S.

Consider the scenario where the "air," initially distributed out to R_S, undergoes gravitational contraction down to the radius R_gs. In this final state, the "system" is the sphere defined by 0 ≤ r < R_gs. The surrounding shell, R_gs < r < R_S, now effectively becomes the "exterior" relative to this new, contracted system.

Crucially, the binding energy released during this contraction is not ejected from the black hole itself. Instead, it is radiated into this exterior shell. Therefore, the total mass and energy of the black hole as a whole (out to R_S) remain conserved.

The State of the Zero-Energy System

What, then, is the state of the system within the radius RgsRgs? Its total energy is zero:
E_T = M_frc^2 - (3/5)G(M_fr)^2/R_gs = 0

However, a total energy of zero does not imply a vacuum or an absence of matter. This state is not an empty E_T = 0 + 0 = 0.

Rather, it is a state where matter M and its associated negative gravitational potential energy coexist, with their sum being zero. Since matter M is still present, it can exhibit all the properties of matter, such as thermal radiation.

At any local point occupied by the "air," the local energy density is still positive. This is because the rest mass energy of the air particles in an infinitesimal volume is greater than the negative gravitational potential energy contained within that same volume. It is only the total energy of the entire system (from r=0r=0 to RgsRgs) that integrates to zero. Consequently, the net gravitational influence of this zero-energy system on an external observer is null.

To put it another way, this state can be visualized as islands of positive-mass "air" existing within a sea of negative gravitational energy. This is analogous to our own universe, where the total energy of dark energy is vast, yet in the local vicinity of planets and galaxies, the matter-energy density is significantly higher than that of the surrounding dark energy.

For a more in-depth treatment of the black hole interior problem, I recommend consulting the following paper.
https://www.researchgate.net/publication/313314666_Solution_of_the_Singularity_Problem_of_Black_Hole

*In previous black hole papers, R_gp was used instead of R_gs. R_gs=R_gp.

Figure 2: Internal structure of a black hole according to the radius of the mass distribution a) Existing Model. b) New Model. The area of within R_gp (or R_gp-vir) has gravitational self-energy (potential energy) of negative value, which is larger than mass energy of positive value. If R is less than R_gp(or R_gp-vir), this area becomes negative energy (mass) state. There is a repulsive gravitational effect between the negative masses, which causes it to expand again. This area (within R_gp(or R_gp-vir) exercises anti-gravity on all particles entering this area, and accordingly prevents all masses from gathering to r=0. Therefore, the mass (energy) distribution cannot be reduced below the radius R_gp (or R_gp-vir).

Figure 6: Temporarily, when the mass M contracts more than R_gp, the central region of the black hole becomes a negative mass state. a) is the case where the mass M is compressed into a region smaller than R_gp, and the negative gravitational potential energy corresponds to -2Mc^2. In this case, the total energy of the system (0 ≤ r ≤ R) will be -1Mc^2, and the total energy outside (R < r ≤ R_S' ) the system will be 2Mc^2. The total energy inside the black hole will remain +1Mc^2. b) is a case where the mass M is compressed into a smaller region, so that the total energy of the system (0 ≤ r ≤ R) is -100Mc^2, and the total energy outside (R < r ≤ R_S' ) the system will be +101Mc^2. R is a value obtained through calculation in individual situations.

Figure 7. Over time, when the energy distribution inside the black hole is stabilized, the internal structure of the black hole. The 0 ≤ r ≤ R_gp region is a region where the positive mass energy and the negative gravitational potential energy have the same size, and the total energy is 0. The released binding energy exists in the region outside R_gp. The total mass of the black hole is M.
The mass or energy distribution at the center of a black hole is,
M + (-M)=0;
M(Equivalent mass of matter and energy) + (-M)(Equivalent mass of gravitational self-energy) = 0

So, how can we determine if the internal structure of a black hole is as described? Can it be proven?

My answer is yes. I believe and argue that this internal structure of a black hole can be proven.

The basis for this claim is my hypothesis that our own universe is structurally analogous to the interior of a black hole. Consequently, I propose that the universe we observe today is, in fact, a direct representation of what the inside of a black hole looks like.

We reside within an exceptionally large black hole, and it is because we are currently located in the region where R < R_gs that we experience an accelerating expansion of the universe.

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We live inside a black hole. I think that the accelerated expansion of the universe and the dark energy effect are evidence that we live inside a black hole. Therefore, I claim that by verifying the dark energy term, we can prove that we live inside a black hole.

You may think that the Hubble-Lemaitre expansion of the universe conflicts with the Black Hole Cosmology, but before you make a definitive judgment on this, let's hear a little more!

1.The size of the event horizon based on the total mass of the observable universe

According to Shell Theorem and Birkhoff’s Theorem, in a spherically symmetric system, the gravitational effect at a given radius is determined only by the mass or energy content surrounded within that radius, and contributions from outside the shell do not affect the internal dynamics.

R_obs=46.5Gly
ρ_c=8.64x10^-27kgm^-3
R_S=2GM/c^2=491.6Gly

The size of the event horizon created by the total mass distribution of the observable universe is 491.6 Gly. The event horizon created by the total mass of the observable universe is roughly 10 times larger than the observable universe. Therefore, the observable universe exists inside the event horizon of a black hole created by its own mass.

There are several fatal weaknesses in this black hole cosmology.

2.Weaknesses of the Black Hole Cosmology

1) In a black hole, all matter is compressed into a singularity, so there is no space for humans to live. There is no almost flat space-time that could contain the observable universe inside a black hole.

2) In the black hole, singularity exist in the future, and in the universe, singularity exist in the past. Black hole and the universe are opposites.

3) The universe is expanding. Inside a black hole, all matter must contract to a singularity. The two models show opposite phenomena. It is difficult to explain the expansion of the universe inside a black hole. In addition, the universe is expanding at an accelerated rate.

Problems such as strong tidal force enough to disintegrate people, the movement of all matter in the direction of the singularity, and the expanding universe have been pointed out as fatal weaknesses of the Black Hole Cosmology. If our universe was a black hole, all galaxies should have collapsed into a singularity or exhibit motion in the direction of the singularity, but the real universe does not exhibit such motion characteristics. Therefore, the Black Hole Cosmology was judged to be inconsistent with the current observations, and the Black Hole Cosmology did not become a mainstream cosmological model.

Although this objection (Weaknesses) appears to be clear and well-grounded, in fact, this objection also has its own fatal weaknesses.

Most physicists and astronomers believe that the singularity problem will be solved by quantum mechanics or some other unknown method. In other words, most scientists think that singularities don't exist. They are using it as a basis for criticism, even though they think that singularity will not exist.

We think that the singularity problem will ultimately be solved by some mechanism. Therefore, in the process of solving the singularity problem, there is a possibility that the singularity problem of the Black Hole Cosmology will also be solved.

For the singularity to disappear, there must be a repulsive force inside the black hole. Due to this repulsive force, an uncompressed region inevitably exists inside the black hole.

The remaining question is, 'Can the uncompressed region be larger than the observable universe?'

If the singularity disappears due to quantum mechanics, the uncompressed region will be very small compared to the observable universe. But what if the singularity disappears by some other means than quantum mechanics?
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A full exposition of this 'Black Hole Cosmology' is too extensive for this post. However, for those interested, I invite you to read the article at the link below. Additionally, Chapter 7 of the 'Sphere Theory' paper referenced in the main text provides further detail.

https://www.reddit.com/r/blackholes/comments/1icshzw/we_live_in_a_black_hole_the_accelerated_expansion/

Edited June 25Jun 25 by icarus2

21 hours ago, icarus2 said:

Your argument rests on a fundamental misapplication of Birkhoff's theorem. You are incorrectly extending it to dynamic, energy-radiating processes as if the total mass of the system were immutable.

1)Birkhoff's Theorem Describes a State, Not a Process: The theorem states that the external spacetime of a spherically symmetric mass is the Schwarzschild metric for a given total mass, M. It does not state that M must remain constant if the system radiates energy.

No. I was not applying Birkhoff's theorem to energy-radiating processes. I was applying it to the case of a spherically symmetric object that expands, contracts, or arbitrarily pulsates due to its own internal processes, without absorbing or emitting energy. For any spherically symmetric spacetime such that outside of some sphere, the Ricci tensor is everywhere zero, the metric outside of this sphere will be the Schwarzschild metric of constant mass irrespective of anything that happens inside the sphere (provided spherical symmetry is maintained). The condition that the Ricci tensor is zero everywhere outside the sphere ensures that there is no addition of energy to or subtraction of energy from the inside of the sphere.

21 hours ago, icarus2 said:

Any system that becomes more tightly bound radiates energy

That is not necessarily true. It may be that the release of gravitational binding energy as the system contracts is absorbed by a chemical reaction. And even if radiation is emitted, this emission is not generally concomitant to the release of gravitational binding energy. Thus, the decrease in total mass does not necessarily correspond to the change in gravitational binding energy.

One form of radiation that is never emitted from spherically symmetric distribution of energy-momentum is gravitational radiation.

21 hours ago, icarus2 said:

Your claim that the total mass remains unchanged contradicts the law of conservation of energy.

I didn't make that claim with regards to the emission of radiation. In my second post, I said the opposite.

I can’t help but notice that your response to me seems to be some prepared manuscript, that does not answer my questions. How about doing that?