How to renormalize gravity and Why Planck scale cutoffs exist!

[icarus2]

Solution to Gravity Divergence, Gravity Renormalization, and Physical Origin of Planck-Scale Cut-off

[Abstract]
By incorporating gravitational binding energy into an effective mass M_eff, we derive the RG flow and the running gravitational coupling G(k). The gravitational coupling is given by G(k) = G_N[1 - (3G_NM_fr/5R_mc^2)(1 + (15/14)(G_NM_fr/R_mc^2)] = G_N[1 - (3G_N/5R_mc^3)k(1 + (15/14)(G_N/R_mc^3)k)], where R_m is the radius of the mass or energy distribution, and R_{gp-GR} ~ 1.16(G_NM_fr/c^2)~0.58R_S is the critical radius derived from general relativity at which the negative gravitational self-energy (binding energy) balances the mass-energy.

At R_m = R_{gp-GR}, where G(k) = 0 and the gravitational coupling vanishes, G(k) resolves gravitational divergences without quantum corrections and provides effective renormalization.

This study proves that the QFT cut-off Λ ~ M_Pc^2 serves as a physical boundary across all energy scales in quantum gravity. Quantum fluctuations (ΔE~M_Pc^2) with Δt~t_P yield an energy distribution radius R_m ~ l_P, where negative gravitational self-energy balances (or offsets) mass-energy, yielding E_T ~ 0 and thus eliminating divergences via G(k) = 0 and preventing negative energy states. In contrast, for proton or electron masses, R_m >> R_{gp-GR} (or R_gp), leading to E_T ~ Mc^2, rendering gravitational effects negligible and unsuitable for a cut-off. This affirms the Planck scale's unique role in quantum gravity.

For R_m < R_{gp-GR}, G(k) < 0, inducing a repulsive force that prevents singularity formation in black holes. This framework unifies solutions to gravitational divergences and singularity issues, offering new insights into cosmological phenomena such as cosmic acceleration.


The Central Idea: Effective Mass and Running Gravitational Coupling G(k)

Any entity possessing spatial extent is an aggregation of infinitesimal elements. Since an entity with mass or energy is in a state of binding of infinitesimal elements, it already has gravitational binding energy or gravitational self-energy. And, this binding energy is reflected in the mass term to form the mass M_eff. It is presumed that the gravitational divergence problem and the non-renormalization problem occur because they do not consider the fact that M_eff changes as this binding energy or gravitational self-energy changes.

One of the key principles of General Relativity is that the energy-momentum tensor (T_μν) in Einstein's field equations already encompasses all forms of energy within a system, including rest mass, kinetic energy, and various binding energies. This implies that the mass serving as the source of gravity is inherently an 'effective mass' (M_eff), accounting for all such contributions, rather than a simple 'free state mass'. My paper starts from this very premise. By explicitly incorporating the negative contribution of gravitational self-energy into this M_eff, I derive a running gravitational coupling constant, G(k), that changes with the energy scale. This, in turn, provides a solution to long-standing problems in gravitational theory.

M_eff = M_fr − ∣U_gp∣/c^2

where M_fr is the free mass and U_gp is the gravitational self-energy (or binding energy).

From this concept of effective mass, I derive a running gravitational coupling constant, G(k). Instead of treating Newton's constant G_N as fundamental at all scales, my work shows that the strength of gravitational interaction effectively changes with the momentum scale k (or, equivalently, with the characteristic radius R_m of the mass/energy distribution). The derived expression, including general relativistic (GR) corrections for the self-energy, is:


1.Vanishing Gravitational Coupling and Resolution of Divergences

1)In Newtonian mechanics, the gravitational binding energy and the gravitational coupling constant G(k)



2)In the Relativistic approximation, the gravitational binding energy and the gravitational coupling constant G(k)



For R_m >>R_{gp-GR} ~ 0.58R_S, the gravitational self-energy term is negligible, and the running gravitational coupling G(k) returns to the gravitational coupling constant G_N.

As the radius approaches the critical value R_m = R_{gp-GR} ~ 0.58R_S, the coupling G(k) smoothly goes to zero, ensuring that gravitational self-energy does not diverge. Remarkably, this mechanism allows gravity to undergo self-renormalization, naturally circumventing the issue of infinite divergences without invoking quantum modifications.

For R_m < R_{gp-GR} ~ 0.58R_S, the gravitational coupling becomes negative (G(k)< 0), indicating a repulsive or antigravitational regime. This provides a natural mechanism preventing further gravitational collapse and singularity formation, consistent with the arguments in Section 2.

4.5. Solving the problem of gravitational divergence at high energy: Gravity's Self-Renormalization Mechanism

At low energy scales (E << M_Pc^2, Δt >>t_P), the divergence problem in gravity is addressed through effective field theory (EFT). However, at high energy scales (E ~ M_Pc^2, Δt~t_P), EFT breaks down due to non-renormalizable divergences, leaving the divergence problem unresolved.

Since the mass M is an equivalent mass including the binding energy, this study proposes the running coupling constant G(k) that reflects the gravitational binding energy.

At the Planck scale (R_m ~ R_{gp-GR} ~ 1.16(G_NM_fr/c^2) ~ l_P), G(k)=0 eliminates divergences, and on higher energy scales than Planck's (R_m < R_{gp-GR}), a repulsion occurs as G(k)<0, solving the divergence problem in the entire energy range. This implies that gravity achieves self-renormalization without the need for quantum corrections.

4.5.1. At Planck scale
If, M ~ M_P

R_{gp-GR} ~ 1.16(G_NM_P/c^2) = 1.16l_P

This means that R_{gp-GR}, where G(k)=0, i.e. gravity is zero, is the same size as the Planck scale.

4.5.2. At high energy scales larger than the Planck scale



In energy regimes beyond the Planck scale (R_m<R_{gp-GP}), where G(k) < 0, the gravitational coupling becomes negative, inducing a repulsive force or antigravity effect. This anti-gravitational effect prevents gravitational collapse and singularity formation while maintaining uniform density properties, thus mitigating UV divergences across the entire energy spectrum by ensuring that curvature terms remain finite.
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#Paper
Solution to Gravity Divergence, Gravity Renormalization, and Physical Origin of Planck-Scale Cut-off

 

[icarus2]

4.5.3. Resolution of the two-loop divergence in perturbative quantum gravity via the effective mass framework

A crucial finding is that at a specific critical radius, R_{gp−GR}≈1.16(G_NM_fr/c^2) ≈ 0.58R_S (where R_S is the Schwarzschild radius based on M_fr), the negative gravitational self-energy precisely balances the positive free mass-energy. At this point, M_eff→0, and consequently, the effective gravitational coupling G(k)→0. This vanishing of the gravitational coupling has profound implications for quantum gravity. Perturbative quantum gravity calculations, which typically lead to non-renormalizable divergences (like the notorious 2-loop R^3 term identified by Goroff and Sagnotti), rely on the coupling constant κ=(32πG)^(1/2).
If G(k)→0 at high energies (Planck scale), then κ→0. As a result, all interaction terms involving κ diminish and ultimately vanish, naturally eliminating these divergences without requiring new quantum correction terms or exotic physics. Gravity, in this sense, undergoes a form of self-renormalization.

In perturbative quantum gravity, the Einstein-Hilbert action is expanded around flat spacetime using a small perturbation h_μν, with the gravitational field expressed as g_μν = η_μν+ κh_μν, where κ= \sqrt {32πG(k)} and G_N is Newton’s constant. Through this expansion, interaction terms such as L^(3), L^(4), etc., emerge, and Feynman diagrams with graviton loops can be computed accordingly.



At the 2-loop level, Goroff and Sagnotti (1986) demonstrated that the perturbative quantization of gravity leads to a divergence term of the form:

Γ_div^(2) ∝ (κ^4)(R^3)

This divergence is non-renormalizable, as it introduces terms not present in the original Einstein-Hilbert action, thus requiring an infinite number of counterterms and destroying the predictive power of the theory.

However, this divergence occurs by treating the mass M involved in gravitational interactions as a constant quantity. The concept of invariant mass pertains to the rest mass remaining unchanged under coordinate transformations; this does not imply that the rest mass of a system is intrinsically immutable. For instance, a hydrogen atom possesses different rest masses corresponding to the varying energy levels of its electrons. Both Newtonian gravity and general relativity dictate that the physically relevant source term is the equivalent mass, which includes not only rest mass energy but also binding energy, kinetic energy, and potential energy. When gravitational binding energy is included, the total energy of a system is reduced, yielding an effective mass:

M_eff = M_fr - M_binding

At this point R_m = R_{gp-GR} ~ 1.16(G_NM_fr/c^2), G(k) = 0, implying that the gravitational interaction vanishes.

As R_m --> R_{gp-GR}, κ= \sqrt {32πG(k)} -->0

Building upon the resolution of the 2-loop divergence identified by Goroff and Sagnotti (1986), our model extends to address divergences across all loop orders in perturbative gravity through the running gravitational coupling constant G(k). At the Planck scale (R_m=R_{gp-GR}), G(k)=0, nullifying the coupling parameter κ= \sqrt {32πG(k)} . If G(k) --> 0, κ --> 0.

As a result, all interaction terms involving κ, including the divergent 2-loop terms proportional to κ^{4} R^{3}, vanish at this scale. This naturally eliminates the divergence without requiring quantum corrections, rendering the theory effectively finite at high energies. This mechanism effectively removes divergences, such as the 2-loop R^3 term, as well as higher-order divergences (e.g., R^4, R^5, ...) at 3-loop and beyond, which are characteristic of gravity's non-renormalizability.

In addition, in the energy regime above the Planck scale (R_m<R_{gp-GR} ~ l_P), G(k)<0, and the corresponding energy distribution becomes a negative mass and negative energy state in the presence of an anti-gravitational effect. This anti-gravitational effect prevents gravitational collapse and singularity formation while maintaining uniform density properties, thus mitigating UV divergences across the entire energy spectrum by ensuring that curvature terms remain finite.

However, due to the repulsive gravitational effect between negative masses, the mass distribution expands over time, passing through the point where G(k)=0 due to the expansion speed, and reaching a state where G(k)>0. This occurs because the gravitational self-energy decreases as the radius R_m of the mass distribution increases, whereas the mass-energy remains constant at Mc^2. When G(k)>0, the state of attractive gravity acts, causing the mass distribution to contract again. As this process repeats, the mass and energy distributions eventually stabilize at G(k)=0, with no net force acting on them.

Unlike traditional renormalization approaches that attempt to absorb divergences via counterterms, this method circumvents the issue by nullifying the gravitational coupling at high energies, thus providing a resolution to the divergence problem across all energy scales. This effect arises because there exists a scale at which negative gravitational self-energy equals positive mass-energy.

~~~

 

[icarus2]

2.Physical Origin of the Planck-Scale Cut-off

4.6. The physical origin of the cut-off energy at the Planck scale

In quantum field theory (QFT), the cut-off energy Λ or cut-off momentum is introduced to address the infinite divergence problem inherent in loop integrals, a cornerstone of the renormalization process. However, this cut-off has traditionally been viewed as a mathematical convenience, with its physical origin or justification remaining poorly understood.

This work proposes that Λ represents a physical boundary determined by the scale where the sum of positive mass-energy and negative gravitational self-energy equals zero, preventing negative energy states at the Planck scale. This mechanism, rooted in the negative gravitational self-energy of positive mass or energy, provides a physical explanation for the Planck-scale cut-off.

At R_m = R_{gp-GR} ~ 1.16(G_NM_fr/c^2), G(k)=0

For a mass M_fr ~ M_P, the characteristic radius is :

R_{gp-GR} ~ 1.16(G_NM_P/c^2) = 1.16l_P

At R_m=R_{gp-GR}, G(k)=0, marking the Planck scale where divergences vanish.

If R_m<R_{gp-GR}, then G(k)<0, which means that the system is in a negative mass state. Therefore, the Planck scale acts as a boundary energy where an object is converted to a negative energy state by the gravitational self-energy of the object. In a theoretical analysis, a negative mass state may be allowed, although the system can temporarily enter a negative mass state, the mass distribution expands again because there is a repulsive gravitational effect between the negative masses. Thus, the Planck scale (l_P) serves as a boundary preventing negative energy states driven by gravitational self-energy.

4.6.2. Uncertainty principle and total energy with gravitational self-energy

ΔEΔt≥hbar/2

ΔE≥hbar/2t_P=(1/2)M_Pc^2

During Planck time t_P, let's suppose that quantum fluctuations of (5/6)M_P mass have occurred.

Since all mass or energy is combinations of infinitesimal masses or energies, positive mass or positive energy has a negative gravitational self-energy. The total energy of the system, including the gravitational self-energy, is

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

Here, the factor 3/5 arises from the gravitational self-energy of a uniform mass distribution. Substituting (5/6)M_P and R=ct_P/2 (where cΔt represents the diameter of the energy distribution, constrained by the speed of light (or the speed of gravitational transfer). Thus, Δx = 2R= cΔt.



This demonstrates that at the Planck scale, the negative gravitational self-energy balances (or can be offset) the positive mass-energy, defining a cut-off energy Λ ~ M_Pc^2. For energies E>Λ, the system enters a negative energy state (E_T<0), which is generally prohibited due to the repulsive gravitational effects of negative mass states. Repulsive gravity prevents further collapse, dynamically enforcing the Planck scale as a minimal length.

1)Case of Planck scale



This negative E_T indicates that R_m(= (1/2) l_P ) < R_{gp−GR}(= 1.16 l_P ), where R_{gp−GR} ∼ l_P is the critical radiusat which E_T = 0. Increasing ∆t ∼ t_p, R_m → R_{gp−GR}, and E_T → 0, suggesting that the Planck scale is where gravitational self-energy can balance the mass-energy, supporting a physical cut-off at Λ ∼ M_Pc^2

2)Case of Electron & Proton


The Planck scale exhibits a unique characteristic: only for M ~ M_P, t ~ t_P , and R ~ l_P does the gravitational self-energy (U_{gp-GR}) approach the mass-energy, enabling E_T ~ 0. This balance (or offset) suggests that the QFT cut-off Λ ~ M_Pc^2 acts as a physical boundary where quantum and gravitational effects converge. In contrast, for proton or electron masses, R_m >> R_{gp-GR}, rendering gravitational effects negligible and aligning with QED/QCD cut-offs (Λ ~ GeV).

3.Resolution of the Black Hole Singularity
For radii smaller than the critical radius, i.e., R_m<R_{gp−GR}, the expression for G(k) becomes negative (G(k)<0). This implies a repulsive gravitational force, or antigravity. Inside a black hole, as matter collapses, it would eventually reach a state where R_m<R_{gp−GR}. The ensuing repulsive gravity would counteract further collapse, preventing the formation of an infinitely dense singularity. Instead, a region of effective zero or even repulsive gravity would form near the center. This resolves the singularity problem purely within a gravitational framework, before quantum effects on spacetime structure might become dominant.

4.Cosmological Implications – A Potential Source for Dark Energy
The anti-gravitational regime (G(k)<0) predicted by my model has observable results on a cosmological scale. If the observable universe's average mass-energy distribution has an effective radius R_m that is less than its own critical radius R_{gp−GR}, then the universe itself would be in a state of repulsive gravitational self-interaction. This could provide a natural explanation for the observed accelerated expansion of the universe, attributing it to a fundamental property of gravity rather than an exotic dark energy component. My calculations suggest that for the estimated mass-energy of the observable universe, its current radius(R_m=46.5BLY) is indeed smaller than its R_{gp−GR}(275.7BLY), placing it in this repulsive regime.

The observable universe is in the R_m=46.5BLY < R_{gp-GR}=275.7BLY state, and therefore, an accelerated expansion exists.

Unifying Perspective
My research suggests that these seemingly disparate problems – gravitational divergences, the nature of the Planck scale cut-off, black hole singularities, and potentially even the mystery of dark energy – might share a common origin rooted in the proper accounting of gravitational self-energy. By incorporating this fundamental aspect of gravity, we can achieve a more consistent and predictive theory.

#Paper
Solution to Gravity Divergence, Gravity Renormalization, and Physical Origin of Planck-Scale Cut-off

 

[James R]

icarus2:

This is all very interesting to experts in the general theory of relativity, I'm sure.

Has your paper been peer reviewed and published in a journal?

What feedback are you hoping for from sciforums?

 

[icarus2]

icarus2:

This is all very interesting to experts in the general theory of relativity, I'm sure.

Has your paper been peer reviewed and published in a journal?

What feedback are you hoping for from sciforums?

I didn't submit it to the journal, and therefore, it wasn't peer reviewed.

I'm doing research as a hobby, and in the past I've written a few papers and sent them to a journal, and I've had the experience of being rejected.

In order to register these papers in arXiv.org, I have registered papers with the guarantee of another physicist, but even if the papers were guaranteed by other physicists, the paper was deleted by the manager's arbitrary judgment.

And, in some scientific communities, non-mainstream theories are banned from posting themselves

Since the above experiences, even when I conduct new research and write papers, I do not attempt to have them published in journals.

I think it's difficult for me to cross the barriers because I lack my knowledge, the journals have their own barriers to entry, and the physics community has considerable prejudice and discrimination.

So, I'm writing to some generous physics communities.

I'm just asking you to read it because I studied it with affection,

If you have any opinions or thoughts on my research or argument, please don't hesitate to tell me.

This is because I can expand the paper by looking at other people's opinions or reactions, studying additionally or getting something to add to the paper.