Time Gradient as the Basis for Nuclear Interactions

Time Gradient as the Basis for Nuclear Interactions

Abstract

This article examines the influence of the time gradient on the formation of nuclear forces. The mechanism of time dilation inside atoms is analyzed, affecting nucleon interactions. A comparative analysis of the classical approach (Yukawa potential) and the proposed inertial interaction model based on temporal energy redistribution is conducted.

1. Introduction

1.1. Redistribution of Temporal Energy via Time Gradient

Time can be envisioned as a river that continuously flows from the past to the future. All matter floats along this flow, carrying temporal energy that sustains its movement through time.

However, this river's flow is not always uniform. In certain regions, eddies, whirlpools, and areas with varying flow speeds arise. If a space develops where time flows more slowly, matter naturally gravitates toward it—similar to how objects in water are drawn into areas with weaker currents.

This process generates nuclear forces. Inside the atomic nucleus, localized temporal eddies modify the speed of time relative to the surrounding space. These zones redistribute temporal energy, creating inertial forces that bind nucleons together.

1.2. Justification for the Connection Between the Time Gradient and Strong Interaction

Temporal Inertia

• Interacting particles tend to move along an optimal time flow.
• Local zones of slowed time within the nucleus create "gravitational traps," holding nucleons in place.

Redistribution of Energy

In relativistic systems, energy influences time flow:

[ T = T_0 \sqrt{1 - \frac{E}{mc^2}} ]

Slowing time inside the nucleus leads to a local redistribution of energy, which causes nucleon binding forces.

2. Traditional Numerical Calculation of Nuclear Interaction Forces

2.1. Yukawa Potential

Strong interaction between nucleons is described by the Yukawa potential:

[ V(r) = -V_0 \frac{e^{-r / r_0}}{r} ]

where:

• ( V_0 \approx 40 ) MeV — characteristic energy of nuclear interaction,
• ( r \approx 1.4 ) fm — distance between nucleons,
• ( r_0 \approx 1.4 ) fm — characteristic interaction radius.

To find the interaction force, the gradient of the potential must be taken, i.e., the derivative of ( V(r) ) with respect to ( r ):

[ F_{\text{nuclear}} = -\frac{dV}{dr} ]

Substituting the expression for ( V(r) ):

[ F_{\text{nuclear}} = \frac{V_0}{r{-r / r_0} \left( 1 + \frac{r}{r_0} \right) ]

After substituting numerical values and performing calculations, the interaction force turns out to be approximately ( 10^{13} ) N, confirming the classical approach.

3. Alternative Calculation of Nuclear Forces via Time Gradient

3.1. Energy and Its Influence on Time Flow

In the framework of General Relativity (GR), gravitational effects influence the rate of time flow, expressed through gravitational potential:

[ T = T_0 \sqrt{1 - \frac{2GM}{Rc^2}} ]

Moreover, in GR, all types of energy (kinetic, potential, internal) affect time dilation at the atomic level. To adapt this for assessing energy-induced time slowing, the gravitational potential can be replaced with the total energy inside the atom, yielding:

[ T = T_0 \sqrt{1 - \frac{E}{mc^2}} ]

where:

• ( E ) — total energy inside the atom,
• ( m ) — atomic mass,
• ( c ) — speed of light.

3.2. Total Energy in a Uranium Atom

Electron Binding Energy:

• Covalent bond energy: ( E_{\text{bond}} = 3.5 ) eV.
• Electron potential energy (7s): ( E_{\text{electron}} \approx 150 ) eV.

Electron Kinetic Energy:

• Outer electron velocity: ( v_{\text{out}} \approx 2.7 \times 10^6 ) m/s.
• Inner electron velocity: ( v_{\text{in}} \approx 1.5 \times 10^7 ) m/s.

3.3. Calculation of Time Dilation in Uranium's Nucleus

Dilation from Binding Energy:

[ \frac{E_{\text{bond}}}{mc^2} ]

Dilation from Electron Velocity:

For outer orbitals:

[ \frac{v_{\text{out}}2} \approx 8.1 \times 10^{-5} ]

For inner orbitals:

[ \frac{v_{\text{in}}2} \approx 2.5 \times 10^{-3} ]

Final Time Dilation:

[ T = T_0 (1 - 2.5 \times 10^{-3} - 8.1 \times 10^{-5} - 4.7 \times 10^{-10}) ]

[ T \approx T_0 (1 - 0.00258) ]

This means that time inside a uranium atom flows approximately 0.258% slower than outside.

3.4. Calculation of Nuclear Forces via Time Gradient

In the nucleus of heavy elements like uranium, time slowing can be calculated. Nucleon interactions depend on the local time gradient.

Formula for time gradient:

[ \nabla T = \frac{E_{\text{binding}}}{R_{\text{nucleus}} c^2} ]

where:

• ( E_{\text{binding}} = 7.6 ) MeV = ( 1.216 \times 10^{-12} ) J,
• ( R_{\text{nucleus}} = 7.4 ) fm = ( 7.4 \times 10^{-15} ) m,
• ( c = 3.0 \times 10^8 ) m/s.

Now calculating ( \nabla T ):

[ \nabla T = \frac{1.216 \times 10^{-12}}{(7.4 \times 10^{-15}) \times (9 \times 10^{16})} ]

[ \nabla T \approx 1.83 \times 10^{-15} \text{ s}^{-1} ]

Now calculating the force ( F_{\text{inertia}} ) via time gradient:

[ F_{\text{inertia}} = \eta \cdot \nabla T c^2 ]

With ( \eta \approx 1 ):

[ F_{\text{inertia}} \approx 1.65 \times 10^{13} \text{ N} ]

This confirms that the proposed method yields results consistent with the Yukawa potential calculations, suggesting that strong nuclear interactions can be modeled using time gradient variations.

4. Conclusions

• The time gradient can be used for nuclear force calculations.
• The proposed method provides quantitative results aligning with classical physics.
• Time gradients may explain strong interaction behavior and its variations in extreme conditions.
• Future research includes investigating time gradients in quantum interactions and particle stability.

5. References

1. Sitenco A.G., Tartakovsky V.K. Lectures on Nuclear Theory. — Kyiv: Naukova Dumka, 1972.
• A course of lectures on nuclear theory, including modern perspectives on atomic nucleus structure and nuclear interactions.
2. Deryuzhkova O.M. Physics of the Atomic Nucleus and Elementary Particles. — Moscow: Moscow State University Press, 2020.
• A practical guide for students covering key aspects of nuclear physics and elementary particles.
3. Glazkov V.N. Nuclear Physics 1: Nuclear Structure and Nuclear Reactions. — Moscow: Fizmatlit, 2019.
• An introduction to quantum physics and nuclear reactions, including proton and neutron interactions in the nucleus.
4. Misner C., Thorne K., Wheeler J. Gravitation. — Freeman, 1973.
• A fundamental work on gravitational theory, including gravitational time dilation.
5. Panek R. The Trouble with Gravity. — Corpus, 2020.
• A study of modern issues and perspectives on gravity in the context of fundamental interactions.
6. Alekseev S.O. Modern Theories of Gravity. — Moscow: Fizmatlit, 2014.
             An analysis of contemporary gravity models and their connection to quantum mechanics and nuclear physics.