Investigating the hypothesis that gravitational force emerges from quantum entanglement between matter with shared primordial origins
Traditional gravity assumes universal attraction between all masses. Our hypothesis suggests gravity emerges from quantum entanglement correlations, making it dependent on shared primordial origins.
Standard Gravity (Scalar):
$$F = G \frac{m_1 m_2}{r^2}$$
Entanglement Gravity (Scalar):
$$F = G_{eff}(E_{1,2}) \frac{m_1 m_2}{r^2}$$
where $G_{eff}(E_{1,2}) = G_0 \sqrt{E_1 \cdot E_2}$ depends on entanglement correlation $E_{1,2}$ between masses
Object positions:
$$\vec{r_1} = x_1\hat{i} + y_1\hat{j} + z_1\hat{k}$$
$$\vec{r_2} = x_2\hat{i} + y_2\hat{j} + z_2\hat{k}$$
Relative position vector:
$$\vec{r_{12}} = \vec{r_2} - \vec{r_1} = (x_2-x_1)\hat{i} + (y_2-y_1)\hat{j} + (z_2-z_1)\hat{k}$$
Distance (magnitude):
$$r = |\vec{r_{12}}| = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$$
Unit vector:
$$\hat{r_{12}} = \frac{\vec{r_{12}}}{|\vec{r_{12}}|} = \frac{(x_2-x_1)\hat{i} + (y_2-y_1)\hat{j} + (z_2-z_1)\hat{k}}{r}$$
Gravitational force on object 1:
$$\vec{F_1} = G_{eff} \frac{m_1 m_2}{r^2} \hat{r_{12}} = G_{eff} \frac{m_1 m_2}{r^3} \vec{r_{12}}$$
Gravitational force on object 2:
$$\vec{F_2} = -\vec{F_1} = -G_{eff} \frac{m_1 m_2}{r^3} \vec{r_{12}}$$
Acceleration of object 1:
$$\vec{a_1} = \frac{\vec{F_1}}{m_1} = G_{eff} \frac{m_2}{r^3} \vec{r_{12}}$$
Acceleration of object 2:
$$\vec{a_2} = \frac{\vec{F_2}}{m_2} = -G_{eff} \frac{m_1}{r^3} \vec{r_{12}}$$
For object 1:
$$a_{1x} = G_{eff} \frac{m_2(x_2-x_1)}{r^3}, \quad a_{1y} = G_{eff} \frac{m_2(y_2-y_1)}{r^3}$$
For object 2:
$$a_{2x} = G_{eff} \frac{m_1(x_1-x_2)}{r^3}, \quad a_{2y} = G_{eff} \frac{m_1(y_1-y_2)}{r^3}$$
Effective gravitational constant:
$$G_{eff} = G_0 \times \begin{cases} 1.0 & \text{if objects are entangled} \\ \alpha < 1 & \text{if objects are not entangled} \end{cases}$$
Orbital velocity:
$$v = \sqrt{\frac{G_{eff}(m_1 + m_2)}{d}}$$
Distances from barycenter:
$$r_1 = \frac{m_2 d}{m_1 + m_2}, \quad r_2 = \frac{m_1 d}{m_1 + m_2}$$
Individual velocities:
$$v_1 = v \frac{m_2}{m_1 + m_2}, \quad v_2 = v \frac{m_1}{m_1 + m_2}$$
Advanced two-body gravitational simulation with 4th-order Runge-Kutta integration and entanglement effects.
Gravity emerges when photons split into correlated pairs, creating quantum correlations that manifest as attractive forces between regions where split photons are absorbed.
Photon Splitting Process:
$$\gamma(\hbar\omega) \rightarrow \gamma_1(\hbar\omega/2) + \gamma_2(\hbar\omega/2)$$
Correlation Creation Rate:
$$\Gamma_{split} = \frac{\alpha^2 c}{a_0} \cdot P_{split}$$
where $P_{split} \sim \frac{\hbar}{E_{photon} \tau_{split}}$
Step 1: Hydrogen atom photon emission rate
$$\Gamma = \frac{\alpha^2 c}{a_0} = 6.27 \times 10^8 \text{ s}^{-1}$$
Step 2: Splitting probability per photon
$$P_{split} = \frac{\hbar}{E_{photon}} \cdot \frac{1}{\tau_{Planck}} \sim 10^{-43}$$
Step 3: Correlation density in space
$$n_{corr}(r) = \frac{N_1 N_2}{4\pi r^2} \cdot \Gamma^2 \cdot P_{split}^2 \cdot \tau_{coherence}$$
Step 4: Gravitational constant derivation
$$G = \frac{\hbar^2 c}{\ell_P^2 m_H^2 m_e a_0} \cdot f(\alpha) \approx 6.67 \times 10^{-11} \text{ m}^3\text{kg}^{-1}\text{s}^{-2}$$
Laboratory Tests:
Mathematical Extensions:
$$\frac{d^2G}{dt^2} = \frac{\partial}{\partial t}\left(\frac{\hbar \Gamma_{split}}{m_H^2 c}\right)$$
Matter from different primordial origins may be gravitationally "private" - lacking mutual attraction while maintaining self-attraction within each family.
Entanglement Family Classification:
$$G_{AB} = G_0 \cdot \delta_{family(A),family(B)}$$
Mixed Material Test:
$$F_{net} = \sum_{i \in A, j \in A} F_{ij} + \sum_{i \in B, j \in B} F_{ij} + \underbrace{\sum_{i \in A, j \in B} 0}_{no\ cross\ attraction}$$
Step 1: Define entanglement genealogy function
$$\mathcal{E}(A,B) = \begin{cases} 1 & \text{if A and B share primordial origin} \\ 0 & \text{if A and B from different origins} \end{cases}$$
Step 2: Gravitational coupling matrix
$$G_{ij} = G_0 \sqrt{\mathcal{E}(i,family) \cdot \mathcal{E}(j,family)}$$
Step 3: Mixing separation prediction
If materials A and B are mixed, separation time scale:
$$\tau_{sep} = \sqrt{\frac{6\pi\eta r^3}{G_0 \rho_{self} m_{particle}}}$$
Step 4: Observable consequences
$$\frac{d\vec{r}_{AB}}{dt^2} = \mathcal{E}(A,B) \cdot G \frac{M_A M_B}{|\vec{r}_{AB}|^3}\vec{r}_{AB}$$
Cosmological Observations:
Theoretical Implications:
$$S_{universe} = \sum_{families} S_{family} + \underbrace{0}_{no\ cross\ correlations}$$
At galactic scales, gravitational privacy could explain dark matter effects. Galaxies from different entanglement families would appear to have missing gravitational interactions.
Apparent Missing Mass:
$$M_{missing} = M_{total} \cdot (1 - \mathcal{E}_{cross})$$
Galaxy Collision Dynamics:
$$\vec{F}_{galaxy1 \rightarrow galaxy2} = \mathcal{E}(g1,g2) \cdot G \frac{M_1 M_2}{r^2}\hat{r}$$
Step 1: Observational velocity dispersion
$$\sigma_v^2 = \frac{GM_{visible}}{R} \cdot f_{entanglement}$$
where $f_{entanglement} < 1$ for mixed galaxy clusters
Step 2: Dark matter interpretation
$$M_{dark} = M_{visible} \cdot \left(\frac{1}{f_{entanglement}} - 1\right)$$
Step 3: Collision cross-section
$$\sigma_{collision} = \pi R^2 \cdot \mathcal{E}(galaxy1, galaxy2)$$
Step 4: Large-scale structure prediction
$$\xi(r) = \xi_0 \left(\frac{r}{r_0}\right)^{-\gamma} \cdot \langle\mathcal{E}\rangle_{families}$$
Observational Targets:
Modified Dynamics:
$$\frac{d^2\vec{r}}{dt^2} = -\nabla\Phi_{visible} - \nabla\Phi_{entangled}$$
where $\Phi_{entangled}$ only couples to same-family matter
In the early universe, all matter existed as photons in thermal equilibrium. This creates a universal entanglement ancestry for all matter within our cosmic horizon, explaining local gravitational universality.
Photon-Matter Transition:
$$\gamma + \gamma \leftrightarrow e^+ + e^- \text{ at } T > 10^{10} \text{ K}$$
Entanglement Inheritance:
$$S_{matter}(t) = S_{photons}(t_0) \cdot \eta_{conservation}$$
Horizon-Limited Entanglement:
$$\mathcal{E}(A,B) = \Theta(c \cdot t_{universe} - |\vec{r}_A - \vec{r}_B|)$$
Step 1: Photon number density in early universe
$$n_\gamma = \frac{2\zeta(3)}{\pi^2} \left(\frac{k_B T}{\hbar c}\right)^3 \approx 20 \left(\frac{T}{2.7K}\right)^3 \text{ cm}^{-3}$$
Step 2: Entanglement entropy per photon
$$S_{photon} = k_B \ln(\text{accessible states}) = k_B \ln\left(\frac{4\pi c^3}{3h^3}t^3\right)$$
Step 3: Matter condensation preserves correlations
$$\rho_{corr,matter} = \rho_{corr,photons} \cdot \frac{\eta_{baryon}}{n_\gamma} \cdot f_{preservation}$$
Step 4: Gravitational constant from primordial entropy
$$G = \frac{\hbar c^3}{M_P^2} \cdot \frac{S_{primordial}}{k_B N_{horizon}}$$
Cosmological Predictions:
Fundamental Physics:
$$H^2 = \frac{8\pi G}{3}\rho_{total} \cdot \langle\mathcal{E}^2\rangle_{cosmic}$$
where entanglement affects the expansion rate itself