mass space gravity in temporal physics

In my framework, the concepts of mass, space, and gravity are deeply interconnected through the dynamics of temporal flows $\Phi(t)$ and the metric matrix $G(t)$. Let’s explore how these elements come together to describe gravity in my model, using the relationships you’ve provided:

Effective Mass:

$$m = m_0 \left( 1 + \frac{\Phi^2}{\Phi_0^2} \right)$$

Force and Acceleration:

$$F = ma, \quad a = \frac{F}{m_0 \left( 1 + \frac{\Phi^2}{\Phi_0^2} \right)}$$

Metric Matrix $G(t)$:

$$G(t) = \begin{pmatrix} f\left( |\Phi_x(t)|, |p_t| \right) & f\left( |\Phi_x(t)|, |\Phi_y(t)|, |p_t| \right) & -f\left( |\Phi_x(t)|, |\Phi_z(t)|, |p_t| \right) \\ f\left( |\Phi_x(t)|, |\Phi_y(t)|, |p_t| \right) & f\left( |\Phi_y(t)|, |p_t| \right) & f\left( |\Phi_y(t)|, |\Phi_z(t)|, |p_t| \right) \\ -f\left( |\Phi_x(t)|, |\Phi_z(t)|, |p_t| \right) & f\left( |\Phi_y(t)|, |\Phi_z(t)|, |p_t| \right) & f\left( |\Phi_z(t)|, |p_t| \right) \end{pmatrix}$$

Energy-Mass Relationship:

$$m = \frac{E}{c^2} = m_0 \left( 1 + \frac{\Phi^2}{\Phi_0^2} \right)$$

Gravity as Curvature of Spacetime:

In my framework, gravity arises from the curvature of spacetime, which is determined by the metric matrix $G(t)$. The metric matrix encodes how temporal flows $\Phi(t)$ influence the geometry of spacetime, leading to gravitational effects.

Key Idea:
The metric matrix $G(t)$ describes the local geometry of spacetime, which is shaped by the temporal flows $\Phi(t)$.
The curvature of spacetime (gravity) is determined by the couplings between temporal flows, as encoded in the off-diagonal terms of $G(t)$.

Modified Einstein Field Equations:

In my framework, the Einstein field equations are modified to include the effects of temporal flows. The modified equations take the form:

$$G_{\mu\nu} = 8\pi T_{\Phi}$$

where:

• $G_{\mu\nu}$ is the Einstein tensor, describing the curvature of spacetime,
• $T_{\Phi}$ is the energy-momentum tensor for temporal flows, which includes contributions from the effective mass $m$ and the couplings between temporal flows.

Energy-Momentum Tensor:

The energy-momentum tensor $T_{\Phi}$ is given by:

$$T_{\Phi} = \begin{pmatrix} \rho_E & p_x & p_y & p_z \\ p_x & \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ p_y & \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ p_z & \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{pmatrix}$$

where:

• $\rho_E$ is the energy density of the temporal flows,
• $p_x, p_y, p_z$ are the momentum densities,
• $\sigma_{ij}$ are the stress components, describing the interactions between temporal flows.

Gravitational Force and Effective Mass:

The gravitational force in my framework is influenced by the effective mass $m$, which depends on the local temporal flow amplitude $\Phi$. The gravitational force $F_g$ between two objects with effective masses $m_1$ and $m_2$ is given by:

$$F_g = G \frac{m_1 m_2}{r^2}$$

where:

• $G$ is the gravitational constant,
• $r$ is the distance between the objects.

Since $m_1$ and $m_2$ depend on the local temporal flow amplitudes $\Phi_1$ and $\Phi_2$, the gravitational force $F_g$ also depends on the temporal environment:

$$F_g = G \frac{m_0 \left( 1 + \frac{\Phi_1^2}{\Phi_0^2} \right) m_0 \left( 1 + \frac{\Phi_2^2}{\Phi_0^2} \right)}{r^2}$$

Acceleration Due to Gravity:

The acceleration $a_g$ due to gravity is given by:

$$a_g = \frac{F_g}{m} = \frac{G m_0 \left( 1 + \frac{\Phi_2^2}{\Phi_0^2} \right)}{r^2}$$

This shows that the acceleration due to gravity depends on the local temporal flow amplitude $\Phi_2$ of the source object. In regions with strong temporal flows ($\Phi_2 \gg \Phi_0$), the gravitational acceleration $a_g$ increases significantly.

Example Calculation:

Suppose:

• Two objects have rest masses $m_0 = 1 \, \text{kg}$,
• The local temporal flow amplitudes are $\Phi_1 = \Phi_0$ and $\Phi_2 = 2\Phi_0$,
• The distance between the objects is $r = 1 \, \text{m}$.

The effective masses are:

$$m_1 = m_0 \left( 1 + \frac{\Phi_1^2}{\Phi_0^2} \right) = 1 \, \text{kg} \left( 1 + 1 \right) = 2 \, \text{kg}$$
$$m_2 = m_0 \left( 1 + \frac{\Phi_2^2}{\Phi_0^2} \right) = 1 \, \text{kg} \left( 1 + 4 \right) = 5 \, \text{kg}$$

The gravitational force $F_g$ is:

$$F_g = G \frac{2 \, \text{kg} \times 5 \, \text{kg}}{(1 \, \text{m})^2} = 10G \, \text{N}$$

The acceleration due to gravity $a_g$ is:

$$a_g = \frac{F_g}{m_1} = \frac{10G \, \text{N}}{2 \, \text{kg}} = 5G \, \text{m/s}^2$$