Overview of Tensors and Fields

This is a summation of my thoughts on Tensors and fields in Temporal Physics. I feel I could compound this concept even further but it gives a fairly brawd conceptual starting point.
 
Field Definition

Fields can be viewed as a summation of amplitudes across dimensions. For example, the amplitude of temporal flows can be defined as:

F(x,y,z,τ) = ∫A(τ) dV

where $dV$ is a volume element across spatial dimensions, and $A(τ)$ is the amplitude of the temporal flow.

Tensors Representing Temporal Flows

Temporal Flow Tensor $T(τ)$

Captures amplitude and derivatives of temporal flow:

$$T(\tau) = \begin{bmatrix} \alpha_m \cdot \left(\frac{\partial \tau}{\partial t}\right) & 0 & 0 \\ 0 & \beta_m \cdot \frac{\partial^2 \tau}{\partial t^2} & \gamma_m \cdot \frac{\partial \tau}{\partial t} \\ \gamma_m \cdot \frac{\partial \tau}{\partial t} & \delta_m \cdot \frac{\partial^2 \tau}{\partial t^2} & \epsilon_m \cdot \frac{\partial^3 \tau}{\partial t^3} \end{bmatrix}$$

Curvature Tensor $K(τ)$

Relates curvature of temporal flow to its derivatives:

$$K(\tau) = \begin{bmatrix} \frac{\partial^2 \tau}{\partial t^2} & 0 \\ 0 & \frac{\partial^2 \tau}{\partial t^2} \end{bmatrix}$$

Stress-Energy Tensor $T_{\mu\nu}(τ)$

Connects mass-energy to temporal flows:

$$T_{\mu\nu}(\tau) = \begin{bmatrix} \alpha_m \cdot \left(\frac{\partial \tau}{\partial t}\right) & 0 & 0 \\ 0 & \beta_m \cdot \frac{\partial^2 \tau}{\partial t^2} & \gamma_m \cdot \frac{\partial \tau}{\partial t} \\ \gamma_m \cdot \frac{\partial \tau}{\partial t} & \delta_m \cdot \frac{\partial^2 \tau}{\partial t^2} & \epsilon_m \cdot \frac{\partial^3 \tau}{\partial t^3} \end{bmatrix}$$

Connections to Maxwell’s Equations

Field Tensor $F_{\mu\nu}$

The electromagnetic field tensor can be expressed as:

$$F_{\mu\nu} = \begin{bmatrix} 0 & -E_x & -E_y & -E_z \\ E_x & 0 & -B_z & B_y \\ E_y & B_z & 0 & -B_x \\ E_z & -B_y & B_x & 0 \end{bmatrix}$$

Relation to Stress-Energy Tensor

The stress-energy tensor for electromagnetic fields:

$$T_{\mu\nu} = F_{\mu\alpha} F_{\nu}^{\alpha} - \frac{1}{4} g_{\mu\nu} F_{\alpha\beta} F^{\alpha\beta}$$

Electric and Magnetic Field Sources

Components $γ_m \cdot \frac{\partial \tau}{\partial t}$ can act as sources of electric fields, while $κ_m \cdot \frac{\partial^2 \tau}{\partial t^2}$ can relate to magnetic fields.

Divergence and Curl Relations

From the tensors, divergence equations (Gauss's law and Ampère's law) can be derived as:

$$\nabla \cdot E = \frac{\rho}{\epsilon_0}, \quad \nabla \cdot B = 0$$
$$\nabla \times E = -\frac{\partial B}{\partial t}, \quad \nabla \times B = \mu_0 J + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}$$

Gravity and Temporal Flows

Gravity Tensor $G_{\mu\nu}(τ)$

Defines gravitational effects based on temporal flows:

$$G_{\mu\nu}(\tau) = \begin{bmatrix} \alpha_g \cdot \left(\frac{\partial \tau}{\partial t}\right) & 0 & 0 \\ 0 & \beta_g \cdot \frac{\partial^2 \tau}{\partial t^2} & \gamma_g \cdot \frac{\partial \tau}{\partial t} \\ \gamma_g \cdot \frac{\partial \tau}{\partial t} & \delta_g \cdot \frac{\partial^2 \tau}{\partial t^2} & \epsilon_g \cdot \frac{\partial^3 \tau}{\partial t^3} \end{bmatrix}$$

Additional Concepts

Distance as Temporal Flow

Consider distance as a function of time, suggesting that as distance increases, temporal flow also increases.

Location Tensor $L_{\mu\nu}$

Represents spatial coordinates and temporal flows:

$$L_{\mu\nu} = \begin{bmatrix} x_1 & x_2 & x_3 & \tau \\ 0 & 0 & 0 & \frac{\partial \tau}{\partial t} \\ 0 & 0 & \frac{\partial^2 \tau}{\partial t^2} & \frac{\partial^3 \tau}{\partial t^3} \end{bmatrix}$$

Information Density Tensor

Represents information saturation in a region:

$$I_{\mu\nu} = \rho[g_{\mu\nu} + \frac{1}{c^2 t^2} T_{\mu\nu}]$$

Visibility Function

Captures observability past singularities in time:

$$V(t) = \int L(t') \Phi(t') \left(1 + z(t')\right)^2 dt'$$

Temporal Flow Dynamics

Incorporates interactions of multiple flows:

$$F_{\text{total}}(t) = F_1(t) + F_2(t) + \Gamma(t)$$