Simplifying the Temporal spacetime metric. (Edited)

Temporal spacetime metric. Scenario: Gravitational Field with Increasing Effect Suppose τ(t) represents a gravitational field effect that increases over time, such that τ(t) = g(t) = t². The generalized metric tensor would be: (edit removed the temproal aspect from the old model, minkowski to temporal) ημν(t) = [ 1 + α₀ · t² 0 0 0 1 + α₁ · t² 0 0 0 1 + α₂ · t² ] The spacetime interval is: ds² = (1 + α₀ · t²) dx² + (1 + α₁ · t²) dy² + (1 + α₂ · t²) dz² In this case, as time t increases, the spatial distances (dx², dy², dz²) are scaled more significantly due to the time-dependent factors, than in Minkowski space. This reveals new insights into how gravitational fields or other temporal factors alter spatial dimensions, which is not captured in the standard Minkowski spacetime model. To see how gravity is described by my metric, let’s examine the spacetime interval and its implications. ds² = (1 + α₀ · τ(t)) dx² + (1 + α₁ · τ(t)) dy² + (1 + α₂ · τ(t)) dz² General temporal Component: - (1 + α₀₁₂ · τ(t)) dt² This term shows how the passage of time is influenced by τ(t). If τ(t) increases, this term decreases, implying that time intervals get stretched due to the gravitational field’s effect. Spatial Components: (1 + α₀ · τ(t)) dx², (1 + α₁ · τ(t)) dy², (1 + α₂ · τ(t)) dz² These terms show how spatial distances are scaled by τ(t). As τ(t) changes, spatial distances are adjusted, indicating how gravity affects space itself. where 1+α_i⋅τ(t)≤1+α_i⋅c resultng in ds^2 =(1+α_0⋅c)dt^2+(1+α_1⋅c)dx^2+(1+α_2⋅c)dy^2 Given that τ(t) is bounded by the speed of light c, the metric components that depend on τ(t) will also be bounded: Metric Tensor: The metric tensor is represented as: η_μν(t) = diag( 1 + α₀ · τ(t), 1 + α₁ · τ(t), ...) If τ(t) is less than or equal to c, then: 1 + αᵢ · τ(t) will be less than or equal to 1 + αᵢ · c Thus, the components of the metric tensor are bounded and cannot become infinite. Spacetime Interval: The spacetime interval is given by: ds² = (1 + α₀ · τ(t)) dt² + (1 + α₁ · τ(t)) dx² + (1 + α₂ · τ(t)) dy² η_μν(t) = diag( 1 + α_0 · c, 1 + α_1 · c, ...) ds² =(1 + α₀ · c) dt² + (1 + α₁ · c) dx² + (1 + α₂ · c) dy² τ(t)=g(t)=t^2 η_μν(t) = diag( 1 + α₀ · t^2, 1 + α₁ · t^2, ...) spacial extraction of x y z If τ(t) = t², the equations become: For dx²: dx² = [(1 + α₀ · t²) dt² - (1 + α₂ · t²) dy² - (1 + α₃ · t²) dz²] / [1 + α₀ · t²] For dy²: dy² = [(1 + α₀ · t²) dt² - (1 + α₁ · t²) dx² - (1 + α₃ · t²) dz²] / [1 + α₂ · t²] For dz²: dz² = [(1 + α₀ · t²) dt² - (1 + α₁ · t²) dx² - (1 + α₂ · t²) dy²] / [1 + α₃ · t²] Gravitational constant G′∝ 1/α_i ​τ(t)=m⋅t^2 Consdiering the Hamiltonian H = 1/2 [πx²/(1 + α₀ · t²) + πy²/(1 + α₁ · t²) + πz²)] M(t) in a form similar to n_mv(t), but focused on interactions Further onto Dirac ψantiparticle(φ) = ψ(-φ) γ~0=γ 0 𝛾~𝑖=𝛾𝑖⋅(1+𝛼_𝑖⋅𝑡^2)^1/2 Incorporate the modified gamma matrices into the Dirac equation: (iℏ * Γ~₍μ₎ * ∂₍μ₎ - m * c + Φ(φ)) * ψ(φ) = 0 Where Γ~₍μ₎ represents the modified gamma matrices that include the effects of time-dependent spatial coordinates. Example Application: Let’s apply these concepts to an electron: Dirac Equation with Temporal Dynamics: [iℏ * (γ₀ * ∂/∂t + Σᵢ=1³ γᵢ * ∂/∂xᵢ) - mₑ * c + Φₑ(φ)] * ψₑ(φ) = 0 Here, γ₀ and γᵢ include the time-dependent factors. Temporal Potential: The term Φₑ(φ) represents the potential related to temporal flows and spatial coordinates. Wavefunction Adaptation: The wavefunction ψₑ(φ) will be influenced by these dynamically changing gamma matrices, reflecting how particles behave in a spacetime where spatial dimensions are not static but evolve with time. The partial derivatives with respect to time (∂/∂t) and spatial coordinates (∂/∂x) need to account for the spatial dependence of the time component, and vice versa. Incorporating the Metric Tensor: We adapt the Dirac equation by incorporating the metric tensor directly into the differential operators. If we denote the modified partial derivatives by ∂̃μ, the Dirac equation becomes: [iħ Γμ ∂̃μ - mc + Φ(φ)] ψ(φ) = 0 where ∂̃μ accounts for the dynamic metric: ∂̃μ = (∂/∂xμ) * (1 + αi(t) * τ(t)) Example Application Using this in the context of an electron, the modified Dirac equation is: [iħ (Γ₀ ∂̃t + Γ₁ ∂̃x + Γ₂ ∂̃y) - me c + Φe(φ)] ψe(φ) = 0 Here: Γ₀, Γ₁, and Γ₂ include the effects of the dynamic metric tensor. ∂̃t and ∂̃x reflect the adjusted partial derivatives considering the new metric. Kinetic Energy: The kinetic energy might be expressed as: K = (1/2) m v² * f(t) where f(t) is a function derived from the temporal dynamics and spacetime metric. Momentum: The momentum could be adjusted as: p = m v * g(t) where g(t) accounts for the dynamic changes in the spacetime and temporal flows. where g(t) = 1/(1+α+i⋅τ(t)) ​Δ= 1/2λ(ψ^2−c^2) λ is a coupling constant related to the interaction strength. ψ represents the dynamic wavefunction. the Lagrangian L = 1/2 * [(1 + α₁ ⋅ m ⋅ t²) * (∂ψ/∂x)² - (1 + α₂ ⋅ m ⋅ t²) * (∂ψ/∂y)² - (1 + α₃ ⋅ m ⋅ t²) * (∂ψ/∂z)²] - λψ² - λc²)