Linearized Theory of Gravity (Temporal Physics)

 Linearized Theory of Gravity

In light of the temporal dependence of the metric tensor and the need for a dynamic approach, the metric perturbation in a weak field might be updated to:

ημν(t) =

[

1 + α₀ · t² 0 0

0 1 + α₁ · t² 0

0 0 1 + α₂ · t²

]

g_μν = η_μν + h_μν

Where:

h_μν = Perturbation influenced by τ(t)

The perturbation h_μν should reflect the time-dependent scaling. For a more accurate representation, the perturbation might be:

h_00 = -2 * (k * q1 * q2) / (r * (1 + α * t²))

Modified Potential

The Coulomb potential in the context of time-dependence is:

V(r, t) = (k * q1 * q2) / (r * (1 + α * t²))

The Newtonian gravitational potential ϕ is then:

ϕ(r, t) = (k * q1 * q2) / (r * (1 + α * t²))

Metric Perturbation

With the modified potential, the perturbation in the metric tensor due to the gravitational field is:

h_00 = -2 * (k * q1 * q2) / (r * (1 + α * t²))

Wave Propagation and Polarization

The linearized Einstein field equations for the perturbation now include the time-dependent term:

□h_μν = -16πG * T_μν

With the perturbation:

□h_00 = -16πG * (k * q1 * q2) / (r * (1 + α * t²))

Where:

□ = ∂² / ∂t² - ∇²

Effects on Polarization

The time-dependent scaling affects how gravitational waves propagate, including their polarizations. The perturbations h_+ and h_× should be adjusted to account for:

h_+ and h_× influenced by 1 / (1 + α * t²)

Modified Lagrangian and Hamiltonian

Lagrangian Density:

L = 1/2 * g^μν * ∂_μ φ * ∂_ν φ - V(φ)

Hamiltonian Density:

H = 1/2 * g_μν * π^μ * π^ν + 1/2 * g^μν * ∂_μ φ * ∂_ν φ + V(φ)

Interaction Terms:

Given that constants like G and k are functions of dimensionality D(t):

L_int = g(D(t)) / 4 * φ^4

Where:

g(D(t)) = G₀ + γ * D(t)

Klein-Gordon Equation:

With the dynamic metric:

(1 / sqrt(1 + α * t²)) * ∂^μ ∂_μ φ - m² * φ = 0

consider

(1 / sqrt(c * t²)) * ∂^μ ∂_μ φ - m² * φ = 0

Coupling Constant for Scalar Field Theory:

λ = λ₀ * (1 + δ * D(t))

Revised Equations for Perturbation Analysis

Temporal and Spatial Equations:

Temporal Function f(t):

f(t) = λ² / (-16πG * k * q1 * q2)

Spatial Function f(r):

f(r) = n * (n + 1) / (-16πG * k * q1 * q2) * r^(-n - 2)

Dimensionality Adjustment:

n * (n + 1) / r^(n + 2) = -16π * (G₀ + γ * D(t)) * (k₀ + δ * D(t)) * q1 * q2 * r^(-n - 2)