Name: Alfred IDE
Author: GoSiteMe

The Action

S[φ] = ∫ d⁴x √(-g) [
R/(16πG) + ℒ(φ,∇φ)
+ αR² + βR_μνR^μν ]

The complete quadratic gravity action with matter. Each term has its own color in the visualization.

Parameters

Mass (M) 3.00

α (R² coupling) 0.50

β (R_μνR^μν coupling) 0.20

φ₀ (scalar field) 1.00

Speed 1.0×

Presets

Display

Spacetime Grid

Curvature Heatmap

Scalar Field φ

Action Formula

Physics

Einstein-Hilbert

S_EH = (1/16πG) ∫ R √(-g) d⁴x

Standard general relativity. The Ricci scalar R encodes spacetime curvature from mass-energy. This is what bends the grid.

Starobinsky R²

S_R² = α ∫ R² √(-g) d⁴x

Drives cosmic inflation — the rapid expansion of the early universe. Best-fit to Planck CMB data. The golden ripples expanding outward.

Ricci Tensor Squared

S_Ric = β ∫ R_μνR^μν √(-g) d⁴x

Makes gravity renormalizable at the quantum level (Stelle, 1977). Introduces a massive spin-2 graviton. The green ripples emanating from the center.

Scalar Field

ℒ = ½(∂_μφ)(∂^μφ) - ½m²φ²

The matter content — a Klein-Gordon scalar field φ oscillating in spacetime. The purple glow pulsing around the mass.

Modified Potential

Φ(r) = -GM/r [1 + ⅓e^-m₀r - ⁴⁄₃e^-m₂r]

Yukawa corrections to Newton's gravity from the quadratic terms. Particles follow orbits shaped by this potential.

Geodesic Equation

d²x^μ/dτ² + Γ^μ_αβ dx^α/dτ dx^β/dτ = 0

The paths of free-falling particles through curved spacetime. Every orbiting dot follows this equation.

Built by GoSiteMe
Danny William Perez
S[φ] = ∫ d⁴x √(-g) [ R/(16πG) + ℒ + αR² + βRμνRμν ]

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