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Quadratic Gravity — S[φ] = ∫ d⁴x √(-g) [ R/(16πG) + ℒ + αR² + βRμνRμν ]
The Action
S[φ] = ∫ d⁴x √(-g) [
  R/(16πG) + ℒ(φ,∇φ)
  + αR² + βRμνRμν ]
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 (RμνRμν 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
SEH = (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 = 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
SRic = β ∫ RμνRμν √(-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
d²xμ/dτ² + Γμαβ dxα/dτ dxβ/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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