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Gravitational Wave: Regge Calculus

This example simulates gravitational wave propagation on a discrete spacetime mesh using Regge Calculus.

How to Run

cargo run -p physics_examples --example gravitational_wave

Engineering Value

Regge Calculus is useful for:

  • Numerical Relativity: Simulating black hole mergers, gravitational waves
  • Quantum Gravity: Discrete approaches to spacetime quantization
  • Mesh-Based Physics: Games, simulations with dynamic geometry

This example shows how calculate_ricci_curvature computes spacetime curvature on a simplicial mesh.

Physics Background

Regge Calculus

Instead of continuous curvature, Regge Calculus uses:

  • Simplicial Complex: Spacetime as triangles/tetrahedra
  • Edge Lengths: Metric encoded in edge lengths
  • Deficit Angles: Curvature concentrated at "bones" (n-2 simplices)

Deficit Angle

For a 2D surface, curvature at a vertex is:

δ = 2π - Σ(angles at vertex)

If angles sum to less than 2π → positive curvature (like a sphere).

Gravitational Waves

Metric perturbations propagate as waves:

  1. Curvature creates stress
  2. Stress changes edge lengths
  3. Changed lengths create new curvature
  4. Wave propagates

Geometry Setup

        1 ─── 2
       / \ / \
      6 ─ 0 ─ 3
       \ / \ /
        5 ─── 4

Hexagonal mesh with internal vertex 0

Output Interpretation

[t=0] Center Curvature: +0.0000  ← Flat spacetime
[t=3] Center Curvature: +0.4311  ← Wave peaks
[t=6] Center Curvature: -0.3545  ← Wave troughs

Oscillating curvature represents the gravitational wave.

Adapting This Example

  1. 3D mesh: Use tetrahedra for full 3D+1 simulation
  2. Different topologies: Try torus, sphere, or hyperbolic meshes
  3. Source terms: Add matter/energy sources
  4. Wave detection: Implement "LIGO-like" detector nodes

Key APIs Used

  • SimplicialComplexBuilder - Construct discrete spacetime
  • ReggeGeometry::calculate_ricci_curvature() - Deficit angles
  • CausalTensor - Edge length storage
  • BaseTopology trait - Mesh navigation