4D Regge calculus with U(1) gauge field for Einstein-Maxwell theory.

Discretizes general relativity on a simplicial complex (Regge calculus) coupled to electromagnetism via a U(1) lattice gauge field. The combined action S = S_R + α·S_M is differentiable w.r.t. all degrees of freedom:

• Edge lengths {l_e}: encode the metric (gravitational DOF)
• Edge phases {θ_e}: encode the U(1) gauge field (electromagnetic DOF)

The coupling between gravity and EM enters through the metric weights in the Maxwell action, which depend on the edge lengths.

Architecture

• complex: Simplicial complex data structure with full incidence relations
• geometry: Cayley-Menger determinants, areas, volumes, dihedral angles
• regge: Regge action S_R = Σ A_t δ_t and its gradient (Schläfli identity)
• gauge: U(1) gauge field, field strengths, Maxwell action on curved background
• action: Combined Einstein-Maxwell action, gradients, symmetry search utilities
• mesh: Mesh generation (flat hypercubic, Reissner-Nordström deformation)

Example

use phyz_regge::{mesh, action::{Fields, ActionParams, einstein_maxwell_action}};

// Flat spacetime on a 2×2×2×2 periodic lattice.
let (complex, lengths) = mesh::flat_hypercubic(2, 1.0);
let phases = vec![0.0; complex.n_edges()];

let fields = Fields::new(lengths, phases);
let params = ActionParams::default();
let s = einstein_maxwell_action(&complex, &fields, &params);

// Flat vacuum → zero action.
assert!(s.abs() < 1e-8);