Enhancement: Full Einstein-Matter Coupling Solver

[arcticoder]

Implement a self-consistent Einstein-matter coupling solver to replace prescribed potentials with dynamically determined solutions.

Motivation

Current implementation uses prescribed Rodal-style potentials. A full solver would:

• Ensure consistency between geometry and matter distribution
• Enable exploration of physically realizable configurations
• Provide stronger validation against known exact solutions

Technical Approach

1. Implement iterative ADM constraint solver
2. Add matter stress-energy tensor computation from prescribed sources
3. Couple potential φ evolution to Einstein equations
4. Validate against:
   • Schwarzschild metric (static limit)
   • Boosted Schwarzschild (kinematic check)
   • Known warp-drive solutions

Dependencies

• Robust PDE solver (consider FEniCS or custom finite-difference)
• Convergence criteria for constraint satisfaction
• Comprehensive test suite with known solutions

References

• Baumgarte & Shapiro (2010): Numerical Relativity
• Alcubierre (2008): Introduction to 3+1 Numerical Relativity

Priority: High (v0.2.0 target)
Complexity: High