# BSSN Z4c Implementation Pathway **Status (2026-05-15):** First cut implemented and validated. Z4c is wired into the CPU solver (Theta field in `state.rs`, d_t Theta plus the two coupling terms in `rhs.rs`, kappa_1/kappa_2 in `params.rs`) and the GPU path. Validated bounded to 7 light crossings of the gauge wave; see [bssn_validation_results.md](bssn_validation_results.md) section 5. Remaining: complete the Z4c equation set (a few d_t A-tilde and d_t chi terms are simplified), then the kappa_1/kappa_2 sweep to reach the full 1000-crossing Apples-with-Apples horizon. The plan below is the spec for that completion work. **Why:** Vacuum BSSN's bounded-evolution horizon is ~5 light crossings (literature-standard; reproduced and validated by us). Beyond that, the canonical formulation-level constraint-violating eigenmode appears. Z4c is the standard NR-community fix; it converts non-propagating constraint drift into radiative decay by promoting the Hamiltonian and momentum constraints to dynamical fields with their own propagation + damping. **Reference:** Bernuzzi & Hilditch, *"Constraint violation in free evolution schemes: Comparing BSSNOK with a conformal decomposition of Z4"*, Phys. Rev. D 81, 084003 (2010); Hilditch et al., *"Compact binary evolutions with the Z4c formulation"*, Phys. Rev. D 88, 084057 (2013). --- ## 1. What changes from BSSN to Z4c Z4c adds **one scalar field** () and **modifies the existing K and \tilde{} evolution equations**. No new spatial dimensions, no new tensors, no new gauge variables. ### New evolution variable - `theta: Vec` per grid point (one scalar). Goes into `Fields` next to `chi` and `k_trace`. ### Modified equations (vs current vacuum BSSN) **_t K:** ``` _t K = _t K|_BSSN + (1 ) ``` The (1) term is the constraint-propagation feedback. 0.020.1, 01. **_t \tilde{}:** ``` _t \tilde{} = _t \tilde{}|_BSSN 2 \tilde{Z} + (advection of \tilde{Z}) ``` Where \tilde{Z} is the conformal momentum-constraint auxiliary, often computed as `\tilde{Z} = (1/2) \tilde{} (\tilde{}_geometric \tilde{}_evolved)` a measure of momentum-constraint violation. **_t (new equation):** ``` _t = /2 (R + (2/3)K _ij ^ij 2 K ) (2 + ) + ^k _k ``` The first parenthesized term is exactly the local Hamiltonian constraint H is forced to track H. The term damps (and through , damps H). **_t :** unchanged (no coupling) **_t \tilde{}_ij:** unchanged **_t _ij:** unchanged **_t / / B:** unchanged (gauge sector untouched) ### Algebraic constraint - After every RK4 substep, enforce `_clean = ` (no projection needed) it's a free dynamical scalar. --- ## 2. Implementation plan (~200300 lines of Rust) ### 2.1 `state.rs` (30 lines) - Add `theta: Vec` to `Fields`. - Update `new`, `zeros_like`, `axpby`, `rk_stage`, `rk4_combine` to handle . - No change to `enforce_algebraic` ( has no projection constraint). ### 2.2 `params.rs` (10 lines) - Add `kappa_1: f64` (default 0.02; standard NR value 0.020.10). - Add `kappa_2: f64` (default 0.0; range 02). - Default `formulation: "bssn"` switches to `"z4c"` to enable. ### 2.3 `initial_data.rs` (10 lines) - Initialize ` = 0` for all initial-data presets (consistent with H = 0 at t=0 if initial data is constraint-satisfying). ### 2.4 `rhs.rs` (100 lines) - Compute contributions to _t K (one term). - Compute \tilde{Z} at each grid point (uses the geometric-vs-evolved \tilde{} difference) contributes to _t \tilde{}. - Add _t equation as a new RHS block (uses already-computed `r_scalar`, `kt`, `a_sq`, `alpha`, `advect_chi`-style advection). - The existing `kappa_h` Hamiltonian damping in _t K is **superseded** by Z4c's (1) term remove or zero it when formulation=="z4c". ### 2.5 `engine.rs` (10 lines) - Thread , through all four RK4 calls. - Initialize from initial data. ### 2.6 `compute_constraints` (5 lines) - Add the \tilde{Z} momentum-constraint diagnostic alongside H. ### 2.7 `compute_ham_field` (no change) - Still useful as a standalone diagnostic; H = _t source term, easy to extract. --- ## 3. Validation plan for Z4c (the test that earns "research-grade asymptotic") Once Z4c is implemented: 1. **Re-run the full validation suite** in `bssn_validation_results.md` with `formulation="z4c"`. The first three gates should still pass at the same numbers (Z4c reduces to BSSN when =0 and =0). 2. **Long-time gauge wave to 1000 light crossings** the gold-standard Apples-with-Apples test. Bernuzzi-Hilditch 2010 demonstrated Z4c is stable for this. Acceptance: `||H||_L2` bounded over the full run, no slicing collapse, no NaN. 3. **Polarized Gowdy T** (the 5 test deferred from the original plan). Z4c is what makes Gowdy work for collapsing-direction evolutions (the hard part). 4. **Robust stability test** (random small noise on Minkowski) Z4c should kill the noise within a few crossings; pure BSSN doesn't. 5. **Resolution scaling regression** should still be O(N) ( is a single scalar; per-step cost increases by maybe 5%). --- ## 4. Effort estimate - **Implementation:** 12 focused days for a working first cut. Most of the work is careful translation of the equations into the existing `compute_rhs` structure + adding the field everywhere `chi` is referenced. - **Tuning + validation:** 1 day to find the , sweet spot for the gauge wave + run the full validation suite + write the updated validation document. - **Total:** 23 days to claim "asymptotic-stability research-grade vacuum BSSN/Z4c on the GMDBS toroid." --- ## 5. What this unblocks - Long-time vacuum evolutions (gauge wave, Gowdy T, Brill waves) at full literature horizon (~1000 crossings). - Initial step toward strong-field evolutions: with Z4c stable, the next pieces (puncture initial data, AH finder, extraction at periodic boundary or matched outer boundary) become plumbing rather than research. - The Apples-with-Apples full battery becomes attainable as a single cargo-test run. - Citable claim: "Z4c-grade vacuum NR on a periodic substrate." That's the line that gets reviewers to take the toroidal substrate seriously. --- ## 6. Why we're shipping vacuum BSSN first instead of going straight to Z4c The current solver passes the convergence + scaling + bounded-stability gates with literature-matching numbers. That's a real, verifiable, citable artifact. Z4c is the obvious next step but it's the kind of work that should land on a fully-validated foundation not as the first thing built. We did it in the right order: vacuum BSSN validated, gap diagnosed and pinned to the canonical Z4c-motivating mode, Z4c scoped as the next phase. This document is the bridge between today's `bssn_validation_results.md` and the next phase's expanded validation. When that next phase lands, this doc gets updated to point at the new validation results, and the BSSN map's Phase B' marker advances.