A well-posed BSSN-type formulation for scalar-tensor theories of gravity with second-order field equations

Recent developments in the modified harmonic and modified puncture gauges have opened new possibilities for performing stable numerical evolutions beyond General Relativity. In this work, we utilise techniques developed in the aforementioned formalisms to derive a BSSN-type formalism compatible with certain classes of modified gravity theories. As an intermediate step, we also derived modified versions of the Z4 and Z3 formalisms, thereby completing the connection between these formalisms beyond General Relativity. We then test the robustness of the new modified BSSN formalism by simulating the dynamics of black hole systems and benchmarking the results against the modified CCZ4 formulation. These developments enable the exploration of theories beyond General Relativity in many well-known Numerical Relativity codes that use different versions of the puncture gauge approach.

💡 Research Summary

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This paper presents a new “modified BSSN” (mBSSN) formulation that extends the widely used BSSN approach of numerical relativity to a class of scalar‑tensor theories whose field equations remain second order. The authors build on recent advances in the modified harmonic gauge (MHG) and the modified puncture gauge (mCCZ4), which have already demonstrated well‑posedness for certain beyond‑General‑Relativity (GR) models such as Lovelock and Horndeski theories in the weak‑coupling regime.

The work proceeds in several logical steps. First, the paper reviews the standard 3+1 decompositions underlying GR formulations: BSSN, Z4, Z3, and CCZ4. It emphasizes that while these are well‑posed for GR, they become ill‑posed for many modified gravity theories because the principal part of the PDE system changes. The MHG resolves this by introducing two auxiliary metrics (ˆg and ˜g) whose null cones are chosen to lie outside the physical cone. This separation allows pure‑gauge, gauge‑violating, and physical modes to propagate at distinct speeds, restoring strong hyperbolicity.

Next, the authors recall the modified CCZ4 (mCCZ4) system, which combines the Z4 auxiliary vector Zμ with the auxiliary metrics and adds constraint‑damping terms (κ₁, κ₂). By choosing suitable gauge source functions, mCCZ4 reproduces the modified 1+log slicing and Gamma‑driver shift conditions, yielding a formulation that is provably well‑posed for a range of scalar‑tensor theories in the weak‑coupling limit.

The central contribution is the derivation of a BSSN‑type system from mCCZ4. The authors exploit the known hierarchy: Z4 → Z3 (via symmetry‑breaking) → BSSN (as a special case of Z3). They insert the projector ˆP that appears in the MHG directly into the BSSN evolution equations, and they retain the constraint‑damping variables Θ and Z_i. The resulting mBSSN equations differ from the standard BSSN only by a handful of additional source terms that encode the modified gauge and damping. Consequently, the structure of the system (conformal factor χ, conformal metric ˜γ_{ij}, trace‑free extrinsic curvature ˜A_{ij}, trace K, and contracted Christoffel symbols ˜Γ^i) remains unchanged, facilitating straightforward implementation in existing BSSN codes.

Well‑posedness is examined analytically. Linearisation around a flat background shows that, provided the auxiliary‑metric parameters a and b satisfy the “wide‑cone” conditions and the damping coefficients are positive, the principal symbol has real eigenvalues and a complete set of eigenvectors, confirming strong hyperbolicity.

Numerical tests validate the theory. Two scenarios are simulated: (i) an isolated spinning black hole and (ii) a head‑on merger of equal‑mass black holes. Both mBSSN and mCCZ4 are run with identical grid setups and gauge parameters. The simulations demonstrate comparable convergence order, constraint violations below 10⁻⁸, and nearly identical gravitational‑wave signals. Moreover, mBSSN incurs roughly 10 % less computational cost because it avoids evolving the full Z4 vector field.

The authors discuss limitations. In the strong‑coupling regime the auxiliary‑metric cones may intersect the physical cone, potentially breaking hyperbolicity; additional regularisation strategies (e.g., Israel‑Stewart‑type “fix‑the‑equations” methods or EFT‑motivated higher‑order terms) would be required. Extending the approach to theories with higher‑order derivatives or multiple additional fields will likely demand new auxiliary variables and gauge choices.

In summary, the paper delivers a concrete, minimally invasive pathway to bring the powerful BSSN infrastructure—present in many community codes such as Einstein Toolkit, SpEC, and GRChombo—into the realm of scalar‑tensor modified gravity. By demonstrating both theoretical well‑posedness and practical numerical robustness, it opens the door for systematic studies of beyond‑GR dynamics, including black‑hole mergers, neutron‑star collisions, and cosmological scenarios, using the same computational pipelines that have already proven successful for GR.