Conformal Decompositions of Initial Data

Conformal transverse–traceless and conformal thin-sandwich decompositions can both produce constraint-satisfying slices, but they parameterize the freely chosen geometry differently. Numerical agreement of their residuals therefore does not make their solutions physically equivalent. A useful comparison holds the intended binary or isolated object as fixed as possible, solves each elliptic system to demonstrated accuracy, and then compares invariant charges, orbital diagnostics, and subsequent junk radiation. The continuum constructions belong to Formulations of General Relativity; this page addresses their computational consequences.

Required background. Hypersurfaces and Foliations supplies the intrinsic metric, extrinsic curvature, lapse, and shift on a slice.

Helpful background. Formulations of General Relativity develops the constraint decompositions, Ordinary Differential Equations supplies nonlinear solver concepts, and Relativistic Matter and Compact Objects supplies material sources and equilibrium criteria.

Conformal decompositions and free data

Use $K_{ij}=-\tfrac12\mathcal L_n\gamma_{ij}$ and $A_{ij}=K_{ij}-\gamma_{ij}K/3$. In a conformal transverse–traceless construction, one writes

$$\gamma_{ij}=\psi^4\tilde\gamma_{ij},\qquad A^{ij}=\psi^{-10}\left(\tilde A_{\rm TT}^{ij}+ (\tilde L W)^{ij}\right),$$

and solves for $\psi$ and a vector potential $W^i$ after choosing $\tilde\gamma_{ij}$, $K$, and a transverse–traceless seed. Here $\tilde D_j\tilde A_{\rm TT}^{ij}=0$, $\tilde\gamma_{ij}\tilde A_{\rm TT}^{ij}=0$, and

$$(\tilde LW)^{ij}=\tilde D^iW^j+\tilde D^jW^i -\frac23\tilde\gamma^{ij}\tilde D_kW^k.$$

Taking the trace shows why the coefficient is $2/3$: the first two terms give $2\tilde D_iW^i$, which the final term cancels in three spatial dimensions. In vacuum the momentum equation becomes

$$\tilde D_j(\tilde LW)^{ij} =\frac23\psi^6\tilde D^iK.$$

For constant $K$ this equation decouples from the Hamiltonian constraint; otherwise the factor $\psi^6$ couples the solves. Boundary conditions and conformal Killing-vector kernels still matter: a zero right-hand side does not by itself determine $W^i$ uniquely.

In the original conformal thin-sandwich construction choose a trace-free conformal-metric derivative $\tilde u_{ij}=\partial_t\tilde\gamma_{ij}$ and a conformal lapse $\tilde\alpha=\alpha\psi^{-6}$, as well as $\tilde\gamma_{ij}$ and $K$. The unknowns are $\psi$ and $\beta^i$, with

$$\tilde A^{ij}=\frac{(\tilde L\beta)^{ij}-\tilde u^{ij}} {2\tilde\alpha}.$$

Indices on $\tilde u_{ij}$ are raised with $\tilde\gamma^{ij}$; $\tilde u^{ij}$ is therefore not $\partial_t\tilde\gamma^{ij}$. The extended thin-sandwich system instead specifies $\partial_tK$ and solves an additional lapse equation. Thus a mean-curvature derivative is free data of the extended system, not a mandatory input of original CTS. These choices follow Baumgarte and Shapiro 2010, §§3.2–3.3, Boxes 3.1–3.3, pp. 68–79.

These formulations expose different controls. Transverse–traceless data make momentum content convenient to seed; thin-sandwich data make approximate helical symmetry and quasi-equilibrium boundary conditions convenient to express. Cook 2000, §§2.2.1–2.3 compares the decomposition choices and equations. York 1999, Eqs. (14)–(15) and following discussion shows how lapse–shift variables enter the thin-sandwich construction.

Comparable is not identical

To compare methods, specify a common physical target: component horizon masses, spins, coordinate separation, orbital frequency prescription, asymptotic frame, and outer boundary. Then document the free conformal metric, $K$, inner-boundary or puncture treatment, and lapse data separately. Even after matching nominal targets, the slices can contain different gravitational-wave content because their freely specified fields differ.

Each solve requires its own refinement and boundary study. Track independent Hamiltonian and momentum residuals, ADM energy and angular momentum on multiple surfaces, horizon charges, and the elliptic iteration history. A comparison table should distinguish solver error from formulation difference. If the CTT and CTS values differ by one percent while each has a numerical uncertainty of $10^{-4}$, the difference is physical or gauge content of the chosen data, not discretization noise.

The most discriminating downstream check evolves both slices with the same evolution formulation, gauge, mesh policy, and extraction method. Compare early constraint transients, orbital eccentricity, and junk-radiation energy. This evolution is a diagnostic of the initial slice, not a proof that either decomposition is uniquely correct.

Benchmark design

Start with time-symmetric Schwarzschild, for which both constructions can reduce to the same conformally flat solution when matching free data and boundary conditions are selected. Recover the analytic conformal factor, ADM mass, and horizon mass across at least three resolutions and outer radii. This verifies the common solver path.

Then choose one equal-mass, nonspinning binary target. With $c=1$, let $M_1$ and $M_2$ be physical component masses and $J_{\rm ADM}$ the physical angular momentum. Solve at matched resolution sequences and report

$$E_b=E_{\rm ADM}-M_1-M_2, \qquad j=\frac{J_{\rm ADM}}{G(M_1+M_2)^2},$$

together with horizon masses and spins, coordinate separation, orbital-frequency prescription, constraint norms, and junk-radiation energy after evolution. The assumptions include asymptotic flatness, the chosen quasi-equilibrium approximation, and sufficient separation for the target labels to be meaningful. Near merger, no conformal ansatz should be assumed to represent a preferred astrophysical slice.

Vary nonlinear tolerance, grid or basis order, outer boundary, and free-data resolution independently. Compare Richardson- or spectral-extrapolated values, not one-grid numbers. Nonuniqueness of extended thin-sandwich equations can allow distinct branches; continuation from multiple initial guesses is required when the parameter regime approaches such behavior.

Environment and failure modes

Pin the decomposition equations, conformal weights, free-data files and checksums, inner and outer conditions, domain maps, solver and preconditioner, precision, stopping norms, charge integrators, evolution diagnostic, and code revision. A clean interface might be initial-data-compare --config equal-mass.toml --output run/, producing both solutions, constraint fields, charge surfaces, refinement sweeps, and evolution summaries. This is a reproducibility specification, not a claim that a program is supplied.

Reject a comparison when one solve is under-resolved, free data are silently mismatched, ADM quantities are evaluated in different asymptotic frames, or junk radiation is extracted with different gauges or radii. A small constraint residual cannot resolve these failures. Likewise, tuning parameters until global charges agree does not establish equivalence if local geometry and radiative content differ.

Diagnostic

CTT and thin-sandwich binaries have identical coordinate separation and solve constraints to $10^{-10}$. Their individual horizon masses agree, but their ADM angular momenta differ by two percent and early waveforms have different junk bursts. Should the difference be called numerical error?

Solution

Not without evidence. First show that each result converges under resolution, boundary, and tolerance changes and that charges use one asymptotic convention. If the two-percent difference survives those tests, it reflects distinct free data or quasi-equilibrium prescriptions. Coordinate separation and small residuals do not uniquely fix a physical binary slice.

References

• Baumgarte, T. W., and Shapiro, S. L. (2010). Numerical Relativity: Solving Einstein’s Equations on the Computer. Cambridge University Press. DOI.
• Cook, G. B. (2000). “Initial data for numerical relativity.” Living Reviews in Relativity 3, 5. DOI.
• York, J. W., Jr. (1999). “Conformal ‘thin-sandwich’ data for the initial-value problem of general relativity.” Physical Review Letters 82, 1350–1353. arXiv.