The BSSN Continuum Formulation

BSSN is a continuum free-evolution formulation obtained from ADM by a conformal and trace decomposition, promotion of contracted conformal connection functions to independent variables, and deliberate use of constraints in their evolution equations. The algebraic constraints make the conformal split invertible; recovering the same Einstein evolution additionally requires the connection, Hamiltonian, and momentum constraints. Constraint additions change the evolution away from that full constraint surface. BSSN is therefore more than ADM written with new symbols. Its hyperbolicity nevertheless depends on the lapse and shift drivers, the reduction used for the second-order spatial terms, and parameter choices; “BSSN” alone is not a well-posedness theorem.

Required background. Partial differential equations and Green functions supplies principal symbols and reduction choices. The Einstein equation supplies the covariant dynamics represented by the conformal variables.

Helpful background. Hypersurface normals and tangential projectors supplies the induced metric and extrinsic curvature. Formulations of General Relativity explains why off-constraint changes matter. Differential equations reviews evolution systems and continuous dependence.

Conformal variables and their constraints

Work locally in Cartesian spatial coordinates, in vacuum with $\Lambda=0$; a reference-metric version replaces partial derivatives and the unit-determinant condition accordingly. Use $K_{ij}=-\gamma_i{}^\mu\gamma_j{}^\nu\nabla_\mu n_\nu$ for the future unit normal. Define

$$\gamma_{ij}=e^{4\phi}\widetilde\gamma_{ij}, \qquad \det\widetilde\gamma_{ij}=1,$$

and split the extrinsic curvature into its trace and trace-free parts,

$$K=\gamma^{ij}K_{ij}, \qquad \widetilde A_{ij} =e^{-4\phi}\left(K_{ij}-\frac13\gamma_{ij}K\right).$$

The conformal factor is positive because $e^{\phi}$ is positive. The definition implies the algebraic constraint

$$\widetilde\gamma^{ij}\widetilde A_{ij}=0.$$

Finally promote the contracted Christoffel symbols of the conformal metric,

$$\widetilde\Gamma^i =\widetilde\gamma^{jk}\widetilde\Gamma^i{}_{jk} =-\partial_j\widetilde\gamma^{ij},$$

to independent evolved variables. Their defining, or connection, constraint is

$$\mathcal G^i =\widetilde\Gamma^i+\partial_j\widetilde\gamma^{ij}=0.$$

The last equality in the definition uses the unit-determinant condition and Cartesian coordinates. In curvilinear coordinates, the difference between the conformal connection and a chosen reference connection is the tensorial object to evolve. Gourgoulhon 2007, §10.4.2, equations (10.39)–(10.54), pp. 181–184, PDF gives that reference-metric construction and shows exactly where the determinant constraint enters.

The physical fields are reconstructed without solving a differential equation:

$$\gamma_{ij}=e^{4\phi}\widetilde\gamma_{ij}, \qquad K_{ij}=e^{4\phi} \left(\widetilde A_{ij}+\frac13\widetilde\gamma_{ij}K\right).$$

Thus the variable change is invertible when the conformal factor is finite and positive and the determinant and trace constraints hold. The Hamiltonian and momentum constraints remain in addition to these algebraic and connection constraints.

How the promoted connection changes the principal part

The conformal Ricci tensor contains a Laplace-type term plus derivatives that would otherwise mix longitudinal metric components. In BSSN it is organized as

$$\widetilde R_{ij} =-\frac12\widetilde\gamma^{kl} \partial_k\partial_l\widetilde\gamma_{ij} +\widetilde\gamma_{k(i}\partial_{j)}\widetilde\Gamma^k +Q_{ij},$$

where the terms with no second derivatives are

$$\begin{aligned} Q_{ij}={}&\widetilde\Gamma^k\widetilde\Gamma_{(ij)k}\\ &+\widetilde\gamma^{kl}\left( 2\widetilde\Gamma^m{}_{k(i}\widetilde\Gamma_{j)ml} +\widetilde\Gamma^m{}_{il}\widetilde\Gamma_{mkj}\right). \end{aligned}$$

Here $\widetilde\Gamma_{ijk} =\widetilde\gamma_{im}\widetilde\Gamma^m{}_{jk}$, and parentheses average over the enclosed free indices. The three-index connections are calculated from $\widetilde\gamma_{ij}$; the one-index $\widetilde\Gamma^k$ is the evolved variable. On $\mathcal G^i=0$, this is exactly the Ricci tensor of the conformal metric. Away from that constraint surface, using the evolved connection defines the BSSN continuation of the curvature expression. The explicit quadratic terms are given in Baumgarte and Shapiro 2010, §11.5, Eq. (11.41), p. 388. The physical Ricci tensor is $R_{ij}=\widetilde R_{ij}+R^{\phi}_{ij}$, with

$$R^{\phi}_{ij} =-2\widetilde D_i\widetilde D_j\phi -2\widetilde\gamma_{ij}\widetilde D^k\widetilde D_k\phi +4\widetilde D_i\phi\widetilde D_j\phi -4\widetilde\gamma_{ij}\widetilde D^k\phi\widetilde D_k\phi.$$

Promoting $\widetilde\Gamma^i$ isolates the conformal-metric Laplacian in the principal part. The derivation and its precise second-derivative split appear in Gourgoulhon 2007, §§10.4.2–10.4.3, equations (10.49)–(10.60), pp. 183–186, PDF.

To see the evolution structure, define $\partial_0=\partial_t-\beta^k\partial_k$. This removes only advection, so the explicit shift-gradient terms below are still required. Conformal indices on $\widetilde A_{ij}$ are raised using $\widetilde\gamma^{ij}$, and $\mathrm{TF}$ removes the physical trace with $\gamma_{ij}$. The four metric and curvature equations are

$$\partial_0\phi =-\frac16\alpha K+\frac16\partial_k\beta^k,$$ $$\partial_0\widetilde\gamma_{ij} =-2\alpha\widetilde A_{ij} +2\widetilde\gamma_{k(i}\partial_{j)}\beta^k -\frac23\widetilde\gamma_{ij}\partial_k\beta^k,$$ $$\partial_0 K =-D^iD_i\alpha +\alpha\left(\widetilde A_{ij}\widetilde A^{ij}+\frac13K^2\right),$$ $$\begin{aligned} \partial_0\widetilde A_{ij} ={}&e^{-4\phi}\left[-D_iD_j\alpha+\alpha R_{ij}\right]^{\mathrm{TF}}\\ &+\alpha\left(K\widetilde A_{ij}-2\widetilde A_{ik}\widetilde A^k{}_j\right)\\ &+2\widetilde A_{k(i}\partial_{j)}\beta^k -\frac23\widetilde A_{ij}\partial_k\beta^k. \end{aligned}$$

The connection equation follows by differentiating $\widetilde\Gamma^i=-\partial_j\widetilde\gamma^{ij}$, then using the momentum constraint to replace $\partial_j\widetilde A^{ij}$. With the usual unit coefficient for that constraint substitution, it is

$$\begin{aligned} \partial_0\widetilde\Gamma^i ={}&-2\widetilde A^{ij}\partial_j\alpha\\ &+2\alpha\left(\widetilde\Gamma^i{}_{jk}\widetilde A^{jk} -\frac23\widetilde\gamma^{ij}\partial_jK +6\widetilde A^{ij}\partial_j\phi\right)\\ &+\widetilde\gamma^{jk}\partial_j\partial_k\beta^i +\frac13\widetilde\gamma^{ij}\partial_j\partial_k\beta^k\\ &+\frac23\widetilde\Gamma^i\partial_k\beta^k -\widetilde\Gamma^j\partial_j\beta^i. \end{aligned}$$

Here $\widetilde\Gamma^i{}_{jk}$ is computed from the conformal metric, whereas the contracted $\widetilde\Gamma^i$ is the independently evolved field. The $\partial_jK$ term exposes where the momentum constraint has changed the principal part. This substitution is not an algebraic rewriting valid for arbitrary off-constraint fields. The full continuum equations and all five groups of constraints are collected in Gourgoulhon 2007, §10.4.4, equations (10.61)–(10.70), pp. 186–187, PDF.

Reconstruction and frozen-symbol checks

Two checks should precede any hyperbolicity claim. First, substitute the reconstruction formula into the definitions. The trace is $K$, the trace-free remainder is $\widetilde A_{ij}$, and the physical determinant is $e^{12\phi}$. Failure of the unit determinant or trace-free conditions means that the nominal BSSN variables no longer encode the intended ADM pair uniquely without a projection convention.

Second, freeze about Minkowski spacetime with $\phi=0$, $\widetilde\gamma_{ij}=\delta_{ij}$, $K=0$, and $\widetilde A_{ij}=0$. The conformal Ricci principal part becomes

$$\widetilde R^{(1)}_{ij} \simeq-\frac12\partial^k\partial_k\widetilde h_{ij} +\partial_{(i}\widetilde\Gamma_{j)}.$$

Unlike raw ADM, the longitudinal derivative combination is represented by an evolved connection variable. Whether the combined scalar, vector, and tensor blocks possess a complete uniform characteristic basis now depends on the equations chosen for $\alpha$ and $\beta^i$ and on any reduction parameters. Fixed gauge, harmonic slicing, moving-puncture drivers, and reference-metric variants are different systems. The continuum formulation does not justify a universal claim of strong hyperbolicity, and observed numerical robustness does not replace this symbol calculation.

Three common failures are analytically distinct. Drift in the determinant or trace-free constraints breaks the intended conformal decomposition; drift in the connection constraint makes the promoted variable disagree with the derivative of the conformal metric; Hamiltonian or momentum violations move the reconstructed data away from Einstein data. Algebraic projection can correct the first kind without correcting the other two. Constraint damping or modified evolution equations may control differential violations, but each modification changes the off-constraint system and requires a fresh principal-symbol and subsidiary-system analysis. Boundary data add another independent requirement: incoming physical, gauge, and constraint fields must be counted for the chosen gauge-complete formulation, not for the five geometric BSSN variables in isolation.

Exercise

Given fields $\phi$, $\widetilde\gamma_{ij}$, $K$, and $\widetilde A_{ij}$ satisfying $\det\widetilde\gamma=1$ and $\widetilde\gamma^{ij}\widetilde A_{ij}=0$, reconstruct $\gamma_{ij}$ and $K_{ij}$. Verify that the reconstructed trace is $K$ and that applying the BSSN definitions returns the original conformal variables.

Solution

Use

$$\gamma_{ij}=e^{4\phi}\widetilde\gamma_{ij}, \qquad \gamma^{ij}=e^{-4\phi}\widetilde\gamma^{ij},$$

and

$$K_{ij}=e^{4\phi} \left(\widetilde A_{ij}+\frac13\widetilde\gamma_{ij}K\right).$$

The trace is

$$\gamma^{ij}K_{ij} =\widetilde\gamma^{ij}\widetilde A_{ij} +\frac13\widetilde\gamma^{ij}\widetilde\gamma_{ij}K =0+K.$$

The determinant obeys $\det\gamma=e^{12\phi}\det\widetilde\gamma=e^{12\phi}$, so the conformal factor recovered from the physical metric is $\phi=(1/12)\ln\det\gamma$. Subtracting one third of the reconstructed trace gives

$$e^{-4\phi}\left(K_{ij}-\frac13\gamma_{ij}K\right) =\widetilde A_{ij}.$$

Therefore the map is inverse on the algebraic-constraint surface. The connection variable is recovered only if its differential constraint $\mathcal G^i=0$ also holds.

References

• Baumgarte, Thomas W., and Stuart L. Shapiro. Numerical Relativity: Solving Einstein’s Equations on the Computer. Cambridge University Press, 2010. doi:10.1017/CBO9781139193344.
• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. Lecture notes, 2007. arXiv:gr-qc/0703035.