research32 min studyLesson 845 in Physics

BSSN and generalized harmonic formulations

Compare evolved variables and the role of hyperbolicity without equating reformulation with a new gravitational theory.

Before you begin

On this page

The question for today

What you’ll understand

Construct conformal BSSN variables from a simple diagonal metric and curvature.
State a generalized harmonic gauge constraint with an explicit sign.
Explain why real principal eigenvalues alone do not establish strong hyperbolicity.

Equivalent continuum descriptions can behave very differently when evolved numerically. BSSN and generalized harmonic formulations reorganize Einstein's equations, introduce auxiliary variables or constraints, and use gauge choices suited to their systems. Their advantage is not a change in gravitational physics. It is a better-controlled evolution problem whose principal structure, constraints, boundaries, and discretization can be analyzed together. Merely attaching a familiar formulation name to a code does not supply that analysis.

Factor scale and trace into BSSN variables

In a Cartesian version of BSSN, separate the determinant of the spatial metric into a logarithmic conformal factor. The rescaled metric has determinant one, and a rescaled trace-free extrinsic curvature stores the shape-changing part of K. The trace K is evolved separately. Contracted conformal connection functions are promoted to evolved quantities, permitting the spatial Ricci tensor and constraint additions to be organized more effectively than in a direct component evolution.

These definitions introduce algebraic and differential constraints: unit conformal determinant, zero conformal trace, and agreement between the evolved connection functions and derivatives of the conformal metric. Codes may enforce some algebraic conditions by projection, but that projection does not automatically remove the Hamiltonian or momentum residuals. Modern implementations also use alternative conformal variables and curvilinear variants. The formulas here describe a declared Cartesian choice, not every object called BSSN.

\phi=\tfrac1{12}\ln\det\gamma,\quad \widetilde\gamma_{ij}=e^{-4\phi}\gamma_{ij},\quad \widetilde A_{ij}=e^{-4\phi}(K_{ij}-\tfrac13\gamma_{ij}K),\quad \widetilde\Gamma^i=-\partial_j\widetilde\gamma^{ij}

The metric and phi are dimensionless; K, A-tilde, and the coordinate connection functions have inverse-length units. The final expression assumes unit determinant in Cartesian coordinates.↔ Scroll wide equations horizontally.

Impose generalized harmonic coordinates consistently

Generalized harmonic evolution uses coordinate source functions H to replace the harmonic-coordinate condition by a prescribed gauge. Declare H-mu equal to the covariant component of the wave operator acting on coordinates, so it is minus the contracted Christoffel symbol when the gauge constraint holds. The metric equations then have a wave-like principal part. First-order reductions add derivative variables and their own reduction constraints, with parameters affecting characteristics and constraint evolution.

Source functions must be specified in a way compatible with the intended hyperbolic system; arbitrary derivative dependence can change the principal symbol. Constraint-damping terms can control some violations without altering a solution that already satisfies the constraints. Boundary conditions must distinguish gauge, constraint and physical incoming information. Generalized harmonic and BSSN methods can both support accurate simulations under suitable choices, while no short variable list proves robustness for every gauge and boundary setup.

H^\mu=\Box x^\mu=-\Gamma^\mu,\qquad C_\mu=H_\mu+\Gamma_\mu=0,\qquad \Gamma^\mu=g^{\rho\sigma}\Gamma^\mu_{\rho\sigma}

This explicitly declares the generalized harmonic sign convention. H and Gamma have inverse-length units; the reduced equations agree with Einstein evolution on the appropriate constraint surface.↔ Scroll wide equations horizontally.

Test the principal symbol rather than its label

A well-posed hyperbolic initial-value problem requires controlled dependence on initial data. For a frozen first-order principal system, real characteristic eigenvalues are necessary but not sufficient for the usual strong-hyperbolicity estimate. One also needs a complete eigenvector structure with appropriate uniform bounds as the spatial direction varies. A Jordan block can hide a growth factor proportional to frequency, defeating a resolution-independent estimate even though its eigenvalue is real.

This distinction explains why formulation and gauge must be assessed together. Changing the gauge can alter the principal symbol, while adding constraints can improve evolution away from the physical constraint surface. Discrete stability and convergence remain additional questions after continuum well-posedness. A successful low-resolution run might simply fail to excite a damaging short-wavelength mode. The small Fourier example below diagnoses that logical issue without asserting that its Jordan matrix is the full symbol of BSSN or generalized harmonic relativity.

\partial_t u=A\partial_xu,\quad A=\begin{pmatrix}1&1\\0&1\end{pmatrix},\qquad \widehat u(t)=e^{ikt}\begin{pmatrix}1&ikt\\0&1\end{pmatrix}\widehat u(0)

This is a separate dimensionless-coefficient toy PDE. The factor k times t is dimensionless and demonstrates frequency-dependent growth caused by a defective principal matrix.↔ Scroll wide equations horizontally.

Work it through

Audit conformal and trace-free definitions

At one Cartesian point take $\gamma=\operatorname{diag}(4,1,1)$ and $K_{ij}=\operatorname{diag}(2,0,0)/L$ . Find phi, the conformal metric, K, and the unscaled trace-free curvature. Check the conformal determinant and trace condition.

The metric determinant is four, giving $\phi=\ln4/12$ and conformal scale $e^{-4\phi}=4^{-1/3}$ . The conformal metric is diagonal with entries $4^{2/3},4^{-1/3},4^{-1/3}$ , whose product is one.
Raising an index with the physical inverse metric gives $K=(1/4)(2)/L=0.5/L$ . Subtracting one third gamma times K gives diagonal trace-free entries $(4/3,-1/6,-1/6)/L$ .
The physical trace of these entries is $(1/4)(4/3)-1/6-1/6=0$ . Multiplying the tensor by the conformal scale and contracting with the conformal inverse preserves zero trace. A connection function cannot be calculated from point values alone because spatial derivatives are missing.

The result

The specified conformal metric has unit determinant, K is 0.5 per L, and the rescaled trace-free tensor satisfies its algebraic zero-trace condition.

The living notebook

Explore the model

Interactive lab

Interactive lab · matrix

A matrix diagnostic has a specified domain

Reset and set diagonal scale one, shear one; predict final determinant one. Compare a fresh Reset run at diagonal scale one, shear zero; predict determinant one again. At Reset geometry is identity, while readouts describe the selected final matrix and playback interpolates geometry. This restricted family illustrates why equal determinants do not determine a matrix; it is not a principal-symbol or BSSN hyperbolicity analyzer.

Interpolated geometryOriginal unit squareSelected final transform

Area scale · det A: 1.4
Singular values · σmax, σmin: 1.597, 0.876
Condition number · κ₂(A): 1.823

Diagonal scale a1.4Shear b0.6

t = 0 s

The selected matrix is A = [[a,b],[0,1]]. Playback interpolates geometry between the identity and A; readouts always describe A. Its determinant is a and its singular values come from AᵀA. Positive a makes every supported matrix invertible. This two-dimensional family is not a general eigensolver or a learned network. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA coarse matrix diagnostic cannot replace the eigenvector and uniform-bound analysis required for an evolution system.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

The Jordan-block example has a real repeated eigenvalue but a frequency-times-time growth factor. Strong hyperbolicity needs the appropriate complete, uniformly controlled diagonalization or symmetrizer.

Your turn

Try it yourself

For the declared Jordan-block toy use initial Fourier vector $(0,1)$ , $k=10/L$ , and $t=0.2L$ . Find its Euclidean norm at that time. Repeat for $k=20/L$ with the same normalized initial vector, and identify the resolution-related concern.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

The overall complex exponential has unit magnitude. The remaining vector is i k t followed by one.

Compare with the solution

At the first frequency k times t is two, so the norm is $\sqrt{2^2+1}=\sqrt5\approx2.236$ . Doubling k gives norm $\sqrt{4^2+1}=\sqrt{17}\approx4.123$ . The initial norm was one in both cases. Higher-frequency data can therefore be amplified without a frequency-independent bound from this expression. The toy demonstrates the need for a stronger principal-symbol test; it does not classify a full relativity formulation.

Keep the idea

Take it with you

Reformulation changes numerical evolution structure without changing constraint-satisfying Einstein physics.
BSSN definitions introduce constraints that must be monitored or enforced consistently.
Hyperbolicity requires more than real eigenvalues or a familiar formulation name.

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