Hyperbolicity Limitations of ADM Evolution

The ADM evolution equations can reproduce every solution of Einstein’s equations and still provide an ill-posed free-evolution system after the gauge is fixed. The distinction is off the constraint surface: geometric equivalence says what happens when the Hamiltonian and momentum constraints vanish, whereas strong hyperbolicity controls arbitrary high-frequency perturbations in the enlarged space explored by a free evolution. For the standard vacuum ADM equations with prescribed unit lapse and vanishing shift, a frozen-coefficient reduction about Minkowski spacetime contains defective zero-speed sectors. It is therefore weakly, not strongly, hyperbolic. This conclusion belongs to that specified gauge and reduction; it is not a statement about constrained ADM schemes or every constraint-modified descendant.

Required background. Partial differential equations and Green functions supplies principal symbols and Fourier-mode estimates. The Einstein equation supplies the covariant equations from which ADM is projected.

Helpful background. Hypersurface normals and tangential projectors supplies the normal–spatial decomposition. Formulations of General Relativity distinguishes geometric equivalence from PDE equivalence. Differential equations reviews the continuous-dependence criterion used below.

The standard ADM free-evolution system

Let $\gamma_{ij}$ be the spatial metric and use the global convention

$$K_{ij}=-\frac12\mathcal L_n\gamma_{ij}.$$

For vacuum with $\Lambda=0$, prescribed lapse $\alpha$ and shift $\beta^i$, the standard ADM evolution equations are

$$(\partial_t-\mathcal L_\beta)\gamma_{ij}=-2\alpha K_{ij},$$ $$(\partial_t-\mathcal L_\beta)K_{ij} =-D_iD_j\alpha +\alpha\left(R_{ij}+KK_{ij}-2K_{ik}K^k{}_j\right).$$

The initial data must also satisfy

$$\mathcal H=R+K^2-K_{ij}K^{ij}=0, \qquad \mathcal M_i=D_jK^j{}_i-D_iK=0.$$

These formulas and their second-order principal part are derived in Gourgoulhon 2007, §§4.3–4.4, equations (4.63)–(4.94), pp. 59–65, PDF. The equations are first order in time and second order in space. Any hyperbolicity statement must consequently name either a direct second-order definition or a particular first-order or pseudodifferential reduction. It must also state how the lapse and shift are supplied. Treating them as fixed functions, evolving them with gauge drivers, or solving elliptic gauge conditions gives different principal symbols.

A free evolution solves the two evolution equations after imposing the constraints only on the initial slice. The contracted Bianchi identity supplies a subsidiary system that preserves exact zero constraints for exact solutions, provided the matter equations are consistent. Gourgoulhon 2007, §§10.2–10.3, equations (10.3)–(10.38), pp. 175–181, PDF makes the distinction between constrained and free schemes and derives this propagation system. Preservation of the zero solution does not by itself control nearby nonzero constraint violations. A system can have an invariant constraint surface yet amplify perturbations transverse to it without a frequency-uniform bound.

Frozen principal part about Minkowski spacetime

Set $\alpha=1$, $\beta^i=0$, and linearize about

$$\gamma_{ij}=\delta_{ij}+h_{ij}, \qquad K_{ij}=k_{ij},$$

retaining only principal terms. Then

$$\partial_t h_{ij}=-2k_{ij},$$ $$\partial_t k_{ij}\simeq R^{(1)}_{ij} =\frac12\left( \partial^l\partial_i h_{lj} +\partial^l\partial_j h_{li} -\partial^l\partial_l h_{ij} -\partial_i\partial_j h \right),$$

where $h=\delta^{ij}h_{ij}$ and $\simeq$ denotes equality of principal parts. Eliminating $k_{ij}$ gives

$$\partial_t^2h_{ij} \simeq \partial^l\partial_l h_{ij} +\partial_i\partial_j h -\partial_i\partial^l h_{lj} -\partial_j\partial^l h_{li}.$$

Now freeze a real unit covector $s_i$ and decompose indices into the longitudinal direction $s$ and two transverse directions $A,B$. This scalar–vector–tensor split is an algebraic decomposition of the principal symbol, not a gauge fixing. The transverse trace-free tensor components obey a wave equation and have characteristic speeds $+1$ and $-1$. The longitudinal–transverse vector components instead satisfy

$$\partial_t^2h_{sA}\simeq0.$$

Introduce the pseudodifferential first-order variable $d_{sA}=\partial_s h_{sA}$. For a Fourier mode with spatial frequency $k$ parallel to $s_i$, its vector-sector principal evolution is

$$\partial_t \begin{pmatrix} \widehat d_{sA}\\[2pt] \widehat k_{sA} \end{pmatrix} =ik \begin{pmatrix} 0&-2\\ 0&0 \end{pmatrix} \begin{pmatrix} \widehat d_{sA}\\[2pt] \widehat k_{sA} \end{pmatrix}.$$

The displayed symbol has the real eigenvalue zero twice but only one eigenvector. If $\widehat k_{sA}(0)$ is nonzero, then

$$\widehat d_{sA}(t) =\widehat d_{sA}(0)-2ikt\,\widehat k_{sA}(0).$$

Its norm grows like $|k|t$, so no estimate with a constant independent of spatial frequency can hold in the natural reduced norm. This explicit Jordan block is the independent diagnostic of weak hyperbolicity. It also shows why checking only real characteristic roots would miss the failure.

Equivalence, constraints, and the scope of the result

On data satisfying the linearized momentum constraint, the offending amplitudes can be restricted; in a fully constrained scheme, elliptic solves can remove them at each time. That does not repair the unconstrained principal symbol used by standard free evolution. Adding multiples of the constraints to the evolution equations, promoting new variables, or changing the lapse prescription changes the off-constraint system and can change the eigenstructure. The resulting formulation must be analyzed under its own name and hypotheses rather than treated as “ADM” without qualification.

Lehner’s review surveys new variables and additions of the constraints in Lehner 2001, §3.1.1, pp. 7–9, PDF, separates incoming boundary variables in Lehner 2001, §3.1.2, pp. 14–16, PDF, and contrasts first- and second-order reductions in Lehner 2001, §4.4.1, pp. 31–32, PDF. These discussions do not imply that every modified ADM system is well posed; the lapse–shift equations, reduction parameters, background, and allowed range must still be specified.

Nor does this frozen linear calculation prove global ill behavior of every exact solution. Its force is narrower and decisive: strong hyperbolicity is a necessary local condition for the standard quasilinear Cauchy theory in the chosen norm, and the frozen system fails it. Smooth lower-order terms cannot remove a high-frequency Jordan block in the principal symbol.

Exercise

For a Fourier covector along the $x$ direction, verify directly that the two components $h_{xy}$ and $h_{xz}$ form defective vector sectors in fixed-lapse, zero-shift linearized ADM. Construct a family of initial data whose reduced norm at time zero is independent of $|k|$ but whose norm at a fixed later time grows with $|k|$.

Solution

Take one component $h_{xA}$ with $A=y$ or $z$. In the linearized Ricci tensor, all transverse derivatives vanish for a mode depending only on $x$:

$$R^{(1)}_{xA} =\frac12\left( \partial_x^2h_{xA}-\partial_x^2h_{xA} \right)=0.$$

Thus $\partial_t k_{xA}=0$ and $\partial_t h_{xA}=-2k_{xA}$. Choose

$$\widehat d_{xA}(0)=0, \qquad \widehat k_{xA}(0)=1,$$

where $d_{xA}=\partial_xh_{xA}$. The initial reduced norm is one, independent of $k$. At time $t$,

$$\widehat k_{xA}(t)=1, \qquad \widehat d_{xA}(t)=-2ikt,$$

so the norm is at least $2|k|t$. A frequency-independent estimate would require this to remain bounded by a constant depending only on the time interval, which is impossible as $|k|$ tends to infinity. The same calculation applies independently to both transverse directions.

References

• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. Lecture notes, 2007. arXiv:gr-qc/0703035.
• Lehner, Luis. “Numerical Relativity: A Review.” Classical and Quantum Gravity 18 (2001): R25–R86. arXiv:gr-qc/0106072.