Hyperbolic PDE Systems in Relativity

Hyperbolicity is the local property that turns a relativistic evolution system into a predictive Cauchy problem. For a fixed first-order reduction, the decisive test is not merely whether the principal symbol has real eigenvalues: it must possess a complete characteristic basis whose conditioning is bounded uniformly over all spatial directions. That stronger requirement yields a frequency-independent energy estimate and hence continuous dependence on the initial data. This page develops that implication for first-order systems, distinguishes weak, strong, and symmetric hyperbolicity, and explains why no conclusion about global existence or physical signal speeds follows from the symbol test alone.

Required background. Partial differential equations and Green functions supplies Fourier analysis, principal parts, and energy norms. The Einstein equation supplies the field equations whose reductions are being classified.

Helpful background. Hypersurface normals and tangential projectors fixes the normal–tangential split used to interpret Cauchy data. Formulations of General Relativity explains why equivalent reductions can have different PDE properties. Differential equations reviews existence estimates for finite-dimensional evolution.

Principal symbols and characteristic fields

Consider a linear first-order system on a fixed time foliation,

$$\partial_t u+A^i(t,x)\,\partial_i u=B(t,x)u+f,$$

where $u$ has $N$ components. The reduction order, the variables included in $u$, and the norm on those variables are part of the formulation; changing any of them can change its hyperbolicity classification. Freeze the coefficients at one spacetime point and let $s_i$ be a real spatial covector of unit length with respect to a chosen positive-definite background spatial metric. The spatial principal symbol is

$$P(s)=A^i s_i.$$

For a Fourier mode $u=\widehat u(t)e^{ik_jx^j}$, with $k_i=|k|s_i$, the principal evolution is

$$\partial_t\widehat u=-i|k|P(s)\widehat u.$$

An eigenvalue $\lambda_a(s)$ of $P(s)$ is a coordinate characteristic speed, and a left eigenvector $\ell_a(s)$ defines a characteristic field $w_a=\ell_a u$. These speeds depend on the time coordinate, gauge, variables, and reduction. Calling every $\lambda_a$ a physical propagation speed would therefore confuse physical, gauge, and constraint sectors.

The standard hierarchy is as follows.

• The system is weakly hyperbolic if $P(s)$ has only real eigenvalues for every real $s_i\ne0$.

• It is strongly hyperbolic if, in addition, every $P(s)$ is diagonalizable with a choice of eigenvector matrix $S(s)$ for which both $\lVert S(s)\rVert$ and $\lVert S(s)^{-1}\rVert$ are bounded uniformly on the unit sphere. Equivalently, there is a positive Hermitian symmetrizer $H(s)$ satisfying

  $$cI\le H(s)\le CI, \qquad H(s)P(s)=P(s)^\dagger H(s),$$

  with constants $0\lt c\le C\lt\infty$ independent of $s$.

• It is symmetric hyperbolic if one positive Hermitian matrix $H(t,x,u)$, independent of $s$, symmetrizes every $A^i$. Symmetric hyperbolicity is stronger than strong hyperbolicity and gives an energy by direct integration by parts.

Pointwise diagonalizability without a uniform condition-number bound is insufficient. For variable-coefficient or quasilinear systems, the symmetrizer must also depend regularly enough on the point, fields, and direction for the pseudodifferential or energy argument being used. The precise differentiability threshold depends on the chosen Sobolev theorem; it cannot be omitted from a nonlinear existence claim.

Several qualifications in this definition do real work. First, $s_i$ ranges over every real direction, not just the coordinate axes. A symbol can be diagonalizable along each axis and become defective along an oblique direction. Homogeneity reduces the test to the compact unit sphere, but it does not reduce it to finitely many selected covectors. Second, a characteristic crossing need not destroy strong hyperbolicity. Degenerate eigenvalues are harmless if the entire degenerate subspace has a complete, continuously controlled basis; it is the loss of a uniform basis, rather than degeneracy itself, that causes high-frequency growth. A symmetrizer is often the safer object to track near such crossings because it need not select individual eigenvectors.

Third, the result belongs to a specified reduction. If a second-order-in-space equation is converted to first order by introducing $d_i u=\partial_i u$, the enlarged system acquires reduction constraints $d_i u-\partial_i u=0$ and $\partial_{[i}d_{j]}u=0$. Different additions of these constraints alter the principal symbol away from the reduction-constraint surface. A claimed hyperbolicity theorem must therefore state which auxiliary variables and constraint additions it uses. A direct second-order theorem may avoid these variables, but then its principal symbol and energy norm must be defined at second order rather than silently borrowed from a first-order reduction.

Finally, Cauchy well-posedness and initial-boundary well-posedness are separate questions. Strong hyperbolicity supplies the interior Cauchy estimate. On a finite domain one must also identify the characteristic fields relative to the outward boundary normal, prescribe only the incoming data, prove a boundary-stable flux estimate, and maintain the Einstein subsidiary constraints. A complete interior eigenbasis by itself says none of those things.

Why the uniform basis gives continuous dependence

For frozen constant coefficients and $B=f=0$, write $P=S\Lambda S^{-1}$, where $\Lambda$ is real diagonal. The Fourier propagator is

$$e^{-i|k|P(s)t}=S(s)e^{-i|k|\Lambda(s)t}S(s)^{-1}.$$

Because every diagonal phase has modulus one,

$$\lVert\widehat u(t,k)\rVert \le \lVert S(s)\rVert\,\lVert S(s)^{-1}\rVert \lVert\widehat u(0,k)\rVert \le K\lVert\widehat u(0,k)\rVert,$$

with $K$ independent of $|k|$. Multiplying by $(1+|k|^2)^{q/2}$ and integrating over $k$ gives the same estimate in every Sobolev space $H^q$. Lower-order terms and forcing then give, by the usual Grönwall argument,

$$\lVert u(t)\rVert_{H^q} \le C_T\left( \lVert u(0)\rVert_{H^q} +\int_0^t\lVert f(\tau)\rVert_{H^q}\,d\tau \right),$$

where $C_T$ is independent of spatial frequency. This frequency-uniform estimate is the operational content of continuous dependence for the frozen Cauchy problem. It is also the local model behind the energy estimates used for relativistic formulations. A concrete second-order wave initial-boundary problem, including an estimate of boundary traces, is proved by direct integration by parts in Kreiss, Reula, Sarbach, and Winicour 2007, §2, Lemma 1 and Theorems 1–2, pp. 3–7, PDF. Their hypotheses make explicit that strong well-posedness at a boundary requires more than a Cauchy estimate: the boundary operator and its flux must also be controlled.

The same classification guides formulations of Einstein’s equations, but only after lapse and shift prescriptions, constraint additions, and reduction variables have been specified. Lehner 2001, §3.1.1, pp. 7–9, PDF surveys how those choices produce different evolution systems, while Lehner 2001, §3.1.2, pp. 14–16, PDF explains why only incoming characteristic fields may be assigned at a boundary. Equivalence of two systems on the Einstein constraint surface does not imply that their off-constraint principal symbols, and hence their free-evolution well-posedness, agree.

The Jordan-block failure test

Real characteristic speeds are necessary but not sufficient. Take a one-dimensional constant-coefficient system with

$$P= \begin{pmatrix} 1&1\\ 0&1 \end{pmatrix} =I+N, \qquad N^2=0.$$

Its only eigenvalue is the real number $1$, so the system is weakly hyperbolic. It has only one eigenvector, however, and

$$e^{-ikPt}=e^{-ikt}\left(I-iktN\right).$$

Initial data in the generalized-eigenvector direction acquire a term proportional to $|k|t$. No constant $C_T$ independent of $k$ can bound the solution by the initial norm on any fixed interval with $t>0$. Higher Jordan blocks generate higher powers of $|k|t$. This is the quickest independent check of why a complete, uniformly conditioned characteristic basis—not reality of eigenvalues alone—is required.

The principal-symbol test remains local. It does not establish global nonlinear existence, control singularity formation, choose a physically appropriate outer boundary, or show that a discretization is stable. Conversely, a numerical evolution that appears stable at finite resolution does not prove strong hyperbolicity: the unresolved Jordan growth can move to shorter wavelengths as the grid is refined.

Exercise

Consider

$$\partial_t u+A\partial_xu=0, \qquad A=\begin{pmatrix}0&1\\1&0\end{pmatrix}.$$

Find its characteristic fields, construct a symmetrizer, and prove an $L^2$ energy estimate on a periodic interval. Then replace $A$ by the Jordan matrix above and identify the step that fails.

Solution

The eigenvalues of $A$ are $+1$ and $-1$, with normalized left eigenvectors $2^{-1/2}(1,1)$ and $2^{-1/2}(1,-1)$. Thus

$$w_+=\frac{u_1+u_2}{\sqrt2}, \qquad w_-=\frac{u_1-u_2}{\sqrt2},$$

and the equations are $\partial_tw_++\partial_xw_+=0$ and $\partial_tw_- -\partial_xw_-=0$. Since $A=A^T$, $H=I$ is a direction-independent positive symmetrizer. On a periodic interval,

$$\frac{d}{dt}\int u^THu\,dx =-\int\partial_x(u^TAu)\,dx=0,$$

so $\lVert u(t)\rVert_{L^2}=\lVert u(0)\rVert_{L^2}$. For the Jordan matrix there is no complete eigenbasis and no positive Hermitian $H$ satisfying $HP=P^\dagger H$: if such an $H$ existed, $H^{1/2}PH^{-1/2}$ would be Hermitian and hence diagonalizable, contradicting the Jordan form. The characteristic transformation and the frequency-independent energy estimate therefore fail together.

References

• Kreiss, Heinz-Otto, Oscar Reula, Olivier Sarbach, and Jeffrey Winicour. “Well-Posed Initial-Boundary Value Problem for the Harmonic Einstein Equations Using Energy Estimates.” Classical and Quantum Gravity 24 (2007): 5973–5984. arXiv:0707.4188.
• Lehner, Luis. “Numerical Relativity: A Review.” Classical and Quantum Gravity 18 (2001): R25–R86. arXiv:gr-qc/0106072.