Principal Symbols and Hyperbolicity

Hyperbolicity is the principal-symbol property that turns an evolution system into a stable Cauchy problem. Real characteristic speeds alone are not enough: the characteristic variables must also form a uniformly controlled basis. Strong hyperbolicity supplies that control; symmetric hyperbolicity supplies it through a positive energy and is a particularly robust sufficient condition.

Required background. Analysis for Mathematical Relativity supplies norm estimates, The Einstein Equation supplies the geometric system, PDEs and Green Functions for Gravitation supplies characteristics and energy methods, and Tensor Fields and Natural Operations supplies the invariant meaning of the fields.

Helpful background. Causality and Global Structure relates characteristic cones to domains of dependence.

Principal symbols and characteristic covectors

For a first-order quasilinear system

$$A^0(u,x)\,\partial_tu+A^i(u,x)\,\partial_i u=B(u,x),$$

assume $A^0$ is invertible. Freezing the coefficients at a state $u_0$ and replacing $\partial_\mu$ by a covector $\xi_\mu$ gives the principal symbol

$$P(\xi)=A^\mu(u_0,x)\xi_\mu.$$

A nonzero covector is characteristic when $\det P(\xi)=0$. For evolution relative to a spatial covector $s_i$, the relevant matrix is

$$M(s)=(A^0)^{-1}A^i s_i.$$

With a plane wave $u=v e^{i(k s_i x^i-\omega t)}$, the symbol equation is $M(s)v=(\omega/k)v$, so its eigenvalues are the propagation speeds in the $s$ direction. The definition is local in phase space and depends on the chosen reduction, but the characteristic set of a properly reduced Einstein system contains the physical null cone.

For the trace-reversed, second-order wave-gauge equations (the reduced Ricci system), the symbol is diagonal in the symmetric tensor components:

$$\sigma(k)h_{\mu\nu} =-\frac12 g^{\alpha\beta}k_\alpha k_\beta h_{\mu\nu}.$$

Thus $k_\mu$ is characteristic exactly when $g^{\alpha\beta}k_\alpha k_\beta=0$. Before gauge reduction, extra symbol-kernel vectors $k_{(\mu}\xi_{\nu)}$ obscure this conclusion. This is why the wave-gauge reduction precedes a well-posedness theorem.

Weak, strong, and symmetric hyperbolicity

For every unit spatial covector $s_i$:

• Weak hyperbolicity requires real eigenvalues of $M(s)$, but allows defective eigenspaces.
• Strong hyperbolicity requires a complete eigenbasis whose diagonalizer and inverse are bounded uniformly in $s$. Equivalently, under standard smoothness hypotheses, there is a uniformly positive Hermitian symmetrizer $H(s)$ for the frozen symbol.
• Symmetric hyperbolicity requires a positive-definite Hermitian matrix $S(u,x)$, independent of $s$, such that $S(A^0)^{-1}A^i$ is Hermitian for every $i$. Equivalently, multiplying the original equations by $H=S(A^0)^{-1}$ makes $HA^0=S$ positive definite and all $HA^i$ Hermitian. The multiplier $H$ itself need not be positive or Hermitian before $A^0$ has been normalized.

Symmetric hyperbolicity implies strong hyperbolicity. Strong hyperbolicity is the principal-symbol condition normally sufficient for an $L^2$ estimate and hence a stable local Cauchy problem, provided the coefficients have the regularity required by the variable-coefficient theorem. It does not by itself enforce Einstein’s constraints, supply boundary conditions, or prevent nonlinear blow-up. Friedrich and Rendall give the relevant reductions and energy-estimate setting in Friedrich and Rendall 2000, §§2–4.

Uniformity is essential: an eigenbasis that exists for each direction but becomes arbitrarily ill-conditioned as $s_i$ varies does not yield a direction-independent estimate. Strong hyperbolicity therefore controls both spectral reality and the geometry of the eigenspaces.

For a symmetric-hyperbolic linearization, multiplying by $u^\dagger H$ and integrating over a spacelike slice gives an energy

$$E(t)=\frac12\int_{\Sigma_t}u^\dagger H A^0u\,d^3x.$$

After integration by parts, coefficient derivatives and lower-order terms yield

$$\frac{dE}{dt}\le C(t)E+\text{boundary flux}+\text{source work}.$$

With no boundary, or with boundary conditions controlling incoming flux, and $C\in L^1_t$, Grönwall’s inequality controls the solution continuously by its initial data. An actual boundary cannot simply be left without conditions. This estimate is the operational content of hyperbolicity; compare Friedrich and Rendall 2000, §§3.1 and 3.3.

The extra work required at a boundary is visible already for a scalar wave: Kreiss et al. 2007, §2, Theorem 1 derive an estimate controlling both the interior energy and boundary traces for an admissible Sommerfeld operator. That conclusion uses the boundary operator, not just the characteristic roots of the bulk wave equation.

Why real speeds are insufficient

Consider in one spatial dimension

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

In the convention above the propagation matrix is $M=-A$, so the only speed is $-1$, but $A$ has a Jordan block. For a Fourier mode $e^{ikx}$,

$$\widehat u(t,k)=e^{ikAt}\widehat u(0,k) =e^{ikt}\begin{pmatrix}1&ikt\\0&1\end{pmatrix}\widehat u(0,k).$$

At fixed $t>0$, the amplification grows like $|k|$. No estimate in the same Sobolev norm can be uniform over frequency: the system is weakly, not strongly, hyperbolic. This counterexample explains why numerical characteristic speeds or a formal dispersion relation do not prove well-posedness.

Constraints introduce a second distinction. A formulation can be strongly hyperbolic off the constraint surface while constraint violations possess their own characteristic fields. One must establish both evolution well-posedness and a homogeneous, well-posed subsidiary system. Likewise, on a domain with boundary, incoming characteristic fields require admissible boundary data. Neither issue is settled by diagonalizing the bulk symbol alone.

Check your understanding

Write the scalar wave equation as a first-order system using $v=\partial_t\phi$ and $w_i=\partial_i\phi$. What are the characteristic speeds in a unit direction $s_i$?

Solution

The principal equations are $\partial_tv=\partial^iw_i$ and $\partial_tw_i=\partial_iv$. Decompose $w_i=w_s s_i+w_i^\perp$. In a one-directional mode, $\partial_t(v+w_s)-\partial_s(v+w_s)=0$ and $\partial_t(v-w_s)+\partial_s(v-w_s)=0$. Thus $v+w_s$ has speed $-1$, $v-w_s$ speed $+1$, and the two transverse components speed $0$. The standard energy $v^2+w_iw^i$ is positive and symmetrizes the system. The zero-speed fields are artifacts of the first-order reduction; the defining constraints $w_i=\partial_i\phi$ must also hold.

References

• Friedrich, Helmut, and Alan D. Rendall. “The Cauchy Problem for the Einstein Equations.” In Einstein’s Field Equations and Their Physical Implications, Lecture Notes in Physics 540, 127–224. Springer, 2000. arXiv:gr-qc/0002074.
• 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. DOI. Open PDF.

Further reading

• Ringström, Hans. The Cauchy Problem in General Relativity. ESI Lectures in Mathematics and Physics. European Mathematical Society, 2009. Publisher record; Chs. 7–9, pp. 57–91.