Linear Degeneracy of the First-Order Generalized-Harmonic Einstein System

TL;DR

This paper clarifies the conditions for linear degeneracy in the first-order generalized-harmonic Einstein system, which is important for understanding the system's behavior in vacuum spacetime evolution.

Contribution

It identifies specific conditions that ensure the first-order generalized-harmonic Einstein system is linearly degenerate, enhancing theoretical understanding.

Findings

Conditions for linear degeneracy are explicitly characterized.

Clarification improves stability analysis of Einstein evolution systems.

Provides theoretical foundation for future numerical relativity work.

Abstract

The purpose of this note is to clarify the conditions under which the first-order generalize-harmonic representation of the vacuum Einstein evolution system is linearly degenerate.

Equations

$${\partial }_t{u}^{\alpha }+A^{k\alpha }{}_{\beta }{\partial }_k{u}^{\beta }=F^{\alpha },$$

$${\partial }_t{\psi }_{ab}-(1+{\gamma }_1)N^k{\partial }_k{\psi }_{ab}$$

$${\partial }_t{\Pi }_{ab}-N^k{\partial }_k{\Pi }_{ab}+N{g}^{ki}{\partial }_k{\Phi }_{iab}-{\gamma }_1{\gamma }_2{N}^k{\partial }_k{\psi }_{ab}$$

$${\partial }_t{\Phi }_{iab}-N^k{\partial }_k{\Phi }_{iab}+N{\partial }_i{\Pi }_{ab}-{\gamma }_{2}N{\partial }_i{\psi }_{ab}$$

$$n_k{A}^{k\alpha }{}_{\beta }{\partial }_{u^{\alpha }}\otimes d{u}^{\beta }=-(1+{\gamma }_1)n_k{N}^k{\partial }_{{\psi }_{ab}}\otimes d{\psi }_{ab}$$

$$-n_k{N}^k{\partial }_{{\Pi }_{ab}}\otimes d{\Pi }_{ab}+N{n}^k{\partial }_{{\Pi }_{ab}}\otimes d{\Phi }_{kab}-{\gamma }_1{\gamma }_2{n}_k{N}^k{\partial }_{{\Pi }_{ab}}\otimes d{\psi }_{ab}$$

$$-n_k{N}^k{\partial }_{{\Psi }_{iab}}\otimes d{\Psi }_{iab}+N{n}_k{\partial }_{{\Psi }_{kab}}\otimes d{\Pi }_{ab}-{\gamma }_{2}N{n}_k{\partial }_{{\Psi }_{kab}}\otimes d{\psi }_{ab},$$

$${\ell }^{\hat{\alpha }}{}_{\alpha }{n}_k{A}^{k\alpha }{}_{\beta }$$

$$n_k{A}^{k\alpha }{}_{\beta }{r}_{\hat{\alpha }}{}^{\beta }$$

$${\ell }_{ab}^{\hat{0}}{}_{\beta }d{u}^{\beta }$$

$${\ell }_{ab}^{\hat{1}\pm }{}_{\beta }d{u}^{\beta }$$

$${\ell }_{iab}^{\hat{2}}{}_{\beta }d{u}^{\beta }$$

$$r_{\hat{0}}^{ab}{}^{\beta }{\partial }_{u^{\beta }}$$

$$r_{\hat{1}\pm }^{ab}{}^{\beta }{\partial }_{u^{\beta }}$$

$$r_{\hat{2}}^{iab}{}^{\beta }{\partial }_{u^{\beta }}$$

$$v^{(\hat{0})}$$

$$v^{(\hat{1}\pm )}$$

$$v^{(\hat{2})}$$

$$r_{\hat{\alpha }}{}^{\alpha }\frac{\partial v_{(\hat{\alpha })}}{\partial u^{\alpha }}=0,$$

$$r_{\hat{0}}^{ab}{}^{\alpha }\frac{\partial v_{(\hat{0})}}{\partial u^{\alpha }}$$

$$r_{\hat{1}\pm }^{ab}{}^{\alpha }\frac{\partial v_{(\hat{1}\pm )}}{\partial u^{\alpha }}$$

$$r_{\hat{2}}^{iab}{}^{\alpha }\frac{\partial v_{(\hat{2})}}{\partial u^{\alpha }}$$

Full text

††: Class. Quantum Grav.

The formation of coordinate shocks is one of the important problems that must be overcome by any representation of Einstein’s equation that is to be used successfully in numerical relativity. Poor dynamical gauge conditions can and will lead to the formation of shocks (and consequently coordinate singularities) from the evolution of smooth initial data [1]. Linear degeneracy is a mathematical condition that prevents the formation of shocks in a large class of hyperbolic evolution systems [2, 3, 4]. The purpose of this note is to clarify the conditions under which the first-order generalized-harmonic representation of the vacuum Einstein system [5] is linearly degenerate. The original paper on this system claimed, without presenting a proof, that the system was linearly degenerate if a certain constant satisfied the condition $\gamma_{1}=-1$ [5]. Here we demonstrate that this claim is correct. While the proof is fairly straightforward, some readers of the original paper have questioned whether that condition is correct [6]. Consequently it seems appropriate to provide a more complete description of the derivation that demonstrates this fact.

The first-order generalized-harmonic representation of Einstein’s vacuum equation [5] can be written abstractly as a quasi-linear system,

[TABLE]

where $u^{\alpha}=\{\psi_{ab},\Pi_{ab},\Phi_{iab}\}$ is the collection of dynamical fields: $\psi_{ab}$ the spacetime metric, and its time and space derivatives $\Pi_{ab}$ and $\Phi_{iab}$. The quantities $A^{k\alpha}{}_{\beta}$ and $F^{\alpha}$ depend on $u^{\alpha}$ but not its derivatives. We use the notation $s^{\alpha}\,\partial_{u^{\alpha}}$ for vectors, and $t_{\alpha}\,du^{\alpha}$ for co-vectors on the space of dynamical fields. The principal parts of the first-order generalized-harmonic vacuum Einstein system, $\partial_{t}u^{\alpha}+A^{k\alpha}{}_{\beta}\partial_{k}u^{\beta}\simeq 0$, are given explicitly by

[TABLE]

where $N$ and $N^{k}$ are the lapse and shift associated with the standard $3+1$ representation of the metric $\psi_{ab}$, and where $\gamma_{1}$ and $\gamma_{2}$ are constants.111The constants $\gamma_{1}$ and $\gamma_{2}$ multiplied by certain constraints of the vacuum Einstein system were added to the equations in Ref. [5]. The resulting system is symmetric hyperbolic for any values of these constants. As shown here, the constant $\gamma_{1}$ effects the linear degeneracy of the system. The constant $\gamma_{2}$ effects the growth of small constraint violations, and must be positive, $\gamma_{2}>0$, for numerical stability. The characteristic matrix $n_{k}A^{k\alpha}{}_{\beta}$ for this system can be written as

[TABLE]

for waves propagating through a surface (chosen arbitrarily) with spacelike unit normal co-vector $n_{k}$. Summation over repeated indices, e.g., $\scriptstyle{k}$, $\scriptstyle{a}$, and $\scriptstyle{b}$, is implied. Linear degeneracy is a condition on the eigenvalues and eigenvectors of this characteristic matrix.

The left and right eigenvectors, $\ell^{\,\hat{\alpha}}$ and $r_{\hat{\alpha}}$ respectively, of the characteristic matrix $n_{k}A^{k\alpha}{}_{\beta}$ are defined by:

[TABLE]

The left eigenvectors $\ell^{\,\hat{\alpha}}{}_{\beta}du^{\beta}$ of the first-order generalized-harmonic vacuum Einstein system are given by

[TABLE]

while the right eigenvectors $r_{\hat{\alpha}}{}^{\beta}\partial_{u^{\beta}}$ are given by

[TABLE]

The first-order vacuum Einstein system is symmetric hyperbolic, since there exists a symmetric positive definite tensor $S_{\alpha\beta}$ on the space of fields that satisfies the condition $S_{\alpha\mu}A^{k\mu}{}_{\beta}\equiv A^{k}{}_{\alpha\beta}=A^{k}{}_{\beta\alpha}$ [5]. This implies that the left and right eigenvectors are related (up to normalizations) by $\ell^{\hat{\alpha}}{}_{\alpha}=S_{\alpha\beta}r_{\hat{\alpha}}{}^{\beta}$, and the associated eigenvalues must be equal $v_{(\hat{\alpha})}=v^{(\hat{\alpha})}$. These eigenvalues for the vacuum Einstein system are given by

[TABLE]

The quasi-linear hyperbolic evolution system, Eq. (1), is said to be linearly degenerate if all the eigenvalues of the characteristic matrix are constant along the corresponding right eigenvectors of that system, so that

[TABLE]

for each $\scriptstyle{\hat{\alpha}}$ [2]. The eigenvalues of the Einstein system, Eqs. (14)–(16), depend only on the lapse, $N$, the shift, $N^{k}$, and the unit normal vector $n^{k}$. These eigenvalues therefore depend only on the metric, $\psi_{ab}$, and not on its derivatives, $\Pi_{ab}$ or $\Phi_{iab}$. Thus the derivatives of the eigenvalues in the direction of the right eigenvectors are given by,

[TABLE]

These derivatives vanish, and consequently the system is linearly degenerate, if and only if $\gamma_{1}=-1$.

The original paper on the first-order generalized-harmonic vacuum Einstein system did not explicitly give expressions for either the left or the right eigenvectors [5]. The characteristic fields, $\hat{u}^{\hat{\alpha}}\equiv\ell^{\hat{\alpha}}{}_{\beta}u^{\beta}$, of this system were given, however, and from those the left eigenvectors could easily be inferred. The reported confusion about the correct conditions for linear degeneracy for this system may have arisen from an examination of the quantities $\ell^{\hat{\alpha}}{}_{\alpha}\partial v^{(\hat{\alpha})}/\partial u^{\alpha}$, which do not vanish unless $\gamma_{1}=-1$ and $\gamma_{2}=0$ [6]. These quantities involving the left eigenvectors (which are co-vectors, not true vectors) are non-covariant and are therefore meaningless from a fundamental mathematical viewpoint. In any case they are irrelevant because the formal definition of linear degeneracy given by Lax in Ref. [2] specifies that the right eigenvectors are to be used in Eq. (17), and this equation is covariant.

L.L. thanks the Morningside Center for Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China for their hospitality during a visit in which a portion of this research was performed. This research was supported in part by the National Science Foundation grants PHY-1604244 and DMS-1620366 to the University of California at San Diego.