Nonlinear electrodynamics as a symmetric hyperbolic system
Nonlinear electrodynamics as a symmetric hyperbolic system
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作为对称双曲系统的非线性电动力学
DOI：
10.1103/physrevd.92.084024
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发表时间：
2015
影响因子：
5
通讯作者：
Oscar A. Reula
中科院分区：
文献类型：
--
作者：
Fernando Abalos;F. Carrasco;'Erico Goulart;Oscar A. Reula
推广麦克斯韦电磁学并源自拉格朗日形式主义的非线性理论具有色散关系，其中传播平面分解为与取决于电磁场的逐点值的两个有效度量相对应的零平面。这些有效的洛伦兹度量共享电磁场的零方向（通常是两个）。我们证明，当且仅当这些度量产生的锥体具有非空交集时，该理论才是对称双曲的，即存在 Geroch [26] 意义上的对称化族，它们对于锥体交集内部的所有协向量都是正定的。因此，对于这些理论，初值问题是很好提出的。我们用几个具有物理意义的非线性模型来说明这种方法的威力，例如 Born\char21{}Infeld、Gauss\char21{}Bonnet 和 Euler\char21{}Heisenberg。
Nonlinear theories generalizing Maxwell's electromagnetism and arising from a Lagrangian formalism have dispersion relations in which propagation planes factor into null planes corresponding to two effective metrics which depend on the pointwise values of the electromagnetic field. These effective Lorentzian metrics share the null (generically two) directions of the electromagnetic field. We show that the theory is symmetric hyperbolic if and only if the cones these metrics give rise to have a nonempty intersection, namely, that there exist families of symmetrizers in the sense of Geroch [26] which are positive definite for all covectors in the interior of the cones intersection. Thus, for these theories, the initial value problem is well posed. We illustrate the power of this approach with several nonlinear models of physical interest such as Born\char21{}Infeld, Gauss\char21{}Bonnet, and Euler\char21{}Heisenberg.