We report on critical phenomena in the gravitational collapse of electromagnetic waves. Generalizing earlier results that focused on dipole electromagnetic waves, we compare here with quadrupole waves in axisymmetry. We perform numerical simulations of dipole and quadrupole wave initial data, fine tuning both sets of data to the onset of black hole formation in order to study the critical solution and related critical phenomena. We observe that different multipole moments have different symmetries, indicating that the critical solution for electromagnetic waves cannot be unique, at least not globally. This is confirmed in our numerical simulations: while dipole data lead to a single center of collapse, at the center of symmetry, quadrupole data feature two separate centers of collapse on the symmetry axis, above and below the center of symmetry—reminiscent of similar findings reported for critical collapse of vacuum gravitational waves. While the critical solution for neither the dipole nor the quadrupole data is exactly self-similar, we find that their approximate echoing periods appear to differ, as do the critical exponents. We discuss whether the centers of collapse found for dipole and quadrupole data might all have the same properties, which would suggest a “local uniqueness” of the critical solution. Instead, we provide some evidence—including the differing echoing periods and critical exponents—suggesting that the critical solutions are distinct even locally. We speculate on the implications of our findings for critical phenomena in the collapse of vacuum gravitational waves, which share with electromagnetic waves the absence of a spherically symmetric critical solution.

中文翻译：

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电磁偶极子和四极子波引力坍缩的临界现象

我们报告了电磁波引力坍缩的关键现象。概括早期关注偶极电磁波的结果，我们在这里与轴对称的四极波进行比较。我们对偶极子和四极子波初始数据进行数值模拟，将两组数据微调到黑洞形成的开始，以研究临界解和相关的临界现象。我们观察到不同的多极矩具有不同的对称性，表明电磁波的临界解不可能是唯一的，至少不是全局唯一的。这在我们的数值模拟中得到证实：虽然偶极数据导致单个坍塌中心，但在对称中心，四极数据在对称轴上具有两个独立的坍塌中心，对称中心的上方和下方——让人想起真空引力波临界坍缩的类似发现。虽然偶极子和四极子数据的临界解都不是完全自相似的，但我们发现它们的近似回波周期似乎不同，临界指数也是如此。我们讨论为偶极子和四极子数据找到的坍缩中心是否可能都具有相同的属性，这表明关键解的“局部唯一性”。相反，我们提供了一些证据——包括不同的回声周期和关键指数——表明关键解决方案即使在局部也是不同的。我们推测我们的发现对真空引力波坍缩的关键现象的影响，