标题： 施瓦茨希尔德上分离向量波动方程的显式三角解耦为标量雷杰-惠勒方程 显示英文标题

标题： Explicit triangular decoupling of the separated vector wave equation on Schwarzschild into scalar Regge-Wheeler equations

摘要： 我们考虑施瓦茨希尔德时空上的矢量波方程，该方程可以看作是规范固定后的麦克斯韦方程（或洛伦兹规范）的结果。 在变量分离之后，径向模方程形成一个复杂的耦合线性常微分方程组。 我们概述了一个精确的抽象策略，将该系统解耦为三角形式，其中对角块由自旋-$s$标量雷格-惠勒方程组成，具有$s=0$或$1$。 然后，该策略被用于将径向模方程（频率和角动量非零）显式地转换为这种三角形式。 我们的解耦比文献中的先前结果更进一步，通过使三角形式显式化并尽可能地简化它。 此外，借助我们抽象表述的解耦策略，我们显著简化了最终结果的呈现和中间计算。 最后，我们注意到矢量波方程是一个更复杂方程的简单模型，例如规范固定的线性化引力方程（或德东德规范），以及如克尔背景等，我们预计相同的抽象解耦策略同样适用。 显示英文摘要

摘要： We consider the vector wave equation on the Schwarzschild spacetime, which can be considered as coming from the harmonic (or Lorenz) gauge fixed Maxwell equations. After a separation of variables, the radial mode equations form a complicated system of coupled linear ODEs. We outline a precise abstract strategy to decouple this system into triangular form, where the diagonal blocks consist of spin-$s$ scalar Regge-Wheeler equations, with $s=0$ or $1$. This strategy is then implemented to give an explicit transformation of the radial mode equations (with nonzero frequency and angular momentum) into this triangular form. Our decoupling goes a step further than previous results in the literature by making the triangular form explicit and reducing it as much as possible. Also, with the help of our abstractly formulated decoupling strategy, we have significantly streamlined both the presentation of the final results and the intermediate calculations. Finally, we note that the vector wave equation is a simple model for more complicated equations, like harmonic (or de Donder) gauge fixed linearized gravity, and backgrounds, like Kerr, where we expect the same abstract decoupling strategy to work as well.