标题： 弯曲时空中的克莱因-戈登方程的模态解 显示英文标题

标题： Mode solutions of the Klein-Gordon equation in warped spacetimes

摘要： 为了简化克莱因-戈登方程（带最小耦合），我们引入了一种所谓的“模态解”的推广，这在罗伯逊-沃克宇宙的特殊情况中是众所周知的。 变量分离之后，我们得到一个低维的偏微分方程。 出现了约化版的戈登流，并且它是守恒的。 当第一个因子流形是洛伦兹型时，从通常的戈登流得到的共轭线性形式的意义上来看，不同的模态表现为相互正交的。 此外，在约化方程的解空间上定义了一个共轭线性形式。 当第二个因子流形具有常标量曲率时，可以将这一图景扩展到曲率耦合。 显示英文摘要

摘要： In order to reduce the Klein-Gordon equation (with minimal coupling), we introduce a generalization of the so-called "mode solutions" that are well-known in the special case of a Robertson-Walker universe. After separation of the variables, we end up with a partial differential equation in lower dimension. A reduced version of the Gordon current arises and is conserved. When the first factor-manifold is Lorentzian, distinct modes appear as mutually orthogonal in the sense of the sesquilinear form obtained from the customary Gordon current. Moreover, a sesquilinear form is defined on the space of solutions to the reduced equation. Extension of this picture to curvature coupling is possible when the second factor-manifold has a constant scalar curvature.