膨胀宇宙中的光学谐振腔
Optical cavity resonator in an expanding universe
摘要：
研究了放置在由Robertson-Walker度规描述的膨胀流形上的谐振光学腔中电磁驻波的频率演化。我们建立了一个局部坐标系，在这个坐标系中，时空是局部闵可夫斯基的。然而，由于罗伯逊-沃克度规的共形性质，传统的局部惯性坐标变换在局部时间坐标的物理解释中引入了模糊性。因此，与常识预期相反，单独直接实现EEP并不允许我们决定原子钟是否以与基于腔的EM模式的时钟相同的速率滴答。为了解决这一问题，我们利用麦克斯韦方程组分析了膨胀宇宙中腔的刚度及其电磁模式的振荡。我们发现，随着宇宙的膨胀，腔的大小和电磁频率在共形坐标中经历绝热漂移。我们建立了电磁模式的振荡方程，用WKB近似求解，并将坐标相关量约化为本地观测者用原子钟计时所测量的量。结果表明，通过局域坐标变换，腔频率的绝热漂移被完全抵消，基于腔电磁模的原子钟计数的时间与原子钟计数的时间具有相同的速率。我们的结论是，应该没有宇宙学漂移的频率振荡的电磁驻波的谐振腔相比，频率的原子钟。连续比较光学谐振腔的频率与原子钟的频率，产生了一个强大的零测试的局部各向同性的哈勃膨胀和爱因斯坦等效原理在宇宙学中。
英文摘要：
We study evolution of frequency of a standing electromagnetic (EM) wave in a resonant optical cavity placed to the expanding manifold described by the Robertson-Walker metric. One builds a local coordinate system in which spacetime is locally Minkowskian. However, due to the conformal nature of the Robertson-Walker metric the conventional transformation to the local inertial coordinates introduces ambiguity in the physical interpretation of the local time coordinate. Therefore, contrary to a common-sense expectation, a straightforward implementation of EEP alone does not allow us to decide whether atomic clocks ticks at the same rate as the clocks based on EM modes of a cavity. To resolve the ambiguity we analyzed the cavity rigidity and the oscillation of its EM modes in an expanding universe by employing the Maxwell equations. We found out that both the size of the cavity and the EM frequency experience an adiabatic drift in conformal coordinates as the universe expands. We set up the oscillation equation for the EM modes, solve it by the WKB approximation, and reduce the coordinate-dependent quantities to their counterparts measured by a local observer who counts time with atomic clock. The solution shows that there is a perfect cancellation of the adiabatic drift of cavity's frequency by the transformation to local coordinates, and the time counted by the clocks based on EM modes of cavity has the same rate as that of atomic clocks. We conclude that there should be no cosmological drift of frequency of a standing EM wave oscillating in the cavity resonator as compared to the frequency of atomic clocks. Continuous comparison of the frequency of the optical cavity resonator against that of atomic clock yields a powerful null test of the local isotropy of the Hubble expansion and the Einstein equivalence principle in cosmology.
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