Scalar–tensor theories of gravity can be formulated in the Einstein frame or in the Jordan frame (JF) which are related with each other by conformal transformations. Although the two frames describe the same physics and are equivalent, the stability of the field equations in the two frames is not the same. Here, we implement dynamical system and phase space approach as a robustness tool to investigate this issue. We concentrate on the Brans–Dicke theory in a Friedmann–Lemaitre–Robertson–Walker universe, but the results can easily be generalized. Our analysis shows that while there is a one-to-one correspondence between critical points in two frames and each critical point in one frame is mapped to its corresponds in another frame, however, stability of a critical point in one frame does not guarantee the stability in another frame. Hence, an unstable point in one frame may be mapped to a stable point in another frame. All trajectories between two critical points in phase space in one frame are different from their corresponding in other ones. This indicates that the dynamical behavior of variables and cosmological parameters is different in two frames. Hence, for those features of the study, which focus on observational measurements, we must use the JF where experimental data have their usual interpretation.

中文翻译：

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Jordan 和 Einstein 框架之间的相空间对应关系

引力的标量张量理论可以用 Einstein 框架或 Jordan 框架 (JF) 来表述，它们通过保角变换相互关联。虽然这两个框架描述了相同的物理并且是等价的，但是两个框架中的场方程的稳定性并不相同。在这里，我们实施动态系统和相空间方法作为研究这个问题的鲁棒性工具。我们专注于弗里德曼-勒梅特-罗伯逊-沃克宇宙中的布兰斯-迪克理论，但结果很容易被概括。我们的分析表明，虽然两帧中的关键点之间存在一一对应关系，并且一帧中的每个关键点都映射到另一帧中的对应点，但是，一帧中关键点的稳定性并不能保证在另一个框架中的稳定性。因此，一帧中的不稳定点可以映射到另一帧中的稳定点。一帧中相空间中两个临界点之间的所有轨迹与其他帧中的对应轨迹不同。这表明变量和宇宙学参数的动力学行为在两个框架中是不同的。因此，对于那些侧重于观察测量的研究特征，我们必须使用 JF，其中实验数据有其通常的解释。