The Einstein-Hilbert action with a cosmological constant is the most general local four-dimensional action leading to second-order derivative equations of motion that are symmetric and divergence free. In higher dimensions, additional terms can appear. We investigate a generalised metric decomposition involving a scalar degree of freedom to express the higher-dimensional action as an effective four-dimensional scalar-tensor theory. From the higher-dimensional Ricci scalar alone and a subclass of our metric ansatz, we recover the subset of Horndeski theories with luminal speed of gravitational waves. More generally, beyond-Horndeski terms appear. When including a Gauss-Bonnet scalar in the higher-dimensional action, we generate contributions to all cubic-order second-derivative terms present in the degenerate higher-order scalar-tensor theory as well as higher-derivative terms beyond that. We discuss this technique as a way to generate healthy four-dimensional gravity theories with an extra scalar degree of freedom and outline further generalisations of our method.

中文翻译：

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Horndeski 理论及更高维度的其他理论

具有宇宙学常数的爱因斯坦-希尔伯特作用是最一般的局部四维作用，导致对称且无发散的二阶导数运动方程。在更高的维度中，可以出现附加项。我们研究了涉及标量自由度的广义度量分解，以将高维动作表达为有效的四维标量张量理论。仅从高维 Ricci 标量和我们的度量 ansatz 的一个子类中，我们用引力波的光速恢复了 Horndeski 理论的子集。更一般地，出现了超越 Horndeski 的术语。当在高维动作中包含 Gauss-Bonnet 标量时，我们对退化高阶标量张量理论中存在的所有三次二阶导数项以及更高阶导数项产生贡献。我们将这种技术作为一种生成具有额外标量自由度的健康四维引力理论的方法，并概述我们方法的进一步概括。