arXiv 2202.063492022-02-13gr-qcastro-ph.COhep-th

Weyl几何中的耦合物质和曲率：共形不变的$f\left(R,L_m\right)$引力

Coupling matter and curvature in Weyl geometry: conformally invariant $f\left(R,L_m\right)$ gravity

Tiberiu Harko, Shahab Shahidi

备注：: 22 pages, 2 figures, accepted for publication in EPJC
DOI：: 10.1140/epjc/s10052-022-10126-1

摘要：

本文研究了共形二次Weyl引力中物质与几何的耦合，通过假设一个形式为$L_m\tilde{R}^2$的耦合项，其中$L_m$是普通物质的拉格朗日量，$\tilde{R}$是Weyl标量.耦合明确满足共形不变性的理论。借助辅助标量场和外尔标量来表示$\tilde{R}^2$，引力作用量可以线性化，从而在黎曼空间中导出共形不变的$f\left(R,L_m\right)$型理论，其中物质拉格朗日量与里奇标量非极小耦合。得到了该理论的引力场方程和能量动量平衡方程。物质的能量-动量张量的发散不会消失，并且产生了一个额外的力，取决于外尔矢量和物质拉格朗日量。本文还讨论了该理论的热力学解释。导出了广义泊松方程，并详细讨论了运动方程的牛顿极限。还考虑了行星在外力作用下的近日点进动，并从水星的观测数据中得到了太阳系外尔矢量大小的限制。该理论的宇宙学意义也被认为是一个平坦的，均匀的和各向同性的Friedmann-Lemaitre-Robertson-Walker几何的情况下，它表明，该模型可以给出一个很好的描述观测数据的哈勃函数的红移的顺序为$z\approx 3$。

英文摘要：

We investigate the coupling of matter to geometry in conformal quadratic Weyl gravity, by assuming a coupling term of the form $L_m\tilde{R}^2$, where $L_m$ is the ordinary matter Lagrangian, and $\tilde{R}$ is the Weyl scalar. The coupling explicitly satisfies the conformal invariance of the theory. By expressing $\tilde{R}^2$ with the help of an auxiliary scalar field and of the Weyl scalar, the gravitational action can be linearized, leading in the Riemann space to a conformally invariant $f\left(R,L_m\right)$ type theory, with the matter Lagrangian nonminimally coupled to the Ricci scalar. We obtain the gravitational field equations of the theory, as well as the energy-momentum balance equations. The divergence of the matter energy-momentum tensor does not vanish, and an extra force, depending on the Weyl vector, and matter Lagrangian is generated. The thermodynamic interpretation of the theory is also discussed. The generalized Poisson equation is derived, and the Newtonian limit of the equations of motion is considered in detail. The perihelion precession of a planet in the presence of an extra force is also considered, and constraints on the magnitude of the Weyl vector in the Solar System are obtained from the observational data of Mercury. The cosmological implications of the theory are also considered for the case of a flat, homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker geometry, and it is shown that the model can give a good description of the observational data for the Hubble function up to a redshift of the order of $z\approx 3$.

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