We study the variational principle and derivation of the field equations for different classes of teleparallel gravity theories, using both their metric-affine and covariant tetrad formulations. These theories have in common that, in addition to the tetrad or metric, they employ a flat connection as additional field variable, but dthey iffer by the presence of absence of torsion and nonmetricity for this independent connection. Besides the different underlying geometric formulation using a tetrad or metric as fundamental field variable, one has different choices to introduce the conditions of vanishing curvature, torsion, and nonmetricity, either by imposing them a priori and correspondingly restricting the variation of the action when the field equations are derived, or by using Lagrange multipliers. Special care must be taken, since these conditions form non-holonomic constraints. Here, we explicitly show that all of the aforementioned approaches are equivalent, and that the same set of field equations is obtained, independently of the choice of the geometric formulation and variation procedure. We further discuss the consequences arising from the diffeomorphism invariance of the gravitational action, and show how they establish relations between the gravitational field equations.

中文翻译：

---

Teleparallel重力理论中的变分原理

我们使用量度仿射和协变四元公式，研究了不同类别的远平行重力理论的变分原理和场方程的推导。这些理论的共同点是，除了四线或公制以外，它们还采用扁平连接作为附加的场变量，但是由于这种独立连接不存在扭转和非对称性，因此它们会相互影响。除了使用四分之一或度量作为基本场变量的不同基础几何公式外，还有其他选择来引入消失的曲率，扭转和非度量条件，方法是先将其强加于先，然后相应地限制运动时场的变化方程可以导出，也可以使用拉格朗日乘数来推导。必须特别注意，因为这些条件形成了非完整的约束。在这里，我们明确表明，所有上述方法都是等效的，并且获得了相同的场方程组，而与几何公式和变化过程的选择无关。我们进一步讨论了引力作用的微分不变性引起的结果，并说明了它们如何建立引力场方程之间的关系。