标题： 广义自由能景观：Iyer-Wald形式主义 显示英文标题

标题： Generalized Free Energy Landscapes from Iyer-Wald Formalism

摘要： 广义的自由能景观在理解黑洞热力学和相变中起着至关重要的作用。在广义相对论中，可以直接从具有锥形奇点的黑洞贡献中推导出广义自由能。在这项工作中，我们将这一思想扩展到广义协变理论中。通过运用诺特第二定理，我们提出了拉格朗日量的一种替代形式，这可以阐明锥形奇点的作用。我们证明了通常情况下，锥形奇点的贡献依赖于正则化方案的具体实现，并且不是唯一确定的；这一特性在三维新质量引力中被明确展示并得到证实。然而，当引力理论由拉格朗日量 $L(g_{ab},R_{abcd})$ 描述时，这些歧义可以被锥形奇点引起的二阶（及更高阶）修正吸收。此外，对于诸如广义相对论和嗡嗡引力等某些理论，这种贡献简化为一个定义明确的结果。但是，嗡嗡引力中广义自由能的解释有所不同，其极值对应于锥形奇点的几何结构。我们的结果揭示了广义相对论之外广义自由能的特殊性质。 显示英文摘要

摘要： The generalized free energy landscape plays a pivotal role in understanding black hole thermodynamics and phase transitions. In general relativity, one can directly derive the generalized free energy from the contributions of black holes exhibiting conical singularities. In this work, we extend this idea to general covariant theories. By employing Noether's second theorem, we present an alternative formulation of the Lagrangian, which can elucidate the role of conical singularities. We demonstrate that, in general, the contribution from conical singularities depends on the specific implementation of the regularization scheme and is not uniquely determined; this feature is explicitly exhibited and confirmed in three-dimensional new massive gravity. Nevertheless, these ambiguities can be absorbed into the second-order (and higher) corrections induced by conical singularities when the gravitational theory is described by the Lagrangian $L(g_{ab},R_{abcd})$. Moreover, for certain theories such as general relativity and Bumblebee gravity, this contribution simplifies to a well-defined result. However, the interpretation of the generalized free energy in Bumblebee gravity is somewhat different, with its extrema corresponding to the geometry of conical singularities. Our results uncover the particular properties of the generalized free energy beyond general relativity.