相对论引力中的维里恒等式：一维有效作用量和边界项的作用
Virial identities in relativistic gravity: 1D effective actions and the role of boundary terms
摘要：
维里恒等式（又称尺度恒等式）是一种积分恒等式，在非线性场论中有很多用途，包括建立孤立子和黑洞解的不通过定理，以及检查数值解的准确性。在本文中，我们提供了一个教学的理由，从标准的变分粒子力学的治疗，这样的积分恒等式的推导。在一维（1D）有效作用量的框架下，本文给出的处理方法给出了一组计算任何场论维里恒等式的有用公式。然后，我们提出，一个完整的相对论引力维里恒等式的处理必须考虑适当的边界项。对于广义相对论，这是Gibbons-Hawking-York边界项。我们用具体的例子来验证这个建议。我们的分析仅限于球对称的配置，产生一维有效的行动（离开更高的D有效行动，特别是轴对称的情况下，配套文件）。在这种情况下，我们表明，有一个特定的“规范”的选择，$i.e.$的坐标和参数化度量函数的选择，简化了计算的位力恒等式在广义相对论，使爱因斯坦-希尔伯特作用和吉布斯-霍金-约克边界项无贡献。在这种选择下，维里同一性完全来自物质作用。然而，对于一般的“量规”选择，情况并非如此。
英文摘要：
Virial (aka scaling) identities are integral identities that are useful for a variety of purposes in non-linear field theories, including establishing no-go theorems for solitonic and black hole solutions, as well as for checking the accuracy of numerical solutions. In this paper, we provide a pedagogical rationale for the derivation of such integral identities, starting from the standard variational treatment of particle mechanics. In the framework of one-dimensional (1D) effective actions, the treatment presented here yields a set of useful formulas for computing virial identities in any field theory. Then, we propose that a complete treatment of virial identities in relativistic gravity must take into account the appropriate boundary term. For General Relativity this is the Gibbons-Hawking-York boundary term. We test and confirm this proposal with concrete examples. Our analysis here is restricted to spherically symmetric configurations, which yield 1D effective actions (leaving higher-D effective actions and in particular the axially symmetric case to a companion paper). In this case, we show that there is a particular "gauge" choice, $i.e.$ a choice of coordinates and parameterizing metric functions, that simplifies the computation of virial identities in General Relativity, making both the Einstein-Hilbert action and the Gibbons-Hawking-York boundary term non-contributing. Under this choice, the virial identity results exclusively from the matter action. For generic "gauge" choices, however, this is not the case.
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