arXiv 2402.018472024-02hep-thgr-qc

事件的相对性：JT引力与线性化广义相对论中的示例

Relativity of the event: examples in JT gravity and linearized GR

Francesco Nitti, Federico Piazza, Alexander Taskov

备注：: 40 pages, 10 Figures. Comments welcome. v2: improved discussion, references added. v3: minor changes. Matches published version

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AI总结：

该研究指出量子引力定域表述存在事件相对性，参考系变换具有概率性，在JT引力中验证并计算视界位置方差，还提出爱因斯坦引力的计算方法与线性化引力的趣味拓展。

摘要：

众所周知，量子引力中的可观测量是在AdS（反德西特）空间或闵可夫斯基空间的边界上渐近定义的。然而，通过规范固定坐标系，或对场算符进行适当的修饰（dressing），也可以采用一种近似的“准定域”方法，该方法能够解释时空内部的一组观测者所进行的测量。具体而言，人们可以将空间坐标附着在这些观测者的世界线上，并将他们的固有时用作时间坐标。\n本文强调，任何此类定域表述都必须面对事件的相对性：改变参考系（即观测者集合）意味着点事件及其识别方式会发生重组。由此导致的结果是，在量子引力中，不同参考系之间的坐标变换具有概率性。\n我们在Jackiw-Teitelboim（JT）引力中给出了这一机制的具体实现：在该理论中，体空间中的一个点可以通过锚定在边界上的测地线来操作定义。我们描述了多种实现该定义的方式，每种方式对应一个不同的参考系，并计算了部分参考系之间变换的方差。具体来说，我们计算了黑洞视界位置的方差，发现在大多数参考系中视界都呈现弥散状态。\n随后，我们提出了在假设度规波函数已知的前提下，如何在爱因斯坦引力中计算这一效应的思路。核心想法是将度规波函数在半经典态的基矢上展开：该基矢的每个元素都满足标准的、确定性的坐标变换，因此最终结果可通过叠加得到。\n作为一个趣味拓展，我们通过向闵可夫斯基时空添加引力波的量子叠加来改造我们熟悉的洛伦兹 boost（ boost 变换），并计算了线性化引力中到加速参考系的概率变换。最后，我们尝试用修饰算符的语言来阐释事件的相对性。

英文摘要：

Observables in quantum gravity are famously defined asymptotically, at the boundary of AdS or Minkowski spaces. However, by gauge fixing a coordinate system or suitably dressing the field operators, an approximate, "quasi-local" approach is also possible, that can give account of the measurements performed by a set of observers living inside the spacetime. In particular, one can attach spatial coordinates to the worldlines of these observers and use their proper times as a time coordinate. Here we highlight that any such local formulation has to face the relativity of the event, in that changing frame (= set of observers) implies a reshuffling of the point-events and the way they are identified. As a consequence, coordinate transformations between different frames become probabilistic in quantum gravity. We give a concrete realization of this mechanism in Jackiw-Teitelboim gravity, where a point in the bulk can be defined operationally with geodesics anchored to the boundary. We describe different ways to do so, each corresponding to a different frame, and compute the variances of the transformations relating some of these frames. In particular, we compute the variance of the location of the black hole horizon, which appears smeared in most frames. We then suggest how to calculate this effect in Einstein gravity, assuming knowledge of the wavefunction of the metric. The idea is to expand the latter on a basis of semiclassical states. Each element of this basis enjoys standard/deterministic coordinate transformations and the result is thus obtained by superposition. As a divertissement, we sabotage the familiar Lorentz boosts by adding to Minkoswki spacetime a quantum superposition of gravitational waves and compute the probabilistic transformation to a boosted frame in linearized gravity. Finally, we attempt to translate the relativity of the event in the language of dressed operators.

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