In this paper we revisit the double cone wormhole introduced by Saad, Shenker and Stanford (SSS), which was shown to reproduce the ramp in the spectral form factor. As a first approximation we can say that this solution computes Tr[e^−iKT], a trace of the “evolution” operator that generates Schwarzschild time translations on the two sided wormhole geometry. This point of view leads to a simple way to compute the normalization factor of the wormhole. When we have bulk matter fields, SSS suggested using a modified evolution \(\widetilde{K}\) which involves a slightly complex geometry, so that we are really computing \({\text{Tr}}\left[{e}^{-i\widetilde{K}T}\right]\). We argue that, for general black holes, the spectrum of \(\widetilde{K}\) is given by quasinormal mode frequencies. We explain that this reproduces various features that were previously predicted from the spectral form factor on hydrodynamics grounds. We also give a general algebraic construction of the modified boost in terms of operators constructed from half sided modular inclusions. For the special case of JT gravity, we work out the backreaction of matter on the geometry of the double cone and find that it deforms the geometry in an undesirable direction. We finally give some comments on the possible physical interpretation of \(\widetilde{K}\).

在本文中，我们重新审视 Saad、Shenker 和斯坦福 (SSS) 提出的双锥虫洞，该虫洞被证明可以重现光谱形状因子中的斜坡。作为第一个近似，我们可以说该解计算 Tr[ e ^−iKT ]，即“进化”算子的迹，该算子在两侧虫洞几何结构上生成史瓦西时间平移。这种观点导致了一种计算虫洞归一化因子的简单方法。当我们有大块物质场时，SSS 建议使用修改后的演化\(\widetilde{K}\)，其中涉及稍微复杂的几何形状，以便我们真正计算\({\text{Tr}}\left[{e} ^{-i\widetilde{K}T}\right]\)。我们认为，对于一般黑洞，\(\widetilde{K}\)的频谱由准正态模频率给出。我们解释说，这再现了之前根据流体动力学的光谱形状因子预测的各种特征。我们还根据从半边模包含物构造的算子给出了修改后的 boost 的一般代数构造。对于JT引力的特殊情况，我们计算了物质对双锥体几何形状的反作用，发现它使几何形状朝不希望的方向变形。我们最后对\(\widetilde{K}\)可能的物理解释给出一些评论。