Using \( \mathcal{N} \) = 8 supergravity as a theoretical laboratory, we extract the 3PM gravitational eikonal for two colliding massive scalars from the classical limit of the corresponding elastic two-loop amplitude. We employ the eikonal phase to obtain the physical deflection angle and to show how its non-relativistic (NR) and ultra-relativistic (UR) regimes are smoothly connected. Such a smooth interpolation rests on keeping contributions to the loop integrals originating from the full soft region, rather than restricting it to its potential sub-region. This task is efficiently carried out by using the method of differential equations with complete near-static boundary conditions. In contrast to the potential-region result, the physical deflection angle includes radiation-reaction contributions that are essential for recovering the finite and universal UR limit implied by general analyticity and crossing arguments. We finally discuss the real emission of massless states, which accounts for the imaginary part of the 3PM eikonal and for the dissipation of energy-momentum. Adopting a direct approach based on unitarity and on the classical limit of the inelastic tree-level amplitude, we are able to treat \( \mathcal{N} \) = 8 and General Relativity on the same footing, and to complete the conservative 3PM eikonal in Einstein’s gravity by the addition of the radiation-reaction contribution. We also show how this approach can be used to compute waveforms, as well as the differential and integrated spectra, for the different radiated massless fields.

使用\( \mathcal{N} \)= 8 超引力作为理论实验室，我们从相应的弹性双环振幅的经典极限中提取了两个碰撞质量标量的 3PM 引力标量。我们使用 eikonal 相位来获得物理偏转角，并展示其非相对论 (NR) 和超相对论 (UR) 状态如何平滑连接。这种平滑的插值取决于保持对源自完整软区域的循环积分的贡献，而不是将其限制在其潜在的子区域。通过使用具有完整近静态边界条件的微分方程方法，可以有效地执行此任务。与潜在区域结果相反，物理偏转角包括辐射反应贡献，这对于恢复一般分析性和交叉论证所隐含的有限和普遍的 UR 极限至关重要。我们最后讨论了无质量状态的真实发射，它解释了 3PM 特征的虚部和能量动量的耗散。采用基于幺正性和非弹性树级振幅的经典极限的直接方法，我们能够处理\( \mathcal{N} \) = 8 和广义相对论在同一基础上，并通过添加辐射反应贡献来完成爱因斯坦引力中的保守的 3PM eikonal。我们还展示了如何使用这种方法来计算不同辐射无质量场的波形以及微分和积分光谱。