shock waves的$D$维碰撞的辐射：一阶公式和各阶角因式分解的证明
Radiation from a $D$-dimensional collision of shock waves: proof of first order formula and angular factorisation at all orders
摘要：
在以前的两篇论文中，我们计算了两个$D$维Aichelburg-Sexl激波迎头碰撞中的非弹性$ε$，用微扰理论计算了碰撞光锥的几何形状。作为精确数值拟合获得的一阶结果产生了非常简单的公式$ε_{\rm 1st\, order} = 1/2 - 1/D$。在这里，我们表明，分析，这一结果是准确的一阶微扰理论。此外，我们澄清了微扰理论和非弹性的角功率围绕碰撞的对称轴$(θ= 0,π)$的角系列之间的关系。为了建立这些结果，首先，我们表明，在零无穷大的角度依赖factorises秩序秩序的微扰理论，作为一个隐藏的对称性的结果。其次，我们表明，一致的截断的角度系列的权力$\sin^2 θ$在某种程度上$O(n)$需要知识的度量扰动$O(n+1)$。特别是，这证明了一阶微扰理论中使用的各向同性假设。然后，我们计算，分析，所有的条款，有助于非弹性和线性依赖于初始条件（表面条款），包括二阶的贡献。
英文摘要：
In two previous papers we have computed the inelasticity $ε$ in a head-on collision of two $D$-dimensional Aichelburg-Sexl shock waves, using perturbation theory to calculate the geometry in the future light-cone of the collision. The first order result, obtained as an accurate numerical fit, yielded the remarkably simple formula $ε_{\rm 1st\, order} = 1/2 - 1/D$. Here we show, analytically, that this result is exact in first order perturbation theory. Moreover, we clarify the relation between perturbation theory and an angular series of the inelasticity's angular power around the symmetry axis of the collision $(θ= 0,π)$. To establish these results, firstly, we show that at null infinity the angular dependence factorises order by order in perturbation theory, as a result of a hidden symmetry. Secondly, we show that a consistent truncation of the angular series in powers of $\sin^2 θ$ at some order $O(n)$ requires knowledge of the metric perturbations up to $O(n+1)$. In particular, this justifies the isotropy assumption used in first order perturbation theory. We then compute, analytically, all terms that contribute to the inelasticity and depend linearly on the initial conditions (surface terms), including second order contributions.
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