[论文解读] How the green light was given for gravitational wave search

C. Denson Hill, Paweł Nurowski|arXiv (Cornell University)|Aug 30, 2016

Pulsars and Gravitational Waves Research参考文献 16被引用 25

一句话总结

本文追溯了引力波搜寻的理论基础，指出安杰伊·特劳特曼（Andrzej Trautman）1958年的研究是解决广义相对论中非线性引力波存在性长期疑虑的关键突破。通过在完整的非线性理论中严格定义引力波并证明其携带能量，特劳特曼的工作为实验探测提供了‘绿灯’，最终促成LIGO/VIRGO对双黑洞并合事件的探测。

ABSTRACT

The recent detection of gravitational waves by the LIGO/VIRGO team is an incredibly impressive achievement of experimental physics. It is also a tremendous success of the theory of General Relativity. It confirms the existence of black holes; shows that binary black holes exist; that they may collide and that during the merging process gravitational waves are produced. These are all predictions of General Relativity theory in its fully nonlinear regime. The existence of gravitational waves was predicted by Albert Einstein in 1916 within the framework of linearized Einstein theory. Contrary to common belief, even the very \emph{definition} of a gravitational wave in the fully nonlinear Einstein theory was provided only after Einstein's death. Actually, Einstein had arguments against the existence of nonlinear gravitational waves (they were erroneous but he did not accept this), which virtually stopped development of the subject until the mid 1950s. This is what we refer to as the \emph{Red Light} for gravitational waves research. In the following years, the theme was picked up again and studied vigorously by various experts, mainly Herman Bondi, Felix Pirani, Ivor Robinson and Andrzej Trautman, where the theoretical obstacles concerning gravitational wave existence were successfully overcome, thus giving the `Green Light' for experimentalists to start designing detectors, culminating in the recent LIGO/VIRGO discovery. In this note we tell the story of this theoretical breakthrough. Particular attention is given to the fundamental 1958 papers of Trautman, which seem to be lesser known outside the circle of General Relativity experts. A more detailed technical description of these 2 papers is given in the Appendix.

研究动机与目标

• 阐明广义相对论完整非线性框架下引力波被接受的历史与理论路径。
• 解决长期以来关于非线性引力波是否能够存在并携带能量的争议，该争议曾使研究停滞数十年。
• 确立此类波的存在并非线性化近似的产物，而是爱因斯坦完整场方程的物理预言。
• 证明在非线性理论中，辐射导致的能量损失可被严格定义与计算，从而实现对波辐射的物理解释。

提出的方法

• 使用未来 null 无穷远处的辐射边界条件，在完整的非线性爱因斯坦理论中形式化引力波的定义。
• 应用四维标架形式与冯·弗罗贝尼乌斯超势（von Freud superpotential），推导引力场的守恒能量-动量流。
• 在世界管面上应用斯托克斯定理，将时空边界与零类边界上的4-动量通量与总辐射能量联系起来。
• 将特劳特曼的能量损失 $p^0$ 定义为辐射4-动量的时间分量，从而量化引力辐射导致的能量发射。
• 构造满足辐射边界条件且具有非零 $p^0 > 0$ 的显式真空解（罗宾逊-特劳特曼时空），证实能量损失的存在。
• 证明能量损失与坐标系无关，仅依赖于空间超曲面的选择，确保物理一致性。

实验结果

研究问题

• RQ1 完整的非线性爱因斯坦理论是否允许存在可解释为具有非平面波前的引力波的解？
• RQ2 非线性理论中的引力波是否能够携带能量？若能，这种能量如何被严格定义与测量？
• RQ3 此类波的存在是否依赖于线性化近似？还是它们在完整理论中作为物理解自然出现？
• RQ4 能否构造满足真空爱因斯坦方程的精确解，使其表现出由于辐射导致的非零能量损失？
• RQ5 引力辐射导致的能量损失是否在坐标变换下保持不变，从而确保其物理实在性？

主要发现

• 特劳特曼1958年的研究首次在完整的非线性爱因斯坦理论中严格定义了引力波，解决了数十年来的理论不确定性。
• 通过冯·弗罗贝尼乌斯超势推导出引力场的能量-动量流 $\mathfrak{T}^\mu$，从而实现了引力能量的一致定义。
• 证明特劳特曼的能量损失 $p^0$ 非负且与坐标系无关，确认辐射确实携带可测量的能量。
• 构造出满足辐射边界条件且具有 $p^0 > 0$ 的显式真空解（罗宾逊-特劳特曼时空），证明非线性引力波可以携带能量。
• 这些解的黎曼张量在将来 null 无穷远处趋于N型，证实了辐射的波状特性。
• 研究结果表明，线性理论中的预测在非线性 regime 中依然具有物理有效性，从而为LIGO/VIRGO等探测器的设计提供了理论依据。