New invariants for the gravitational two-body problem

目前，我们对宇宙的认知建立在“视觉”的基础上。自从1610年伽利略探测到木星的卫星以来，我们一直在努力提高望远镜的灵敏度和分辨率。这些是我们观察宇宙的眼睛，在整个电磁频谱中工作。然而，只有“眼睛”，我们的感知是有限的；宇宙的大部分仍然是黑暗的，或者被尘埃和气体的云层所笼罩。如果我们能“听到”宇宙的声音呢？这并不像看起来那么离奇。爱因斯坦的广义相对论预言了“引力波”的存在：时空结构中以光速传播的涟漪。爱因斯坦的理论已经有一个世纪的历史了，到目前为止已经通过了所有的实验测试。事实上，对脉冲星轨道长达十年的仔细观察证实了引力波的存在(出色的工作赢得了1993年的诺贝尔奖)。人们普遍预计，我们正站在发展“听力”的尖端。像LIGO和Virgo这样的增强型探测器将在未来几年上线，应该能让我们第一次“听到”引力波。引力波是由宇宙中最强大的过程产生的，例如超大质量黑洞的轨道。与我们的“眼睛”不同，我们的“耳朵”只有微弱的方向性，因此可以听到各种各样的声源和周围环境的噪音。关键的挑战将是将有趣的来源从背景中分离出来。这项挑战类似于在喧闹的派对上试图在喧闹的人群中倾听朋友的声音：耳朵是不够的，高度进化的大脑也是必不可少的！要想取得成功，准确地知道我们在听什么是至关重要的。具体地说，我们需要从致密双星发出的引力波信号的高度精确的模型。在这个项目中，我的目标是改进和提升我们对这类信号的建模。为了实现这一目标，我必须解决相对论中的一个基本问题：预测两个致密物体的运动，例如黑洞，它们在相互引力的作用下运动。为什么这么简单的问题还没有“解决”呢？因为这不是那么简单！爱因斯坦的理论同时描述了舞台如何影响演员以及演员如何影响舞台。用约翰·惠勒的话说，“物质告诉空间如何弯曲，空间告诉物质如何移动”。换句话说，在动态场景中找到“精确”的数学解是困难的，甚至是不可能的！相反，我们必须开发和应用一系列数值和近似工具。我的关键主张是，在引力两体问题的100年历史中，有一些与物理可观测的量(如红移、进动角、潮汐应力等)有关的“不变量”尚未计算出来。不变量是至关重要的，因为它们允许我们比较、校准和增强目前使用的各种数学方法。奇怪的是，我想象这些不变量是罗塞塔石碑的一部分，用于在数学“语言”之间进行翻译。我建议首先通过计算不变量(本身是一项困难的任务)，然后研究它们在解决相同问题的其他三种方法中的作用，来探索这个想法。我希望与加拿大、法国和美国的领先团队合作，帮助英国在这一令人兴奋的领域成为领导者。

Our perception of the Universe, at present, is based on "sight". Ever since Galileo spied the moons of Jupiter in 1610, we have been driven to improve the sensitivity and resolution of telescopes. These are our eyes on the Universe, working across the electromagnetic spectrum. Yet, with only "eyes", our perception is limited; much of the Universe remains dark, or shrouded from view behind clouds of dust and gas. What if we could "hear" the Universe? This is not as fanciful as it may seem. Einstein's theory of General Relativity predicts the existence of "gravitational waves": ripples in the fabric of spacetime which propagate at the speed of light. Einstein's theory, now a century old, has passed every experimental test so far devised. Indeed, the existence of gravitational waves was confirmed by careful decade-long observations of a pulsar's orbit (brilliant work which won the Nobel prize in 1993). It is widely anticipated that we stand on the cusp of developing "hearing". Enhanced detectors such as LIGO and VIRGO, coming online in the next few years, should enable us to "hear" gravitational waves for the first time. Gravitational waves are generated by the most powerful processes in the Universe, such as the orbits of supermassive black holes. Unlike our "eyes", our "ears" will be only weakly directional, and thus will hear a wide variety of sources and ambient noise. The key challenge will be to separate the interesting sources from the background. The challenge is akin to trying to listen to a friend over the hubbub at a noisy party: ears are not sufficient, a highly-evolved brain is essential too! To succeed, it is crucial to know precisely what we are listening for. Specifically, we need highly accurate models of gravitational wave signals emanating from compact binaries. In this project, I aim to improve and upgrade our modelling of such signals. To achieve this aim, I must address a foundational problem in relativity: predicting the motion of two compact bodies, such as black holes, moving under mutual gravitational attraction. Why hasn't such a simple problem been "solved" already? Because it's not so simple! Einstein's theory describes, simultaneously, how the stage affects the actors and how the actors affect the stage. In the words of John Wheeler, "Matter tells space how to curve. Space tells matter how to move". In other words, finding "exact" mathematical solutions in dynamical scenarios is hard or impossible! Instead, we must develop and apply a range of numerical and approximation tools.My key claim is that there are certain "invariants", related to physically-observable quantities (such as redshift, precession angle; tidal stress, etc.), that are yet to be computed for the gravitational two-body problem over its 100-year history. Invariants are crucial, as they allow us to compare, calibrate and enhance the various mathematical methods currently in use. Fancifully, I imagine these invariants to be part of a Rosetta stone for translation between mathematical "languages". I propose to explore the idea by, first, calculating the invariants (itself a difficult task) and, then, investigating their role in three other approaches to the same problem. I hope to work with leading teams in Canada, France and the US, and to help establish the UK as a leader in this exciting area.