Wave Propagation Through Caustics: Applications in Gravitational Wave Physics

引力波——在空间和时间结构中传播的涟漪——是爱因斯坦广义相对论的一个基本预测。爱因斯坦的杰作已经过去了将近一个世纪，但引力波仍然令人沮丧地难以捉摸！通过测量快速旋转的中子星（脉冲星）的旋转速度，天文学家已经收集到了它们存在的有力证据。在地球上，物理学家乐观地认为十年内就能直接探测到。引力波（GWs）很重要，因为它们是由已知宇宙中最剧烈的过程产生的，比如超新星、黑洞合并和星系碰撞。让天文学家感到沮丧的是，这种强大的过程隐藏在尘埃和强磁场的背后。光不能穿透这层寿衣，但引力波可以。通过揭开裹尸布，GWs将揭示这些过程中心的动态心脏。随着2018年（est）激光干涉仪空间天线（LISA）的发射，天文学将进入一个新的时代。LISA的一个关键目标是探测双黑洞系统发出的引力波。理论上，仅从引力波信号就可以重建绕黑洞运行的致密天体的运动。美国宇航局和欧洲航天局最近的一份报告得出结论，LISA将在强场动力机制下为广义相对论提供明确而清晰的测试，并能够绘制黑洞附近的详细时空图。这是一个令人兴奋的前景，然而，如果我们要从嘈杂的背景中分离出小的GW信号，还需要进行大量的理论工作。在这个项目中，我将模拟所谓的极端质量比吸气（EMRI）事件，在这个事件中，一个小的致密体螺旋进入一个大黑洞（例如Sag a *，位于我们银河系的中心）。EMRIs发射的引力波信号是LISA的关键目标；因此，这个项目既及时又具有潜在的重要意义。最有希望的EMRIs建模方法需要计算作用在小物体上的重力自作用力。自作用力导致轨道能量的损失，导致小天体向内旋转，起初缓慢，但速度越来越快。在弯曲的时空中——例如，在黑洞附近——计算自作用力是非常困难的，因为它取决于小物体运动的整个历史！在这个项目中，我将开发新的数学方法来计算自作用力，以补充由南安普顿相对论小组成员开创的现有方法。自作用力产生于物体与其自身引力场中的扰动之间的相互作用。最近的研究表明，对于EMRIs来说，引力的自作用力可能主要是由于一个非局域分量，它是由围绕中心黑洞透镜多次的扰动引起的。为了研究这个想法，我将对引力波通过焦点（也称为焦散）的传播进行数学研究。我的主要动机是澄清自我力的物理起源，但其他应用也可能从这项工作中产生。由于波现象（衍射、折射、彩虹、光晕等）在物理科学中无处不在，我建议对焦散对波传播的影响进行多学科研究。我将借鉴地震学的理论发展；光学中的突变理论；数学中的分布理论；以及其他领域。我还将利用在南安普顿数学学院的广度和深度的经验。我希望与同事们一起，为精密引力波天文学的未来发展做出重大贡献。

Gravitational Waves -- propagating ripples in the fabric of space and time -- are a fundamental prediction of Einstein's theory of General Relativity. Nearly a century has passed since Einstein's masterwork, and yet gravitational waves remain frustratingly elusive! Astronomers have gathered strong evidence for their existence by measuring the rate of rotation of rapidly-spinning neutron stars ( pulsars ). Here on Earth, physicists are optimistic of direct detection within a decade.Gravitational waves (GWs) are important because they are generated by the most violent processes in the known Universe, such as supernovae, black hole mergers, and galaxy collisions. To the frustration of astronomers, such powerful processes are hidden behind shrouds of dust and strong fields. Light cannot penetrate this shroud, but gravitational waves can. By lifting the shroud, GWs will reveal the dynamic heart at the centre of such processes.Astronomy will enter a new era with the launch of the Laser Interferometer Space Antenna (LISA) in 2018 (est). A key aim of LISA is to detect gravitational waves emitted by binary black hole systems. In theory, the motion of a compact body orbiting a black hole can be reconstructed from the gravitational wave signal alone. A recent NASA-ESA report concludes that LISA will provide unambiguous and clean tests of the theory of general relativity in the strong field dynamical regime and be able to make detailed maps of spacetime near black holes. This is an exciting prospect, yet much theoretical work is needed if we are to separate out a small GW signal from a noisy background.In this project I will model so-called Extreme Mass Ratio Inspiral (EMRI) events, in which a small compact body spirals into a large black hole (e.g. Sag A*, residing at the centre of our galaxy). The gravitational wave signal emitted by EMRIs is a key target for LISA; hence this project is both timely and potentially significant.The most promising method for modelling EMRIs requires the calculation of a gravitational self-force which acts upon the small body. The self-force leads to a loss of orbital energy, causing the small body to spiral inwards, slowly at first, but with increasing rapidity. In curved spacetime -- for example, in the immediate vicinity of a black hole -- it turns out to be surprisingly difficult to compute the self-force, because it depends on the entire history of the small body's motion! In this project I will develop new mathematical methods to calculate the self-force, to complement the existing approach pioneered by members of the Southampton Relativity Group.The self-force arises from the interaction between a body and perturbations in its own gravitational field. Recent work suggests that, for EMRIs, the gravitational self-force may be primarily due to a non-local component which arises from perturbations that are lensed multiple times around the central black hole. To investigate this idea, I will make a mathematical study of the propagation of gravitational waves through focal points (also known as caustics). My primary motivation is to clarify the physical origin of the self-force, but other applications may also arise from this work.Since wave phenomena (diffraction, refraction, rainbows, glories, etc.) are ubiquitous across the physical sciences, I propose to make a multi-disciplinary study of the effects of caustics upon wave propagation. I will draw upon theoretical developments in seismology; catastrophe theory in optics; distribution theory in mathematics; and a range of other fields. I will also draw upon the breadth and depth of experience in the School of Mathematics at Southampton. Together with co-workers, I hope to make a significant contribution to the future development of precision Gravitational Wave Astronomy.