[论文解读] A numerical study of boson stars

Chi Wai Lai|arXiv (Cornell University)|Jan 1, 2009

Pulsars and Gravitational Waves Research参考文献 64被引用 11

一句话总结

本论文通过平滑标量场演化模拟，对球对称与轴对称配置下的广义相对论性玻色子星进行了数值研究，首次在轴对称情形下展示了I型临界现象，验证了与线性微扰理论预测一致的稳定振荡终态，并开发了一种高效算法，用于构建具有k = 1和k = 2角动量量子数的旋转玻色子星，揭示了每种情况下的最大质量。

ABSTRACT

In this thesis we present a numerical study of general relativistic boson stars in both spherical symmetry and axisymmetry. We consider both time-independent problems, involving the solution of equilibrium equations for rotating boson stars, and time-dependent problems, focusing on black hole critical behaviour associated with boson stars. Boson stars are localized solutions of the equations governing a massive complex scalar field coupled to the gravitational field. They can be simulated using more straightforward numerical techniques than are required for fluid stars. In particular, the evolution of smooth initial data for a scalar field tends to stay smooth, in sharp contrast to hydrodynamical evolution, which tends to develop discontinuities, even from smooth initial conditions. At the same time, relativistic boson stars share many of the same features with respect to the strong-field gravitational interaction as their fermionic counterparts. A detailed study of their dynamics can thus potentially lead to a better understanding of the dynamics of compact fermionic stars (such as neutron stars), while the relative ease with which they can be treated numerically makes them ideal for use in theoretical studies of strong gravity. In this last vein, the study of the critical phenomena that arise at the threshold of black hole formation has been a subject of intense interest among relativists and applied mathematicians over the past decade. Type I critical phenomena, in which the black hole mass jumps discontinuously at threshold, were previously observed in the dynamics of spherically symmetric boson stars by Hawley and Choptuik [1, 2]. We extend this work and show that, contrary to previous claims, the subcritical end-state is well described by a stable boson star executing a large amplitude oscillation with a frequency in good agreement with that predicted for the fundamental normal mode of the end-state star from linear perturbation theory. We then extend our studies of critical phenomena to the axisymmetric case, studying two distinct classes of parametrized families of initial data whose evolution generates families of spacetimes that "interpolate" between those than contain a black hole and those that do not. In both cases we find strong evidence for a Type I transition at threshold, and are able to demonstrate scaling of the lifetime for near-critical configurations of the type expected for such a transition. This is the first time that Type I critical solutions have been simulated in axisymmetry (all previous general relativistic calculations of this sort imposed spherical symmetry). In addition, we develop an efficient algorithm for constructing equilibrium configurations of rotating boson stars, which are characterized by discrete values of an angular momentum parameter, k (an azimuthal quantum number). We construct families of solutions for k = 1 and k = 2, and demonstrate the existence of a maximum mass in each case.

研究动机与目标

• 通过数值相对论研究球对称与轴对称时空中的玻色子星动力学。
• 在轴对称情形下研究黑洞形成阈值处的临界现象，拓展此前仅限于球对称情形的研究。
• 开发一种高效的数值算法，用于构建具有离散角动量量子数的平衡态旋转玻色子星。
• 确定旋转玻色子星族中最大质量构型的存在性及其性质。
• 验证亚临界终态对线性微扰理论预测的稳定性。

提出的方法

• 求解带有质量的复标量场与引力耦合的爱因斯坦-克莱因-戈登系统。
• 采用时间依赖的演化方法，使用平滑初值以避免流体动力学模拟中常见的不连续性。
• 在空间和时间上采用谱方法与时间推进积分器，实现标量场与度规的高精度演化。
• 开发一种带松弛的打靶法，用于构建k = 1和k = 2的静止旋转玻色子星解。
• 对近阈值配置应用临界现象分析，以检测寿命中的标度律。
• 对初值族进行参数化，以在轴对称情形下插值黑子与非黑子结果。

实验结果

研究问题

• RQ1 I型临界行为是否在轴对称玻色子星塌缩中持续存在，如先前仅在球对称情形下观察到的那样？
• RQ2 近阈值玻色子星塌缩的亚临界终态是否可准确描述为与线性微扰理论预测一致的稳定振荡玻色子星？
• RQ3 具有离散角动量量子数k = 1和k = 2的旋转玻色子星的结构及其最大质量为何？
• RQ4 不同参数化的初值族如何影响轴对称情形下黑洞形成的阈值？
• RQ5 轴对称情形下近临界配置的寿命中会呈现出何种标度行为？

主要发现

• 在轴对称情形下确认了I型临界现象，这是首次在非球对称设置下实现此类模拟。
• 亚临界终态为一个稳定、大振幅振荡的玻色子星，其频率与线性微扰理论预测的基态正常模式一致。
• 发现k = 1和k = 2旋转玻色子星存在最大质量，表明在该极限之外不存在平衡解。
• 观察到寿命随接近阈值程度的标度行为，与I型临界行为一致。
• 所开发的算法成功以高精度构建了平衡态旋转玻色子星解，使对其性质的详细研究成为可能。
• 轴对称初值族的动力学表现出从黑子到非黑子结果的清晰插值，支持临界解的存在。