实验数学遇到引力自作用力
Experimental mathematics meets gravitational self-force
摘要：
现在可以用适用于各种不变量的黑洞微扰理论，在后牛顿（PN）展开式中计算质量比项的线性到非常高的阶数。例如，在线性微扰理论中计算绕黑洞的圆形轨道上的点粒子的红移不变量，给出了具有任意质量比的圆形双星的结合能的质量比线性部分。这种结合能，反过来，编码系统的保守动力学。我们给出了一种方法，用于提取这些PN系数的解析形式，从高精度的数值数据，使用实验数学技术，特别是整数关系算法。当计算进展到相当困难的克尔度规扰动情况时，这种方法应该特别重要。作为一个例子，我们将这种方法应用于Schwarzschild中的红移不变量。在这里，我们得到了12.5PN的解析系数，以及21.5PN的混合解析-数值形式的高阶项，包括完整的13.5PN系数的解析形式，以及13 PN的所有对数项。在这些高阶中，单个系数可以有超过30项，包括各种各样的超越数，当完整地写出来时。通过对展开式结构的仔细研究，我们仍然能够从数值数据中得到这些系数的解析形式。我们发现的结构也使我们能够预测某些“领先的对数”类型的贡献，所有订单。在扩展中，我们得到的额外的条款，提高了精度的PN系列的红移观测，即使在非常强的领域内的最内部的稳定的圆形轨道，特别是当结合指数重。
英文摘要：
It is now possible to compute linear in mass-ratio terms in the post-Newtonian (PN) expansion for compact binaries to very high orders using black hole perturbation theory applied to various invariants. For instance, a computation of the redshift invariant of a point particle in a circular orbit about a black hole in linear perturbation theory gives the linear-in-mass-ratio portion of the binding energy of a circular binary with arbitrary mass ratio. This binding energy, in turn, encodes the system's conservative dynamics. We give a method for extracting the analytic forms of these PN coefficients from high-accuracy numerical data using experimental mathematics techniques, notably an integer relation algorithm. Such methods should be particularly important when the calculations progress to the considerably more difficult case of perturbations of the Kerr metric. As an example, we apply this method to the redshift invariant in Schwarzschild. Here we obtain analytic coefficients to 12.5PN, and higher-order terms in mixed analytic-numerical form to 21.5PN, including analytic forms for the complete 13.5PN coefficient, and all the logarithmic terms at 13PN. At these high orders, an individual coefficient can have over 30 terms, including a wide variety of transcendental numbers, when written out in full. We are still able to obtain analytic forms for such coefficients from the numerical data through a careful study of the structure of the expansion. The structure we find also allows us to predict certain "leading logarithm"-type contributions to all orders. The additional terms in the expansion we obtain improve the accuracy of the PN series for the redshift observable, even in the very strong-field regime inside the innermost stable circular orbit, particularly when combined with exponential resummation.
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