标题： 二阶微扰理论：大尺度上的问题 显示英文标题

标题： Second-order perturbation theory: problems on large scales

摘要： 在广义相对论摄动理论中，由于其引力自力作用，点质量会偏离测地线运动。 由于自力较小，通常可以将运动近似为测地线运动。 然而，众所周知，自力效应会随时间累积，使得测地线近似在长时间尺度上失效。 较少为人知的是，这种长时间尺度上的失效也会导致大距离尺度上的失效。 在二阶摄动理论中，会出现两种大距离病态：虚假的长期增长和红外发散的推迟积分。 这两种病态都会阻碍对二阶自力效应的实际计算。 利用一个简单的平空间标量玩具模型，我开发了克服这些障碍的方法。 通过多尺度展开来捕捉系统的缓慢演化，从而抑制长期增长。 通过在大距离处与正确的推迟解进行匹配来消除发散积分。 我还展示了如何通过取全局解的局部时间“快照”来提取保守的自力效应。 这些方法很容易适应点质量绕黑洞运动的物理相关情况。 显示英文摘要

摘要： In general-relativistic perturbation theory, a point mass accelerates away from geodesic motion due to its gravitational self-force. Because the self-force is small, one can often approximate the motion as geodesic. However, it is well known that self-force effects accumulate over time, making the geodesic approximation fail on long timescales. It is less well known that this failure at large times translates to a failure at large distances as well. At second perturbative order, two large-distance pathologies arise: spurious secular growth and infrared-divergent retarded integrals. Both stand in the way of practical computations of second-order self-force effects. Utilizing a simple flat-space scalar toy model, I develop methods to overcome these obstacles. The secular growth is tamed with a multiscale expansion that captures the system's slow evolution. The divergent integrals are eliminated by matching to the correct retarded solution at large distances. I also show how to extract conservative self-force effects by taking local-in-time "snapshots" of the global solution. These methods are readily adaptable to the physically relevant case of a point mass orbiting a black hole.