Nonlinear studies of modifications to general relativity: Comparing different approaches

Studying the dynamical, nonlinear regime of modified theories of gravity remains a theoretical challenge that limits our ability to test general relativity. Here we consider two generally applicable, but approximate methods for treating modifications to full general relativity that have been used to study binary black hole mergers and other phenomena in this regime, and compare solutions obtained by them to those from solving the full equations of motion. The first method evolves corrections to general relativity order by order in a perturbative expansion, while the second method introduces extra dynamical fields in such a way that strong hyperbolicity is recovered. We use shift-symmetric Einstein-scalar-Gauss-Bonnet gravity as a benchmark theory to illustrate the differences between these methods for several spacetimes of physical interest. We study the formation of scalar hair about initially non-spinning black holes, the collision of black holes with scalar charge, and the inspiral and merger of binary black holes. By directly comparing predictions, we assess the extent to which those from the approximate treatments can be meaningfully confronted with gravitational wave observations. We find that the order-by-order approach cannot faithfully track the solutions when the corrections to general relativity are non-negligible. The second approach, however, can provide consistent solutions, provided the ad-hoc timescale over which the dynamical fields are driven to their target values is made short compared to the physical timescales.

研究修正引力理论的动力学、非线性机制仍然是一个理论挑战，它限制了我们检验广义相对论的能力。在此，我们考虑两种普遍适用但近似的方法来处理对完整广义相对论的修正，这些方法已被用于研究该机制下的双黑洞并合及其他现象，并将通过它们得到的解与通过求解完整运动方程得到的解进行比较。第一种方法是在微扰展开中按阶次演化对广义相对论的修正，而第二种方法是以恢复强双曲性的方式引入额外的动力学场。我们使用平移对称的爱因斯坦 - 标量 - 高斯 - 邦尼特引力作为基准理论，来说明这些方法在几个具有物理意义的时空上的差异。我们研究了初始无自旋黑洞周围标量毛的形成、带有标量荷的黑洞碰撞以及双黑洞的旋进和并合。通过直接比较预测结果，我们评估了近似处理的结果在多大程度上能够有意义地与引力波观测相对照。我们发现，当对广义相对论的修正不可忽略时，按阶次的方法无法忠实地追踪解。然而，第二种方法可以提供一致的解，前提是将动力学场被驱动到其目标值的特定时间尺度设定得比物理时间尺度短。