The investigation of non-vacuum cosmological backgrounds containing black holes is greatly enhanced by the Kiselev solution. This solution plays a crucial role in understanding the properties of the background and its relationship with the features of the black hole. Consequently, the gravitational memory effects at large distances from the black hole offer a valuable means of obtaining information about the surrounding field parameter N and parameters related to the hair of the hairy Kiselev Black hole. This paper investigates the gravitational memory effects in the context of the Kiselev solution through two distinct approaches. At first, the gravitational memory effect at null infinity is explored by utilizing the Bondi–Sachs formalism by introducing a gravitational wave (GW) pulse to the solution. The resulting Bondi mass is then analyzed to gain further insight. Therefore, the Kiselev solution is being examined to determine the variations in Bondi mass caused by the pulse of GWs. The study of changes in Bondi mass is motivated by the fact that it is dynamic and time-dependent, and it measures mass on an asymptotically null slice or the densities of energy on celestial spheres. In the second approach, the investigation of displacement and velocity memory effects is undertaken in relation to the deviation of two neighboring geodesics and the deviation of their derivative influenced by surrounding field parameter N and the hair of hairy Kiselev black hole. This analysis is conducted within the context of a GW pulse present in the background of a hairy Kiselev black hole surrounded by a field parameter N.

中文翻译：

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多毛基谢廖夫黑洞的引力波脉冲和记忆效应及其与 Bondi-Sachs 形式主义的类比

Kiselev 解决方案大大增强了对包含黑洞的非真空宇宙学背景的研究。这个解决方案对于理解背景​​的特性及其与黑洞特征的关系起着至关重要的作用。因此，距黑洞较远距离处的引力记忆效应为获取有关周围场参数 N 和与毛状基谢列夫黑洞的毛发相关的参数的信息提供了一种有价值的手段。本文通过两种不同的方法研究了 Kiselev 解中的引力记忆效应。首先，通过在解中引入引力波（GW）脉冲，利用 Bondi-Sachs 形式来探索零无穷远处的引力记忆效应。然后分析所得的邦迪质量以获得进一步的了解。因此，正在检查 Kiselev 解以确定引力波脉冲引起的 Bondi 质量的变化。研究邦迪质量变化的动机是它是动态的、与时间相关的，并且它测量渐近零切片上的质量或天球上的能量密度。第二种方法是针对两个相邻测地线的偏差及其导数受周围场参数N和毛状基谢廖夫黑洞毛发影响的偏差进行位移和速度记忆效应的研究。该分析是在被场参数 N 包围的多毛 Kiselev 黑洞背景中存在的 GW 脉冲的背景下进行的。