黑洞的非零勒夫係數

Abstract

在每年的農曆八月十五，發生在杭州灣的“錢塘江大潮”總是吸引著無數遊客，而這一自然現象產生的原因就是由於月球對海水產生的潮汐力。在牛頓力學中，物體受到潮汐力而產生的形變程度是由潮汐勒夫數（Tidal Love Number）這一無量綱的耦合常數來描述的。而潮汐勒夫數這一概念還可以被推廣到相對論的框架下。有趣的是，一系列的研究結果表明，四維時空下的施瓦西黑洞的潮汐勒夫數恰好為零。這個神奇的結果讓我們有了以下問題：四維時空下的施瓦西黑洞是否存在著一個隱藏的對稱性，從而迫使它的潮汐勒夫數為零？ 基於施瓦西黑洞的潮汐勒夫數為零這一神奇的事實，與愛因斯坦重力理論只是有效場理論，即在系統的能量足夠高的情形下愛因斯坦重力理論需要添加一些高階修正項。我們在這篇文章中分析了在加入高階修正項後的黑洞近似解，並且在新的黑洞解的背景下做線性的微擾，在適當的邊界條件下提取出新的黑洞解的潮汐勒夫數。結果表明，此時的潮汐勒夫數並不等於零，因此使得原本的廣義相對論中施瓦西黑洞的潮汐勒夫數等於零這一結果便更加撲朔迷離。

On the 15th of August in lunar calendar, the magnificent natural landscape of the Qiantang River Tide always attracts countless tourists. And the reason for this magical phenomenon is the tidal forces of the Earth-Moon interaction. In Newtonian gravity, the tidal deformability of an astronomical body is measured by its tidal Love numbers, dimensionless coupling constants which depend on the body’s composition. The gravitational Love numbers characterize the body’s response to the tidal field through the change in its gravitational potential. The gravitational Love numbers were promoted to a relativistic setting by Damour and Nagar, and Binnington and Poisson. Interestingly, in general relativity (GR), TLNs of black holes (BHs) are precisely zero. This intriguing result motivate us to analyze this property under more general scenarios. In this thesis we analyze black holes solutions under R^3-type corrections, which is the leading correction induced by quantum corrections in four-dimensions. Our methodology starts with the perturbation of our BH solutions using a linear perturbation formalism. We then impose the Regge-Wheeler gauge and solve the perturbed field equations of the theory using appropriate boundary conditions, both at the event horizon and infinity. From the resulting solutions, we identify the induced multipole moments and the tidal fields which allow us to compute the TLNs. Finally, we showed that perturbations around this black hole background will lead to non-zero tidal Love number (TLN). This further accentuates the “unnaturalness” of the vanishing TLN for Schwarzschild black hole under Einstein-Hilbert action.