黑洞之 “洛芙係數”

Abstract

“洛芙係數” 這個物理量涵蓋了天體上一些極緊緻星體的資訊，例如黑洞或中子星。更重要的是我們在未來有機會能測到這個物理量。在廣義相對論中給出的黑洞解，經過計算發現竟然是零，這好比開出了一扇窗讓我們去真正的探索時空的本質。基於黑洞的“洛芙係數” 是零這個奇怪的事實跟愛因斯坦的理論只是有效場理論，也就是在四維時空裡的廣義相對論，我們知道當系統能量夠高時這理論需要一些高階修正項。在這篇文章中，我們分析了在這些高階修正下的黑洞解。我們在新的黑洞解的背景下做線性的微擾，並解了微擾的方程式。在適當的邊界條件下，我們可以提取黑洞的“洛芙係數” 這個資訊。我們發現在這新的黑洞背景下做微擾展開會得出不為零的“洛芙係數”。這更加深了在原本廣義相對論裡，黑洞的“洛芙係數”都是零謎。

The tidal Love numbers (TLNs) encode the internal structure of compact objects, such as black holes (BHs) or neutron stars and, more importantly, it is testable in the future gravitational detector. The vanishing of TLNs in vacuum classical general relativity thus offers a fantastic opportunity to probe the very fabric of spacetime, in the advent of a new era of precision gravity.Motivated by the bizarre features that BHs TLNs are all zero and the fact that Einstein’s gravity may not be well-behaved theory in high enough energy scale, we analyze black holes solutions under R^3-type corrections, which is the leading correction induced by quantum corrections in four-dimensions.Our methodology stars with the perturbation of our BHs solutions using a linear perturbation formalism. We then impose the Regge-Wheeler gauge and solve the perturbed differential equations using appropriate boundary conditions, both at the event horizon and infinity. From the resulting solutions, we can identify the induced multipole moments and the tidal fields which allow us to extract TLNs. We showed that perturbations around this black hole background will lead to non-zero TLNs. This further accentuates the “unnaturalness” of the vanishing TLNs for Schwarzschild black hole under Einstein-Hilbert action.