A New Approach to Black Hole Quasinormal Modes: A Review of the Asymptotic Iteration Method

Table 1

QNMs to 4 decimal places for gravitational perturbations (𝑠=2) where the fifth column is taken from [28]. Note that the imaginary part of the 𝑛=0 and ℓ=2 result in [28] has been corrected to agree with [6]. [*] Note also that if the number of iterations in the AIM is increased, to say 50, then we find agreement with [6] accurate to 6 significant figures.

ℓ	𝑛	𝜔Leaver	𝜔AIM(after15iterations)	𝜔WKB
2	0	0.3737–0.0896i [*]	0.3737–0.0896i	0.3732–0.0892i
			(<0.01%)(<0.01%)	(−0.13%)(0.44%)[*]
	1	0.3467–0.2739i	0.3467–0.2739i	0.3460–0.2749i
			(<0.01%)(<0.01%)	(−0.20%)(−0.36%)
	2	0.3011–0.4783i	0.3012–0.4785i	0.3029–0.4711i
			(0.03%)(−0.04%)	(0.60%)(1.5%)
	3	0.2515–0.7051i	0.2523–0.7023i	0.2475–0.6703i
			(0.32%)(0.40%)	(−1.6%)(4.6%)
3	0	0.5994–0.0927i	0.5994–0.0927i	0.5993–0.0927i
			(<0.01%)(<0.01%)	(−0.02%)(0.0%)
	1	0.5826–0.2813i	0.5826–0.2813i	0.5824–0.2814i
			(<0.01%)(<0.01%)	(−0.03%)(−0.04%)
	2	0.5517–0.4791i	0.5517–0.4791i	0.5532–0.4767i
			(<0.01%)(<0.01%)	(0.27%)(0.50%)
	3	0.5120–0.6903i	0.5120–0.6905i	0.5157–0.6774i
			(<0.01%)(−0.03%)	(0.72%)(1.9%)
	4	0.4702–0.9156i	0.4715–0.9156i	0.4711–0.8815i
			(0.28%)(<0.01%)	(0.19%)(3.7%)
	5	0.4314–1.152i	0.4360–1.147i	0.4189–1.088i
			(1.07%)(0.43%)	(−2.9%) (5.6%)
4	0	0.8092–0.0942i	0.8092–0.0942i	0.8091–0.0942i
			(<0.01%)(<0.01%)	(−0.01%)(0.0%)
	1	0.7966–0.2843i	0.7966–0.2843i	0.7965–0.2844i
			(<0.01%)(<0.01%)	(−0.01%)(−0.04%)
	2	0.7727–0.4799i	0.7727–0.4799i	0.7736–0.4790i
			(<0.01%)(<0.01%)	(0.12%)(0.19%)
	3	0.7398–0.6839i	0.7398–0.6839i	0.7433–0.6783i
			(<0.01%)(<0.01%)	(0.47%) (0.82%)
	4	0.7015–0.8982i	0.7014–0.8985i	0.7072–0.8813i
			(−0.01%)(−0.03%)	(0.81%)(1.9%)