The no-hair theorem and black hole shadows

Abstract

The successful observation of M87 supermassive black hole by the Black Hole Event Horizon Telescope(EHT) provides a very good opportunity to study the theory of gravity. In this work, we obtain the exact solution for the short hair black hole (BH) in the rotation situation, and calculate in detail how hairs affect the BH shadow. For the exact solution part, using the Newman-Janis algorithm, we generalize the spherically symmetric short-hair black hole metric to the rotation case (space-time lie element (2.25)). For the BH shadow part, we study two hairy BH models. In model 1, the properties of scalar hair are determined by the parameters α0 and L (We re-obtained the results in reference [48] for the convenience of discussion in this work). In model 2, the scalar hair of the BH is short hair. In this model, the shape of the BH shadow is determined by scalar charge Qm and k. The main results are as follows: (1) In the case of rotational short-hair BH, the value range of parameter k is k > 1 (2.25), the range of short-hair charge value Qm is greatly reduced due to the introduction of the BH spin a. When 0leqslant {Q}_mleqslant frac{2}{3}times {4}^{frac{1}{3}} , the rotational short-hair BH has two event horizons at this time. When {Q}_m>frac{2}{3}times {4}^{frac{1}{3}} , the rotational short-hair BH has three unequal event horizons, so the space-time structure of the BH is significantly different from that of Kerr BH. (2) For model 1, the effect of scalar hair on the BH shadows corresponds to that of ε > 0 in reference [38, 48], but the specific changes of the shadows in model 1 are different. This is because the BH hair in reference [38] is considered as a perturbation to the BH, while the space-time metric of model 1 is accurate and does not have perturbation property. For model 2, that is, the change of the BH shadow caused by short hairs, the main change trend is consistent with that of ε < 0 in reference [38]. Because of the special structure of the short-hair BH, the specific changes of BH shadows are different. (3) the variation of Rs and δs with L and α0 is not a monotone function in model 1, but in model 2, it is. These results show that scalar hairs (model 1) have different effects on Kerr BH shadows than short hairs (model 2), so it is possible to distinguish the types and properties of these hairs if they are detected by EHT observations. (4) as for the effects of the hairs on energy emissivity, the main results in model 1 [48], different energy emissivity curves have intersection phenomenon, while in model 2 (short-hair BH), there is no similar intersection phenomenon. In general, various BH hairs have different effects on the shadows, such as non-monotonic properties and intersection phenomena mentioned in this work. Using these characteristics, it is possible to test the no-hair theorem in future EHT observations, so as to have a deeper understanding of the quantum effect of BHs. In future work, we will use numerical simulations to study the effects of various hairs on BHs and their observed properties.