Wen-Jie Zhang,Pu-Jian Mao and Jun-Bao Wu

Center for Joint Quantum Studies and Department of Physics,School of Science,Tianjin University,135 Yaguan Road,Tianjin 300350,China

Abstract In this paper,we study the Hawking radiation of Dirac particles via tunneling formalism from linearly supertranslated Schwarzschild black holes.We find that the radiation spectrum and the Hawking temperature remain the same as the one without soft hair.

Keywords:quantum aspects of black holes,quantum field theory in curved spacetime,black hole thermodynamics

1.Introduction

Black holes are shown to carry soft hairs[1–10].Consequently,the vacuum of quantum gravity is not unique and a stationary black hole is physically inequivalent to the one with soft hairs.During the course of Hawking evaporation,soft hairs should be radiated through infinity and the assumption of charge conservation will induce correlations between the early and late time Hawking radiation which are normally not considered in the semiclassical computation.Moreover,the normal(hard)radiated particles are always accompanied by an infinite cloud of correlated soft particles and the quantum purity can be just restored in principle by correlations between the hard and soft radiated particles[11].The whole picture makes the soft hair at the position of a possible resolution of the black hole information paradox,see,e.g.[12]for a review and references therein.Meanwhile,infalling observers will automatically observe the soft radiated particles outside the horizon.Taking into account the soft dynamics,it is argued that an infalling observer will never really meet a firewall before reaching the singularity[13]which indicates that the soft hair can also play an essential role in dissolving the firewall paradox.

A natural question arises about how soft hairs can be observed,in particular macroscopically.Recent studies show that soft hairs in three dimensions are purely microscopic[14,15].In four dimensions,the same feature has been verified by checking the Hawking radiation of soft-hairy black holes[7,16–19].More precisely,the Hawking fluxes from the Schwarzschild black hole are unaffected by their linearized supertranslation hair[7].The spectrum of Hawking radiation of the Schwarzschild black hole is unchanged after including the dressing of asymptotic states with clouds of soft photons and soft gravitons[16].The Unruh effect is invariant under supertranslations[17].The Hawking fluxes from Schwarzschild with non-linear supertranslation hair are still the same as the Schwarzschild’s one[18](see also[19]).However counter examples are presented in[7,20,21].In particular,the Hawking temperature of dynamical black holes is found to be modified by soft hair[7,20].Complementary to these investigations where only scalar particles are involved,we study in this paper the Hawking radiation of Dirac particles from soft-hairy black holes using the tunneling method.The supertranslation is characterized by an arbitrary function of the angular variables.The resulting solution after a supertranslation will be in a rotating frame.The main motivation of the present work of investigating fermions emitted from a supertranslated black hole is to check if there is a correlation between the fermion spin and the supertranslation parameters in the tunneling probability.

The tunneling method proposed by Parikh and Wilczek[22]is a very effective way of studying Hawking radiation.In general,the tunneling method involves calculating the imaginary part of the action for the classically forbidden path to evaluate the radiation spectrum.Applying the tunneling method,we examine the Hawking radiation of Dirac particles from Schwarzschild black holes with linear supertranslation hairs.We find that the radiation spectrum and the Hawking temperature are unchanged after soft hair implant on black holes.When applying the tunneling method for Dirac particles,one needs to properly take care of the angular-dependent part of the action[23–31].The simplest assumption is that solving only the radial equation at a fixed angle may not lead to a solution of the Dirac equation.We find that the linear supertranslation hair dependence in the Hawking spectrum is precisely canceled by taking account of the contribution from the angular-dependent part.Our result is consistent with the scalar particle case[7]and furnishes an important confirmation of the microscopic nature of soft hairs in four dimensions.

2.Schwarzschild black holes with supertranslation hair

The line element of the supertranslated Schwarzschild black hole in the advanced Bondi coordinates reads[6]

There are two different approaches to computing the imaginary part of the action for the emitted particle,namely the null geodesic method used by Parikh and Wilczek[22]and the Hamilton–Jacobi method proposed by Aghebenet al[32].The former relies on a specific choice of coordinates while the latter can apply to any coordinate system that is regular at the horizon.Moreover,the Hamilton–Jacobi equation can be emerged from a WKB approximation of the Klein–Gordon equation[23].For a scalar field,assuming that

one can obtain the usual Hamilton–Jacobi equation from the lowest order in ? expansion of the Klein–Gordon equation when identifyingIas the action of the tunneling particle.Such a treatment can be further applied to the Dirac equation which allows one to study the tunneling of fermions[23].We consider massive spinor field Ψ with mass μ.The covariant Dirac equation reads

where ?μis the spinor-covariant derivative defined byis the spin connection.The co-tetrad fieldsaccording to the metric(1)are chosen as1Notice the choice of ηab in equation(10).

3.Conclusion

In this work,we examine the semi-classical property of soft hair implants on black holes.Applying the tunneling method,we compute the Hawking radiation of Dirac particles from Schwarzschild black holes with linear supertranslation hairs and confirm that the Hawking radiation is not modified by supertranslation hairs.This is further evidence to support the microscopic nature of soft hairs.

Acknowledgments

We would like to thank Chong-Sun Chu and Ran Li for very helpful discussions.This work is supported in part by the National Natural Science Foundation of China under Grant No.11 905 156,No.11 975 164,No.11 935 009,and Natural Science Foundation of Tianjin under Grant No.20JCYBJC00910.

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