Quasinormal Modes And Late-time Tails Of Scalar And Electromagnetic Field In Finslerian-Schwarzschild Black Hole

The black hole,as one of the important predictions of general relativity,has been discussed intensively by physicists and tested by astronomical observations.Recently,the observations on black hole have stepped into a veracious stage,such as the gravitational waves and the Event Horizon Telescope.The black hole possesses strong gravitational field,even light cannot escape from the event horizon.Therefore,researches on black hole have become an important research field for testing the validity of general relativity in strong gravitational fields.The perturbation of black hole is one of the methods to analyze the physical property of black hole.The stability of the perturbed black hole is analyzed by investigating the perturbation of black hole with numerical methods.The perturbation of the black hole includes three stages,which are initial burst,quasinormal modes(QNMs)and the late-time tails.The characteristic frequency of the perturbed black hole can be driven from QNMs,which can be described as the characteristic sound of the black hole.Thus,it is of importance to research the intrinsic properties of the black hole by analyzing the perturbation of the black hole.To unify general relativity with the standard quantum field theory,various modified gravity theories have been proposed,such as f(R)gravity,Gauss-Bonnet gravity theory,extra dimension theory and so on.The geometric background of these modified gravity theories is based on Riemann geometry.Finsler geometry is a natural generalization of Riemann geometry.Professor Chern pointed out that Finsler geometry is Riemann geometry without quadratic restriction.Therefore,it is significant to study the QNMs of the Finslerian black hole to explore the existence of Finsler gravity with general relativity.Firstly,we study the scalar field and electromagnetic field perturbations of Finslerian-Schwarzschild black holes.Based on the solution of Finslerian-Schwarzschild black hole,the equations of motion for scalar field and electromagnetic field have been constructed by Finslerian-Laplace operator and divergence operator.It is found that the angular parts for scalar field and electromagnetic field in Finslerian-Schwarzschild black hole are different with the one in Riemann geometric spacetime.The behaviors of the axial modes and the polar modes of the electromagnetic field perturbation are different,which reflects the difference between the eigenvalues of the angular part of the axial and polar modes of electromagnetic field perturbation.The perturbation of black hole is related with the radial part of equation of the motion,which is directly affected by the eigenvalues of the angular part of equation of motion.Analyzing the eigenvalues of scalar field and electromagnetic field,when l0 and l=1,m=0,the Finslerian-Schwarzschild solutions of scalar field and polar modes of electromagnetic field reduce to the Schwarzschild solution.By analyzing the effective scalar potential and electromagnetic potential in Finslerian-Schwarzschild black hole,the parameters affecting the QNM frequencies are Finslerian parameter?²,multipole quantum number l and magnetic quantum number m.The sixth-order WKB approximation method with higher precision has been utilized to research the QNM frequencies affected by the parameters.We find that the magnetic the real part R e(?₀)of scalar field and electromagnetic field are suppressed with higher?².The magnitude of the imaginary part I m(?₀)of the scalar field increases while the magnitude of the imaginary part I m(?₀)of the electromagnetic field decreases with the increasing of?².It is reflected that the Finslerian parameter?² has a nonnegligible impact on the fundamental frequency of QNMs.Similar with Kerr QNM spectrum,spherical symmetry breaking causes Zeeman-like splitting of QNM spectrum in Finslerian-Schwarzschild spacetime,which depends onm² for Finslerian-Schwarzschild black hole.The impacts ofm² on the fundamental frequencies of QNMs obtained from WKB approximation method are consistent with the theoretical analysis.We have adopted finite difference method to analyze the dynamical evolution of QNMs in Finslerian-Schwarzschild black hole.From the dynamical evolutions,we can find that the periods of oscillation of scalar field and electromagnetic field both increase with higher?².It demonstrates that the magnitude of the real part R e(?₀)of scalar field and electromagnetic field decreases with higher?².The descent speeds of the peak are not obvious.Thus,our numerical results of dynamical evolutions of scalar field and electromagnetic field in Finslerian-Schwarzschild black hole show that such dependence is the same with the numerical results obtained by the sixth-order WKB approximation method.At asymptotic infinity,the behaviors between the Finslerian-Schwarzschild black hole and the Schwarzschild black hole are different,which is embodied on the late-time tails of the perturbation of black hole.The Green function has been utilized to analyzed the late-time tails of scalar field and electromagnetic field perturbation in Finslerian-Schwarzschild black holes.One can find that there exists a discontinues change for scalar field and electromagnetic field perturbation with variation of?²from zero to decimals.With the increasing of?²,the decay of the late-time tails of the scalar field and electromagnetic field perturbation in Finslerian-Schwarzschild black hole slows down,which implies that the decay of the late-time tails for the Finslerian-Schwarzschild black hole is slower than that for the Schwarzschild black hole.The late-time tails of scalar field and electromagnetic field in Finslerian-Schwarzschild black hole do not upturn.The stability of black hole is directly affected with the effective potential.If the effective potential is always positive definite,the black hole is stable.Therefore,the effective potentials of scalar field and electromagnetic field are always positive if Finslerian parameter?²(27)0.8,which implies that the Finslerian-Schwarzschild black hole is stable.