李光杰_量子场论导论课程论文_Reading Note_Mimick Black Hole.pdf

Mimick Black Hole Reading note for introduction to quantum field theory Guangjie Li 14344013 Department of Physics School of Physics and Yat Sen School Sun Yat Sen University September 4 2016 Introduction General relativity predicts lots of fascinating phenomenon But these predictions are really difficult to find Recently the gravitational wave is confirmed by LIGO after a century For the difficulty to find a real black hole some physicist tried to mimick a black hole in laboratory There are two main methods to achieve this idea We can change the index of refraction generally speaking it s and in dielectric which results from the similarity of Maxwell equations in gravitational field and in dielectric Also you can control a irrotational fluid to mimick a massless scalar field The equation of motion for fluid can be linearized to a form like a massless scalar field in a curved geometry 1 Gravitational lensing 1 During the total solar eclipse in 1919 Arthur Eddington and collaborators performed the first direct observation of light deflection from the Sun thereby validating Einstein s theory of general relativity According to Einstein s theory the presence of matter energy results in curved spacetime and the complex motion of matter and light along geodesic trajectories that are no longer straight lines The general theory of relativity has been highly successful with many of its predictions validated including the precession of Mercury gravitational time dilation and gravitational redshift expansion of the universe and frame dragging Given the analogy between the macroscopic Maxwell s equations in complex inhomogeneous media and free space Maxwell s equations for the background of an arbitrary spacetime metric In gravitational field Maxwell equations are 4 0 1 1 4 Where is the determinant of g g g For a static gravitational field 4 0 1 1 4 Which is exactly the Maxwell equations in media The only difference is the relation between and and And it gives analogy between the Maxwell equations in gravitational field and index of refraction varying media 1 g 00 1 1 Spacetime and pressure density relation Consider a centrally symmetric isotropic Schwarzchild like spacetime in homogeneous coordinates d 2 2 2d 2 2 d 2 2d 2 By standard procedure introduce Einstein field equation which gives 1 2 g g Then make 2 n 2 1 4 denote e2 hence ln The meaning of is exactly the index of refraction And we can solve Einstein equation Where 5 2cot 1 2 If we take 2 Finally we got d 2 csc 2 2 2 d 2 2 d 2 2d 2 And it gives complicated pressure density relation At large distance we have Both the pressure and mass density are positive for cot 10 0 8071 1 2 Trajectory of light Derive the equation of motion by Lagrangian 1 2 d d 2 1 2 g00 2 2 2 2 2 Then g00 2 2 g00 2 2 4 2 2 Which can be formulated by 2 2 1 4 2 1 3 Lens equation For large distance we can express this bending light effect as s lens 1 1 1 2 1 Where 4 3 5 is related to metric tensor equivalently the gravitational binding of light 1 4 Transformation optics 2 More general cases are the theorem of transformation optics which gives the relation of and metric tensor g For Schwarzchild metric Transform and according to the coordinate transformations and calculate and This can be done by simulation but for experiment proposal it s not realistic 2 Unruh acoustic black hole 3 The equations of motion for irrotational fluid gives possibility to mimick black hole in acoustic system In Unruh s paper It reveals the similarity between the equation of motion of a massless scalar field in a specific geometry and the motion of irrotational fluid Also this helps us to mimick Hawking radiation Actually Israel scientist Steinhauer mimicked such a balck hole by BEC and they detected phonon Hawking radiation 4 3 Possible idea In optics system one way photon forms laser And in laboratory scientists found such gravitational binding of laser But for a really weak laser the quantum effect of photon will appear to influence the system If we can investigate the behavior of a single photon around a mimicking black hole that will be interesting Reference 1 C Sheng H Liu Y Wang N Zhu and D A Genov Trapping light by mimicking gravitational lensing Nature Photonics 7 902 906 2013 2 Chen H Miao R X Li M Transformation optics that mimics the system outside a Schwarzschild black hole Opt Express 18 15183 15188 201