# coding: utf-8 # # Geodesics in anti-de Sitter spacetime # # This worksheet aims at recovering some geodesics of anti-de Sitter (AdS) spacetime using numerical tools implemented in the SageMath class **IntegratedGeodesic**. # *NB:* a version of SageMath at least equal to 8.1 is required to run this worksheet: # In[1]: version() # First set up the notebook to display mathematical objects using LaTeX rendering: # In[2]: get_ipython().run_line_magic('display', 'latex') # ## Anti-de Sitter manifold # # To set AdS spacetime, declare a 4-dimensional differentiable manifold and a non-negative parameter $b$: # In[3]: AdS = Manifold(4, 'AdS', r'\mathcal{Schw}') b = var('b') ; assume(b > 0) # Define the hyperbolic coordinates: # In[4]: Hyp. = AdS.chart(r'ta:\tau rh:(0,+oo):\rho th:(0,pi):\theta ph:(0,2*pi):\phi') # Finally set the Lorentzian metric of Ads spacetime: # In[5]: g = AdS.lorentzian_metric('g') g[0,0], g[1,1] = -cosh(rh)^2, b^(-2) g[2,2], g[3,3] = b^(-2)*sinh(rh)^2, b^(-2)*sinh(rh)^2*sin(th)^2 g.display() # ## Geodesics of AdS spacetime # ### Defining an affinely parametrised geodesic # Declare the various variables needed to define a geodesic; start with the affine parameter and its extremal values: # In[6]: affine_param = var('s s_0 s_max') # Then, declare the starting point of the geodesic: # In[7]: initial_pt_coords = var('ta_0 rh_0 th_0 ph_0') p_0 = AdS.point(initial_pt_coords, name='p_0') # Declare the initial tangent vector: # In[8]: initial_tgt_vec_comps = var('Dta_0 Drh_0 Dth_0 Dph_0') v_0 = AdS.tangent_space(p_0)(initial_tgt_vec_comps) # The parametrised geodesic may now be initialised: # In[9]: geod = AdS.integrated_geodesic(g, affine_param, v_0, verbose=True) # Display the system of the affinely parametrised geodesic equations: # In[10]: sys = geod.system(verbose=True) # ### Computing and plotting the geodesic # # #### Null radial geodesic # # Set a dictionnary providing numerical values for each of the parameters apprearing in the system defining the geodesic. # # The values suggested below make the initial tangent vector null and radial. # In[11]: params_values_null_radial = {b:1, s_0:0, s_max:15, ta_0:0, rh_0:0.1, th_0:pi/3, ph_0:pi/4, Dta_0:1/cosh(0.1), Drh_0:1, Dth_0:0, Dph_0:0} # Then integrate the geodesic for such values of the parameters: # In[12]: sol_null_radial = geod.solve(step=0.1, parameters_values=params_values_null_radial, solution_key='null_radial', verbose=True) # Analytic expressions of null geodesics in AdS being known, these may be used to check the previous numerical solution: # In[13]: rh_num = [] rh_analyt = [] rh_error = [] for (S,TA,RH,TH,PH) in sol_null_radial: rh_num += [(S,RH)] rh_analyt += [(S, arcsinh(S + sinh(0.1)))] rh_error += [(S, RH - arcsinh(S + sinh(0.1)))] plot_rh_num = line(rh_num, legend_label=r'Numerical solution for coordinate $\rho$', color='green') plot_rh_analyt = line(rh_analyt, color='orange', legend_label=r'Analytical solution for coordinate $\rho$', linestyle='--') plot_rh_error = line(rh_error) (plot_rh_num + plot_rh_analyt).show() plot_rh_error.show(title=r'Error on coordinate $\rho$') # The following results provide another way to check the numerical integration. # # The squared norm $g_{\mu\nu} \dot{x}^{\mu} \dot{x}^{\nu}$ of the vector tangent to any geodesic with respect to any affine parameter $s$ is constant throughout motion. # # In addition, $\partial_{\tau}$ is an obvious Killing vector of AdS metric, so that quantity $e \equiv \cosh^{2}(\rho) \ \dot{\tau}$ is constant as well. # # Therefore, using an interpolation of the previous numerical solution, one may check that these two quantities are conserved (not to be disturbed by initial edge effects, the reference values used to check conservations of the various quantities are updated after 10 steps): # In[14]: interp_null_radial = geod.interpolate(solution_key='null_radial', interpolation_key='null_radial', verbose=True) error_squar_norm_null_radial = [] error_e_null_radial = [] i = 0 for (S,TA,RH,TH,PH) in sol_null_radial: P = geod(S, interpolation_key='null_radial') V = geod.tangent_vector_eval_at(S, interpolation_key='null_radial') squar_norm_null_radial = numerical_approx((g.at(P)(V,V)).substitute({b:1})) e_null_radial = numerical_approx((-g.at(P)[0,0]*V[0]).substitute({b:1})) if i == 0: squar_norm_null_radial_0 = squar_norm_null_radial e_null_radial_0 = e_null_radial if i == 10: squar_norm_null_radial_0 = squar_norm_null_radial e_null_radial_0 = e_null_radial error_squar_norm_null_radial += [(S,squar_norm_null_radial - squar_norm_null_radial_0)] error_e_null_radial += [(S,e_null_radial - e_null_radial_0)] i += 1 plot_error_squar_norm_null_radial = line(error_squar_norm_null_radial) plot_error_e_null_radial = line(error_e_null_radial) plot_error_squar_norm_null_radial.show(title="Null, radial geodesic: error on conservation of squared norm", ymin=-1e-3, ymax=1e-3) plot_error_e_null_radial.show(title="Null, radial geodesic: error on conservation of e", ymin=-1e-3, ymax=1e-3) # One may finally plot the time coordinate of the geodesic with respect to radial coordinate: # In[15]: geod.plot_integrated(interpolation_key='null_radial', ambient_coords=(rh,ta), plot_points=200, color='green') # #### Timelike radial geodesic # # Set a dictionnary providing numerical values for each of the parameters apprearing in the system defining the geodesic. # # The values suggested below make the initial tangent vector timelike and radial, with squared norm equal to -1, so that the curve parameter s is proper time along the (timelike) geodesic. # In[16]: params_values_timelike_radial = {b:1, s_0:0, s_max:3, ta_0:0, rh_0:0.1, th_0:pi/3, ph_0:pi/4, Dta_0:sqrt(2)/cosh(0.1), Drh_0:1, Dth_0:0, Dph_0:0} # Integrate the geodesic for such values of the parameters: # In[17]: sol_timelike_radial = geod.solve(step=0.01, parameters_values=params_values_timelike_radial, solution_key='timelike_radial', verbose=True) # Use the analytic solution to check the numerical integration: # In[18]: rh_num = [] rh_analyt = [] rh_error = [] for (S,TA,RH,TH,PH) in sol_timelike_radial: rh_num += [(S,RH)] rh_analyt += [(S, arcsinh(sinh(0.1)*cos(S) + cosh(0.1)*sin(S)))] rh_error += [(S, RH - arcsinh(sinh(0.1)*cos(S) + cosh(0.1)*sin(S)))] plot_rh_num = line(rh_num, legend_label=r'Numerical solution for coordinate $\rho$', color='green') plot_rh_analyt = line(rh_analyt, color='orange', legend_label=r'Analytical solution for coordinate $\rho$', linestyle='--') plot_rh_error = line(rh_error) (plot_rh_num + plot_rh_analyt).show() plot_rh_error.show(title=r'Error on coordinate $\rho$') # Interpolate the solution to check conservation of norm and quantity $e$, and plot the solution: # In[19]: interp_timelike_radial = geod.interpolate(solution_key='timelike_radial', interpolation_key='timelike_radial', verbose=True) error_squar_norm_timelike_radial = [] error_e_timelike_radial = [] i = 0 for (S,TA,RH,TH,PH) in sol_timelike_radial: P = geod(S, interpolation_key='timelike_radial') V = geod.tangent_vector_eval_at(S, interpolation_key='timelike_radial') squar_norm_timelike_radial = numerical_approx((g.at(P)(V,V)).substitute({b:1})) e_timelike_radial = numerical_approx((-g.at(P)[0,0]*V[0]).substitute({b:1})) if i == 0: squar_norm_timelike_radial_0 = squar_norm_timelike_radial e_timelike_radial_0 = e_timelike_radial if i == 10: squar_norm_timelike_radial_0 = squar_norm_timelike_radial e_timelike_radial_0 = e_timelike_radial error_squar_norm_timelike_radial += [(S,squar_norm_timelike_radial - squar_norm_timelike_radial_0)] error_e_timelike_radial += [(S,e_timelike_radial - e_timelike_radial_0)] i += 1 plot_error_squar_norm_timelike_radial = line(error_squar_norm_timelike_radial) plot_error_e_timelike_radial = line(error_e_timelike_radial) plot_error_squar_norm_timelike_radial.show(title="Timelike, radial geodesic: error on conservation of squared norm", ymin=-1e-3, ymax=1e-3) plot_error_e_timelike_radial.show(title="Timelike, radial geodesic: error on conservation of e", ymin=-1e-3, ymax=1e-3) geod.plot_integrated(interpolation_key='timelike_radial', ambient_coords=(rh,ta), plot_points=200, style='--') # One may thus plot a grid of null and timelike radial geodesics in AdS spacetime: # In[20]: S_MAX_NULL = [10, 10, 10, 1] TH_0_NULL = [0, pi/4, pi/2, 3*pi/4] grid = Graphics() for (S_MAX,TH_0) in zip(S_MAX_NULL,TH_0_NULL): params_values_null_radial = {b:1, s_0:0, s_max:S_MAX, ta_0:TH_0, rh_0:1e-3, th_0:pi/3, ph_0:pi/4, Dta_0:1/cosh(1e-3), Drh_0:1, Dth_0:0, Dph_0:0} sol_null_radial = geod.solve(step=S_MAX/100, parameters_values=params_values_null_radial, solution_key='null_radial-{}'.format(TH_0)) interp_null_radial = geod.interpolate(solution_key='null_radial-{}'.format(TH_0), interpolation_key='null_radial-{}'.format(TH_0)) grid += geod.plot_integrated(interpolation_key='null_radial-{}'.format(TH_0), ambient_coords=(rh,ta), plot_points=200, color='green') S_MAX_NULL_BACK = [1, 10, 10, 10] TH_0_NULL_BACK = [pi/4, pi/2, 3*pi/4, pi] for (S_MAX,TH_0) in zip(S_MAX_NULL_BACK,TH_0_NULL_BACK): params_values_null_radial_back = {b:1, s_0:0, s_max:S_MAX, ta_0:TH_0, rh_0:1e-3, th_0:pi/3, ph_0:pi/4, Dta_0:-1/cosh(1e-3), Drh_0:1, Dth_0:0, Dph_0:0} sol_null_radial = geod.solve(step=S_MAX/100, parameters_values=params_values_null_radial_back, solution_key='null_radial_back-{}'.format(TH_0)) interp_null_radial = geod.interpolate(solution_key='null_radial_back-{}'.format(TH_0), interpolation_key='null_radial_back-{}'.format(TH_0)) grid += geod.plot_integrated(interpolation_key='null_radial_back-{}'.format(TH_0), ambient_coords=(rh,ta), plot_points=200, color='green') S_MAX_TIMELIKE = [3.13, 3.14, 3.14, 3.19] DRH_0_TIMELIKE = [0.4, 1, 2.5, 10] for (S_MAX,DRH_0) in zip(S_MAX_TIMELIKE,DRH_0_TIMELIKE): params_values_timelike_radial = {b:1, s_0:0, s_max:S_MAX, ta_0:0, rh_0:1e-3, th_0:pi/3, ph_0:pi/4, Dta_0:sqrt(DRH_0^2 + 1)/cosh(1e-3), Drh_0:DRH_0, Dth_0:0, Dph_0:0} sol_timelike_radial = geod.solve(step=S_MAX/100, parameters_values=params_values_timelike_radial, solution_key='timelike_radial-{}'.format(DRH_0)) interp_timelike_radial = geod.interpolate(solution_key='timelike_radial-{}'.format(DRH_0), interpolation_key='timelike_radial-{}'.format(DRH_0)) grid += geod.plot_integrated(interpolation_key='timelike_radial-{}'.format(DRH_0), ambient_coords=(rh,ta), plot_points=200, style='--') grid.show()