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AdS3 metrics using SageManifolds

  1. Mar 22, 2017 #1

    ShayanJ

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    Gold Member

    I've written the following code using SageManifolds both for practice and for when I need different quantities related to the ## AdS_3 ## spacetime.
    Code (Text):
    AdS3=Manifold(3,'AdS3')

    AdS3_Polar=AdS3.open_subset('AdS3_Polar')
    #AdS3_Poincare=AdS3.open_subset('AdS3_Poincare')

    Global.<tau,rho,phi>=AdS3_Polar.chart(r'tau:(-oo,+oo) rho:(0,+oo) phi:(0,2*pi)')
    #Poincare.<t,y,z>=AdS3_Poincare.chart(r't:(-oo,+oo) y:(-oo,+oo) z:(0,+oo)')

    R22=Manifold(4,'R22')
    X22.<X,Y,Z,W>=R22.chart()
    h=R22.metric('h',signature=0)
    h[0,0],h[1,1],h[2,2],h[3,3]=-1,-1,1,1

    var('R',domain='real')
    assume(R>0)
    Phi=AdS3.diff_map( R22, [ R*cosh(rho)*cos(tau),R*cosh(rho)*sin(tau),R*sinh(rho)*cos(phi),R*sinh(rho)*sin(phi) ] , name='Phi' )
    #Phi=AdS3.diff_map( R22, [ (R/(2*z))*(y^2-t^2+z^2+1),R*t/z,R*y/z,(R/(2*z))*(y^2-t^2+z^2-1) ], name='Phi' )

    g=AdS3.lorentzian_metric('g')
    g.set( Phi.pullback(h) )

    #g=AdS3.lorentzian_metric('g')
    #g.set( Phi.pullback(h) )
    As you can see, the codes for computing the metric in the Poincare coordinates are commented. That's because I need the metric in both global and Poincare coordinates but I don't know how to associate both of them to the manifold. So one of them is commented at a given time. How can I associate more than one metric to a single manifold?

    Thanks
     
  2. jcsd
  3. Mar 27, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
  4. Jun 30, 2017 #3
    You can find some SageManifolds code for AdS spacetime, including the use of both global and Poincaré coordinates, here.
    This is for ## AdS_4 ##, but I guess you can easily adapt it to ## AdS_3 ##.
     
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