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Metric in Manifold

  1. Sep 22, 2014 #1

    kau

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    1. The problem statement, all variables and given/known data

    I am confused if there is any standard way to check what should be the line element $$ds^{2}$$ when the dimensions are more than three ( since we don't have the option to draw things as we usually do in case of 3d or less dimensional cases. I am following Hobson's book. I am giving you an example. consider a 3sphere embedded in 4d euclidean space. show that line element can be written as following

    2. Relevant equations
    ##ds^{2} = a^2[(d\chi^2)+sin^{2} \chi(d\theta^{2}+sin^{2}\theta d\phi^{2})]##
    now here can you introduce the concept of radial distance?? what it would be??
    In that case if I have a metric ##ds^2=\frac{dr^{2}}{1-(2\mu/r)}+r^{2}[d\theta^2+sin^{2}\theta d\phi^{2}]##
    tell me what should be the radial distance between a sphere at r=2##\mu## and r=3##\mu##??
    these are ques from hobson's book basically.
    if any of you take the pain to explain me little bit in detail to visualise these stuff i would be grateful.
    thanks


    3. The attempt at a solution
     
  2. jcsd
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