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Proper distance, Area and Volume given a Metric

  1. Feb 14, 2015 #1
    1. The problem statement, all variables and given/known data

    (a) Find the proper distance
    (b) Find the proper area
    (c) Find the proper volume
    (d) Find the four-volume

    ricci2.png

    2. Relevant equations


    3. The attempt at a solution

    Part (a)

    Letting ##d\theta = dt = d\phi = 0##:

    [tex] \Delta s = \int_0^R \left( 1-Ar^2 \right) dr = R \left(1 - \frac{AR}{2} \right) [/tex]

    Part (b)

    At ##r=R##:

    [tex]A = R^2 \int_0^{2\pi} d\phi \int_0^{\pi} sin \theta d\theta [/tex]
    [tex]A = 4\pi R^2 [/tex]

    Part (c)

    [tex]V = \int_0^R r^2 \left( 1 - Ar^2 \right) dr \int_0^{\pi} sin \theta d\theta \int_0 ^{2\pi} d\phi [/tex]
    [tex]V = \frac{4}{3} \pi R^3 \left( 1 - \frac{3}{5} AR^2 \right) [/tex]

    Part (d)

    [tex]V_4 = c\int_0^R r^2 \left( 1 - Ar^2\right)^2 dr \int_0^T dt \int_0^{\pi} sin \theta d\theta \int_0^{2\pi} d\phi [/tex]

    [tex]V_4 = \frac{4}{3} \pi R^3 \cdot cT \cdot \left( 1 - \frac{6}{5}AR^2 + \frac{3}{7}A^2R^4 \right) [/tex]


    Is this the correct method? This question seems a little too straightforward..
     
  2. jcsd
  3. Feb 16, 2015 #2

    DEvens

    User Avatar
    Education Advisor
    Gold Member

    I would guess that, if you look forward in your text, you will find this metric gets a lot of work. For example, you may be doing some interesting work studying the place where r^2 = 1/A.

    So quite likely this is just a warm-up getting some interesting features of the metric.
     
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