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Homework Help: Why Cant I Calculate 1-forms?

  1. May 31, 2006 #1
    Why Cant I Calculate 1-forms!?

    Ive been given the following formula to find 1-forms:

    [tex]2\omega_{ab} = e^d i(X_a)i(X_b)\mbox{d}e_d + i(X_b)\mbox{d}e_a - i(X_a)\mbox{d}e_b[/tex]

    and have been asked to find all connection 1-forms. Of course, you cant find these 1-forms without a metric, so here it is:

    [tex]g = -\mbox{d}t\otimes\mbox{d}t + f(t)^2\hat{g}[/tex]

    where [itex]\hat{g}[/itex] is the metric on some 3-dimensional space of constant curvature c. My orthonormal frame is

    [tex]e^0 = \mbox{d}t \quad \quad e^i = f(t)\hat{e}^i[/tex]

    where [itex]\hat{e}^i[/itex] is a [itex]\hat{g}[/itex]-orthonormal frame. And note that [itex]X_i = \frac{1}{f}\hat{X}_i[/itex]. Phew! Hope thats not too hard to comprehend.

    But I dont know what to put in for d in the index of e. Im aware that you have to sum over d but Im not sure what to sum over. 0,1,2,3? EDIT: Yes, I sum over everything.
     
    Last edited: May 31, 2006
  2. jcsd
  3. May 31, 2006 #2
    So here is how I attempted to calculate [itex]\omega_{12}[/itex]:

    First I calculated that

    [tex]\mbox{d}e^1 = f^{\prime}\mbox{d}t\wedge\hat{e}^1 + f\mbox{d}\hat{e}^1[/tex]
    [tex]= \frac{f^{\prime}}{f}e^0\wedge e^1 + f\mbox{d}e^1[/tex]

    and, similarly

    [tex]\mbox{d}e^2 = \frac{f^{\prime}}{f}e^0\wedge e^2 + f\mbox{d}\hat{e}^2[/tex]

    So then, using that equation and substituting d=0 (I dont even know if that's right), I get:

    [tex]2\omega_{12} = e^di(X_1)i(X_2)\mbox{d}e_d + i(X_2)\mbox{d}e_1 - i(X_1)\mbox{d}e_2 [/tex]

    [tex]= e^di(X_1)i(X_2)\mbox{d}e_d + i(X_2)\left[\frac{f^{\prime}}{f}e^0\wedge e^1 + f\mbox{d}\hat{e}^1\right] - i(X_1)\left[\frac{f^{\prime}}{f}e^0\wedge e^2 + f\mbox{d}\hat{e}^2\right] [/tex]

    [tex]= e^di(X_1)i(X_2)\mbox{d}e_d + i(X_2)(f\mbox{d}\hat{e}^1) - i(X_1)(f\mbox{d}\hat{e}^2) + \frac{f^{\prime}}{f}\left[i(X_2)(e^0\wedge e^1) - i(X_1)(e^0 \wedge e^2)\right] [/tex]

    [tex]= e^di(X_1)i(X_2)\mbox{d}e_d + \frac{1}{f}i(\hat{X}_2)(f\mbox{d}\hat{e}^1) - \frac{1}{f}i(\hat{X}_1)(f\mbox{d}\hat{e}^2) + \frac{f^{\prime}}{f}\left[i(X_2)(e^0\wedge e^1) - i(X_1)(e^0 \wedge e^2)\right] [/tex]

    [tex]= \frac{1}{f^2}\hat{e}^di(\hat{X}_1)i(\hat{X}_2)\mbox{d}\hat{e}_d + i(\hat{X}_2)(\mbox{d}\hat{e}^1) - i(\hat{X}_1)(\mbox{d}\hat{e}^2) + \frac{f^{\prime}}{f}\left[i(X_2)(e^0\wedge e^1) - i(X_1)(e^0 \wedge e^2)\right] [/tex]

    [tex]= \frac{2}{f^2}\hat{\omega}_{12} + \frac{f^{\prime}}{f}\left[i(X_2)(e^0\wedge e^1) - i(X_1)(e^0 \wedge e^2)\right]
    [/tex]
     
    Last edited: May 31, 2006
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