1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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]
    Last edited: May 31, 2006
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook