# Why Cant I Calculate 1-forms?

1. May 31, 2006

### Oxymoron

Why Cant I Calculate 1-forms!?

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

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

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:

$$g = -\mbox{d}t\otimes\mbox{d}t + f(t)^2\hat{g}$$

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

$$e^0 = \mbox{d}t \quad \quad e^i = f(t)\hat{e}^i$$

where $\hat{e}^i$ is a $\hat{g}$-orthonormal frame. And note that $X_i = \frac{1}{f}\hat{X}_i$. 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. May 31, 2006

### Oxymoron

So here is how I attempted to calculate $\omega_{12}$:

First I calculated that

$$\mbox{d}e^1 = f^{\prime}\mbox{d}t\wedge\hat{e}^1 + f\mbox{d}\hat{e}^1$$
$$= \frac{f^{\prime}}{f}e^0\wedge e^1 + f\mbox{d}e^1$$

and, similarly

$$\mbox{d}e^2 = \frac{f^{\prime}}{f}e^0\wedge e^2 + f\mbox{d}\hat{e}^2$$

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

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

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

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

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

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

$$= \frac{2}{f^2}\hat{\omega}_{12} + \frac{f^{\prime}}{f}\left[i(X_2)(e^0\wedge e^1) - i(X_1)(e^0 \wedge e^2)\right]$$

Last edited: May 31, 2006