# Pondering basis vectors and one forms

• Matterwave
In summary, the author is discussing the issue of whether or not vectors and one forms can be considered dual to each other. The author thinks that, unless they are using a Cartesian metric, they cannot get the first property from the second quantity.f

#### Matterwave

Gold Member
So, I've been thinking about this for a while...and I can't seem to resolve it in my head. In this thread I will use a tilde when referring to one forms and a vector sign when referring to vectors and boldface for tensors. It seems to me that if we require the basis vectors and one forms to obey the property that:

$$\tilde{\omega}^j(\vec{e}_i)=\delta_i^j$$

Then, we cannot require the basis vectors and one forms to be "dual" to each other in the sense that we raise and lower their indices with the metric tensor. I.e.:

$$\tilde{\omega}^i=\bf{g}(\vec{e}_i,\quad)$$

Since, unless my metric is the Cartesian metric, I cannot get the first property from the second quantity.

Is this true, or have I messed up somewhere?

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The "duality" between vector and one forms that you cite does not require a metric. When the vector basis is changed, the dual forms are also changed.

The "duality" between vectors and one forms that can be defined with a metric is different, as the duals do not change if you change basis.

So you are right. The former has nothing to do with raising and lowering indices with a metric (it's defined even without a metric), the latter involves raising and lowering indices with a metric.

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The natural set of one form basis vectors then, is defined by the first property and not the second one then right?

Yes.

I have another question. If we choose to use the tetrad method to describe our manifold, then the metric is automatically diag(1,1,1,...)? Since we are choosing ortho-normal basis vectors, then it would make sense that that would have to be the case.

I could never figure out Wald's take on the tetrad method...His abstract index notation there makes me really confused what his equations mean...>.>

Just reading wikipedia, the metric is the Minkowski metric times the "square" of the vierbein field. http://en.wikipedia.org/wiki/Frame_fields_in_general_relativity