Fun with the metric tensor in Einstein notation

Posted by on Jan 17, 2013 in Science | 0 comments

In my last post, I introduced the metric tensor and tried to show that it really is not too horrible a beast. It arises quite naturally as a consequence of applying the chain rule of calculus in the context of coordinate transformations. In this post, I want to go back over the same ground, but this time using the Einstein notation. I guess that means talking a little about the Einstein notation. One feature of this notation is that repeated indices are summed over. For example, these two forms are identical: This shows two features of the Einstein notation. First, if any index is repeated,...

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The metric tensor

Posted by on Jan 15, 2013 in Science | 0 comments

The name, metric tensor, is enough to strike fear into the hearts of any physics student, associated as it is with General Relativity. I’m here to tell you that, in its heart of hearts, the metric tensor is not much more complex than the standard vector dot product, of which it is a generalization. In what follows, I hope to remove some of the tension surrounding the metric tensor. That raises a pretty basic question right off the bat: if we already have the machinery of the dot product, why do we need a metric tensor? Where does the dot product break down? Recall that one way to...

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