- #1

- 18

- 0

But I'm having difficulty understanding where this comes from intuitively. How would I know that I need them if I'm trying to reason things out?

More clearly:

1) I have a linear vector space (LVS).

2) I say "I have addition of vectors from condition of LVS. From my experience with the outside world, I know I need my space of vectors to have notions of length and distance, not just rules for combining them."

3) Ok, then I need a way to combine vectors in my space that have something to do with how they're positionally related to one another.

4) ???

Now what?

1) How do I know that the distance operation I'm looking for is the scalar product?

2) Why do I need a notion of distance to specify a scalar product? (v

^{u}w

^{n}g

_{un}=> in order to have my scalar product to work, I need g

_{un})

3) Why do I not need a notion of distance if I use objects from the dual space?

(w

^{n}g

_{un}= w

_{u}=> v

^{u}w

_{u}= scalar)

This might need to be in the GR forum, but I was reading the first chapter of Shankar for the billionth time when I was able to finally articulate all this. as you might be able to tell -- this is *really* bothering me, I'm really confused about the motivation for defining all this stuff. plz hellllllp! <3