Understanding the Difference between Scalar and Dot Product in Tensor Calculus

[matheinste]

Hello all.

In a quite easy to follow short piece by Edmond Bertschinger entitled Introductio to Tensor Calculus for General Relativity on page 6 when speaking of the metric tensor he says, referring to the symbol conventions used in the piece :-

"" We reserve the dot product notation for the metric and inversr metric tensor just as we reserve the angle bracket scalar product notation for the identity tensor---""

In the second case he is referring to the action of the identity tensor on a one form and a vector and in the first case he is referring to the action of the metric and inverse metric tensor on two one forms or two vectors.

What, if any, is the difference between the scalar and dot product.

Thanks. Matheinste

 

[robphy]

Using the terms given,
the dot product involves the metric or its inverse, whereas the scalar product doesn't.
Maybe some abstract-index-notation will clarify:

$$\vec V\cdot \vec W=g(\vec V,\vec W)=V^a g_{ab} V^b$$

$$\widetilde P\cdot \widetilde Q=g^{-1}(\widetilde P,\widetilde Q)=P_a g^{ab} Q_b$$

$$< \widetilde P , \vec V > = P_a \delta^a{}_b V^b = P_b V^b$$

Note the nature of the "factors" involved the various "products".

 

[matheinste]

Thanks for your reply robphy.

Yes I understand the formulae but is the difference in terminology standard. I have seen other places where the terms dot product and scalar product mean the same.

Matheinste.

 

[robphy]

Thanks for your reply robphy.

Yes I understand the formulae but is the difference in terminology standard. I have seen other places where the terms dot product and scalar product mean the same.

Matheinste.

I don't think it's standard...You even hear of "scalar dot products".
There are also "inner products", "contractions", "transvections"...

A purist might argue with the "overloaded" use of the dot
for use with the metric and with its inverse.

 

[matheinste]

Thanks robphy.

I'm happy with that.

Matheinste.