Why Do Subscripts and Superscripts Vary in Einstein Summation Notation?

[iontail]

hi i am just reading some notes on tesor analysis and in the notes itself while representing vectors in terms of basis using einstein summation notation the author switches between subsripts and superscripts at times. are there any different in these notation. if so what are they and when should they be used?

An examples is given on the pdf in section 1.3.

 

[Fredrik]

They're using subscripts to label different members of the same basis, and superscripts to label components of a vector, so A=A^i e_i. The components of the metric are g_{ij}=g(e_i,e_j). The g^{ij} are the components of the inverse of the matrix with components g_{ij}. Therefore g^{ij}g_{jk}=\delta^i_k. The e^i are members of a basis for the dual space of the vector space with basis \{e_i\}. They're defining them by e^i=g^{ij}e_j. An expansion of a member of the dual space in terms of the e^i would appear as \omega=\omega_i e^i, i.e. components of dual vectors are labeled by a subscript.

The dual space V* of a real vector space V is the set of continuous linear functions from V into the real numbers.

 

[iontail]

hi thanks for you reply. what is the underlying benifit of switching between the vectors on covector (indices). I ask this because Feynman introduces the concept of four vectors only using subscripts only.

 

[Fredrik]

In this context, there is no advantage at all. You can think of these equations as matrix equations, put all the indices downstairs, and forget you've ever even heard the word "tensor". (I often do that myself. See this for example. But then you should also be aware of this so that you understand what you read in books. Note in particular the expression for the inverse of a Lorentz transformation).

In differential geometry, the distinction between subscripts and superscripts has the advantage that the notation reveals what sort of tensor you're dealing with. For example, when you see R_{abc}{}^d, you know it's supposed to be acting on three tangent vectors and one cotangent vector.