Why Lowering Index on Components Affects Physical Variables

[enomanus]

Hi everyone! I am trying to self- study GR and I am (very slowly) working through Schutz at the moment ( not the clearest at times for a " first course" text). Anyway I have noticed that several times when working with components of physical variables e.g. momentum/ energy, the 'up' index in say p alpha is deliberately 'lowered' in the workings to get p alpha 'down'.
for example in - deriving the Consevation law on geodesics so that we can spot if the metric is independent of any x component or - in Shutz page 170 where we work out the scalar product - p.p to show that energy is conserved in a stationary gravity field.
i can follow the lines and the equations BUT I don't understand why we keep lowering the index?
Is this something to do with physical variables or with 1-forms??
The answer is probably something very simple but I'm asking anyway
Thanks!

 

[atyy]

We hypothesize that there is a metric - it eats two vectors and spits out a number - this number is the scalar product of the two vectors. When you compute the scalar product, you may choose to have the metric eat one vector before eating the second vector. This process is called lowering the index of the first vector.

 

[DrGreg]

It is related to the fact that

$$\sum_{\alpha=0}^3dx_{\alpha}dx^{\alpha}=\sum_{\alpha=0}^3\sum_{\beta=0}^3g_{\alpha\beta}dx^{\alpha}dx^{\beta}$$​

is invariant (the same in every coordinate system), whereas

$$\sum_{\alpha=0}^3dx^{\alpha}dx^{\alpha}$$​

isn't.