Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I'm having trouble understanding the following sentence from Schutz's

  1. Jul 16, 2010 #1
    I'm having trouble understanding the following sentence from Schutz's A First Course in General Relativity, so I was hoping someone could explain/expound on it. "
    [tex] p_a \equiv \widetilde{p}( \vec{e_{\alpha} ) } ) [/tex] Any component with a single lower index is, by convention, the component of a one-form; an upper index denotes the component of a vector."
  2. jcsd
  3. Jul 17, 2010 #2


    User Avatar
    Science Advisor

    Re: One-Forms

    If I have a vector X, I can write it in component form as

    X = X^{\mu}e_{(\mu)}

    I put round brackets around the mu index of the basis vector e to indicate that for every mu we have a whole vector, not just one component! Now, I can act with this vector X on the dual basis:

    X(e^{(\nu)}) = X^{\mu} e_{(\mu)} (e^{(\nu)}) = X^{\mu}\delta_{\mu}^{\nu} = X^{\nu}
    In words: acting with the vector X on the dual basis means acting with the basis on the dual basis, times the components of X. By definition this gives me a delta function times the components of X, which equals the components of X. This holds for every tensor T in general; you could try it for, say, a rank (2,1) tensor T.

    Components with lower indices transform different from components with upper indices; the latter we sometimes call "contravariant", the former "covariant" ( co goes below ).
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook