0
$\begingroup$

I'm taking a course on general relativity and try to find a good source on spinors. I'm using "General relativity" by Wald as a guideline, but I'm struggling with his notation. I already took a math class on spinor bundles and am no fan of the abstract index notation. It's proablly great for calculations but it's very hard to read if you're used to reading math texts. Can some one point out to me a reference for learning about the use of spinors in GR that does not use the abstract index notation for example $\omega(v,w)$ instead of $\omega_{AB}v^Aw^B$? With all the lowering and uppering of the indices, I'm totally losing track of the calculations, especially since you have to distinguish between capital latin, capital greek, lowercase latin, lowercase greek all of which can be primed and occur in super- and subscript.

$\endgroup$
5
  • $\begingroup$ I haven't used that text, but I took a course on QFT that used Einstein index notation for tensors and 4-vectors. I really recommend you spend some time trying to get used to the notation. There's a reason it's still used over 100 years later - the raising and lowering of indices has physical meaning, and it turns complex matrix algebra into simple sums. To me $\omega_{AB}v^Aw^B$ makes a lot more sense intuitively than $\omega(v,w)$ which looks like it could be a bivariate function, or an inner product of $v$ and $w$, both of which are wrong. $\endgroup$ Feb 4, 2021 at 9:59
  • $\begingroup$ @DanPollard Why should it be wrong to denote a scalar product by $\omega(v,w)$ ? I was thinking of $\omega$ as a section in $\Gamma(T_pM^*\otimes TpM^*)$ for a finite dimensional manifold $M$. So $\omega$ can be a scalar product at every point $p\in M$. $\endgroup$
    – Leroy Od
    Feb 4, 2021 at 11:20
  • $\begingroup$ It's not wrong to denote a scalar product as $(v,w)$, but $v^Aw^B$ isn't a scalar product, it's an outer product. Saying $\omega_{AB}v^Aw^B = \omega(v,w)$ isn't correct. If you want to use that notation, I'd write it as $(\omega, v \otimes w)$, but I've never seen it written that way before. $\endgroup$ Feb 4, 2021 at 11:47
  • $\begingroup$ O'Neill does relativity in math notation, but it doesn't cover spinors. $\endgroup$
    – Javier
    Feb 4, 2021 at 13:50
  • $\begingroup$ @LeroyOd, you got to be fluent in both Roman and Greek languages. See related post here: physics.stackexchange.com/questions/608748/… $\endgroup$
    – MadMax
    Feb 4, 2021 at 17:09

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.