I have a somewhat broad/big question, and I know that there are many references for it available out there. However, so far I couldn't find anything that I can really understand, that's why here is my last resort.

The question is about the spinor notation and its use in general relativity. Reading a paper by Penrose, I saw the following statement (non-verbatim):

Define $\kappa_{AB}=\psi^{-1/3}o_{(A}i_{B)}$, where the Weyl conformal spinor is $\Psi_{ABCD}=\psi o_{(A}o_{B}i_{c}i_{D)}$. Then $H_{ab}=ik_{AB}\epsilon_{A'B'}-i\epsilon_{AB}\overline{\kappa}_{A'B'}$ is a skew tensor. <…>

Now, I don't understand the above objects at all. From what I read (book "Introduction to 2-Spinors in General Relativity" by O'Donnel) we can think of $o$ and $i$ as vectors $o=(1,0)$ and $i=(0,1)$ (is it correct?) Then the argument is, that the spinor indices (here, A B C D A' B') are purely a notation and don't mean anything. Well, this is where I stopped understanding what kind of objects I'm working with. If it's purely some symbolics, how can I actually work with it/calculate something?

In the example above, since on the left hand side he uses "normal notation" $H_{ab}$ I understand that we have a rank 2 tensor. By looking at the right hand side, I don't understand anything. What I know is that $\epsilon$ is like an equivalent of the metric in the spinor notation, in the sense that we can, for example, raise and lower indices with it. However, since everything is supposed to be just a "symbol", I don't understand the object on the right hand side.


1 Answer 1


Before going further, I would suggest you to read Chapter 13 ("Spinors") of R.Wald's book "General Relativity". In that chapter, you will see that 2-spinors are simply vectors living in a two-dimensional complex vector space. The capital letters in the indices are simply the abstract index notation for these vectors (see Section 2.4 in Chapter 2 of the same book).

You will also see that the real spinorial tensors, i.e., the spinorial tensors of type (1,0;1,0) such that $\overline{\varphi}^{A'A}=\varphi^{AA'}$, form a real four-dimensional vector space, $V$. Also, taking the $\epsilon_{AB}$, you can build the following spinorial tensor: $$g_{AA'BB'}=\epsilon_{AB}\overline{\epsilon_{A'B'}} $$ which gives you a multilinear map of the form $V\times V\longrightarrow\mathbb{R}$. It can be verified that this multilinear map defines a Lorentz metric on $V$. This vector space $V$ can be identified with the usual tangent spaces $TM_{p}$ of flat spacetime (via identification between orthonormal bases of $V$ and $TM_{p}$), and that's why it's customary to make the notational identification $a\cong AA' $, where $a$ is the abstract index notation for a tangent vector. For the Minkowski metric of spacetime, $\eta_{ab}$ , we, of course, get:

$$\eta_{ab}\cong g_{AA'BB'}=\epsilon_{AB}\overline{\epsilon_{A'B'}} $$

It was Penrose who invented this notation, and he (as well as most relativists today) uses the notation extensively.


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