Definition of a spinor and applications to GR

Question

1. I understand the construction of the Clifford algebra $C(r,s)$ and in turn the corresponding $Pin$ and $Spin$ groups. I would like first to clarify that $Spin(r,s)^e$ is the universal covering group (a 2 to 1 covering) of $SO(r,s)^e$ ?

2. It is true that for $r+s:=n$ even, there is an isomorphism: $$\rho\ : C(r,s)^\mathbb{C} \rightarrow\ \mathbb{C}(2^{n/2})$$ we then have representations of $Spin(r,s)$.

3. My main questions is whether a spinor is defined to be an element of $Spin(r,s)$ in some representation or if it is a vector in the space upon which the representation of $Spin(r,s)$ acts?

4. Moving on to NP formalism and I am quite frankly very lost. I would like to understand it, if possible in the framework above. What is meant by a '2-spinor' given the above notation?

5. To my understanding a tetrad is a choice of basis $e^{(i)}$, $i=1,...,n$ on each tangent space of our spacetime. How does the null tetrad in the NP formalism relate to this thinking? Some initial questions: if every vector in the tetrad is null how can it span the tangent space?; the vectors involve complex coefficients, does this mean they are spinors?

Sources: 'Differential Geometry, gauge theories, and gravity' - M.Göckeler & T.Shücker

https://en.wikipedia.org/wiki/Newman%E2%80%93Penrose_formalism

Disclaimer: I am a pure mathematician trying to introduce myself to spinors. My aim is to understand the Newman-Penrose (NP) formalism of GR. I know a bit of GR already but have no understanding of the quantum physics language which seems to engulf most of the literature on spinors.

Answer 1

This question should probably be splitted up into several questions. Really quickly, I can answer the bit about the Newman-Penrose formulation and tetrads.

I think the best modern approach to the tetrad formalism is to treat the set of tetrads as a map from an internal orthonormal frame to an ordinary coordinate frame. So, the tetrad is a set of matrices $e_{a}{}^{I}$ satisfying:

$$g_{ab} = \eta_{IJ}e_{a}{}^{I}e_{b}{}^{J}$$

So, $e_{a}^{I}dx^{a}$ is a vector giving an orthonormal basis on the spacetime at each point, as you said.

Now, the Newman-penrose formulation goes a step farther than this, and specifies that $\eta_{IJ}$ must take the form $\eta_{01} = -1$ and $\eta_{23} = 1$ with all others zero. This requires that the $e$ have complex values. For the example of Schwarzschild spacetime, one can choose Kruskal coordinates $u$ and $v$ for the first two $e$, and then the angular coordinates chosen can be $r/\sqrt{2}(1 \pm i \sin \theta)$. Then, it should be clear that all four coordinates are null, and that they satisfy the correct relationship for $\eta_{IJ}$. This is done because there is a natural relationship between null vectors and spinors, and by having a quadruply null basis, the curvature forms take an especially simple form.

Answer 2

• III) A spinor is indeed an element of the vector space on which the representation of the Spin group acts.
• IV) I don't know
• V) The fact that the norm of the vector is null does not imply that has no nonvanishing entries. Remember that in physics applications one has a metric of indefinite signature, so contributions to the norm can (and most of the time will) cancel each other.

I hope this is at least a little help.