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


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.


This question should probably be split up into several questions. Really quickly, I can answer the bit about the Newman-Penrose forumulation 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.

  • 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.


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