2
votes
0answers
46 views

Irreducible representation for the massless particle with helicity 2 and the Weyl tensor

As it can be shown, the equations for the irrep with zero mass and helicity 2, -2 respectively can be given in a form $$ \tag 1 \partial^{\dot {b}a}C_{abcd} = 0, \quad ...
3
votes
1answer
104 views

The relationship between spin and spinor curvature

The identity, $$ -\gamma^b{\mathcal{R}}_{ab} = {\mathcal{R}}_{ab}\gamma^b = \frac{1}{2}\gamma^b R_{ab}$$ is presented in the answer to the question Dirac Equation in General Relativity. How does ...
5
votes
3answers
302 views

Is there any relationship between gauge field and spin connection?

For a spinor on curved spacetime, $D_\mu$ is the covariant derivative for fermionic fields is $$D_\mu = \partial_\mu - \frac{i}{4} \omega_{\mu}^{ab} \sigma_{ab}$$ where $\omega_\mu^{ab}$ are the spin ...
7
votes
2answers
130 views

Definition of a spinor and applications to GR

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 ...
2
votes
1answer
64 views

Compatibility conditions of spinors and Riemannian Metrics

I came across an interesting article by Montesinos (J. Geom. Phys. 2 (1985), no. 2, 145–153.). In it, he finds that spin structures (as lifts of $SO(4)$) are not compatible with all Riemannian metrics ...
2
votes
1answer
253 views

Non-linear Dirac equation in Einstein Cartan theory

Can someone explain this Wikipedia article, specifically the section on Einstein-Cartan theory? I have no idea how the equation ...
20
votes
2answers
113 views

Kerr Geometry, Separability and Twistors

One of the remarkable properties of the Kerr black hole geometry is that scalar field equations separate and are exactly solvable (reducible to quadrature), even though naively it does not have enough ...