vity is restricted to a representation of the Lorentz group; however, as soon as the theory is to be given a coordinate-independent (“generally covariant”) form, then the full tensor concept comes into play.
Then, by a transformation from ,38. Moreover, which, according to Equation (70), expresses the invariance of the space-time distance to the origin:
Now, contravariant 2-spinors () are the elements of a complex linear space, spinor space, on which the matrices are acting39. The spinor is called elementary if it transforms under a Lorentz-transformation ascontravariant dotted spinors are those transforming with the complex-conjugate matrix : inverse matrices,
Higher-order spinors with dotted and undotted indices transform correspondingly. For the raising and lowering of indices now a real, antisymmetric -matrix with components is needed, such that
Next to a spinor, bispinors of the form , etc. are the simplest quantities (spinors of 2nd order). A vector can be represented by a bispinor ,
In order to write down spinorial field equations, we need a spinorial derivative,Weyl equation:
Dirac- or 4-spinors with 4 components , , may be constructed from 2-spinors as a direct sum of contravariant undotted and covariant dotted spinors and : For , we enter and ; for , we enter and In connection with Dirac spinors, instead of the Pauli-matrices the Dirac -matrices (-matrices) appear; they satisfy[53, 54]:
The generally-covariant formulation of spinor equations necessitates the use of -beins , whose internal “rotation” group, operating on the “hatted” indices, is the Lorentz group. The group of coordinate transformations acts on the Latin indices. In Cartan’s one-form formalism (cf. Section 2.1.4), the covariant derivative of a 4-spinor is defined by
Equation (89) is a special case of the general formula for the covariant derivative of a tensorial form , i.e., a vector in some vector space , whose components are differential forms,40.
A Riemannian space is called (locally) stationary if it admits a timelike Killing vector; it is called (locally) static if this Killing vector is hypersurface orthogonal. Thus if, in a special coordinate system, we take then from Equation (91) we conclude that stationarity reduces to the condition . If we take to be the tangent vector field to the congruence of curves , i.e., if , then a necessary and sufficient condition for hypersurface-orthogonality is
A generalisation of Killing vectors are conformal Killing vectors for which with an arbitrary smooth function holds. In purely affine spaces, another type of symmetry may be defined: ; they are called affine motions .
© Max Planck Society and the author(s)