Spinors in Curved Space Erik Olsen December 4, 2008 1 Introduction The main problem that this paper will address is as follows: how can we in- clude a gravitational field g within a given Lagrangian density L? To solve this, we make use of the Principle of General Covariance. The first step is finding the special-relativistic equations of motion for the system where the gravita- tional field is neglected. The next step is to make several substitutions within these equations; all Lorentz tensors, derivatives and Minkowski tensor (η) terms will be replaced with objects that behave like tensors under general coordinate transformations, covariant derivatives and the metric tensor for curved space- time gµν respectively. These substitutions will make the equations generally covariant. It should be noted here that this method is only valid for space- time distances that are small in comparison to the space-time distances of the gravitational field. Also note that: [?] @ξα @ξβ g = η (1) µν @xµ @xν αβ Where ξ represents a coordinate system under the influence of gravity. The rank of the tensor within the Lagrangian density corresponds to the spin of the particle described. For example, a rank 0 tensor describes a spin-0 particle (π meson), a rank 1 tensor describes a spin-1 particle (photon) and a rank 2 tensor describes spin-2 particles. Notice however that these are all particles of integer spin; we are interested in particles with half-integer spin (electrons, quarks, etc). Such particles are described by spinors, not tensors [?]. Unfortunately, the substitution method described above does not work if we use spinors in our Lagrangian density instead of tensors. The reason for this can be seen by observing the group of linear invertible 4 X 4 matrices GL(4) and Lorentz transformations (which form a subgroup of GL(4)). The elements of GL(4) that are represented as tensors will act as tensors under a Lorentz transformation. However, if we look for representations of GL(4) that will act as spinors under an infinitesimal Lorentz transformation, we find that no such elements exist. To alleviate this problem, we shall use what are known as tetrads (or vier- beins). 2 Tetrads α The idea behind this method is to introduce normal local coordinates zX to each point in space-time X in addition to the generalized coordinates xµ one 1 would have to define their system. Using this, we can re-write the metric for curved space-time: @zα (x) @zβ (x) g (x) = X X η (2) µν @xµ @xν αβ Let: α @zX (x) α µ = Vµ (X) (3) @x x=X β @zX (x) β ν = Vν (X) (4) @x x=X α We define Vµ (X) to be a tetrad. Since we have two coordinate systems here (local inertial and general nonin- ertial), let's see what happens when we fix the local one and change the general from xµ to x0µ. This will cause the following change in the tetrad: α 0α Vµ ! Vµ (5) where @zα V 0α = X −! (6) µ @x0µ @zα @xν V 0α = X −! (7) µ @x0µ @xν @xν @zα V 0α = X (8) µ @x0µ @xν Thus we see [?]: @xν V 0α = V α (9) µ @x0µ ν From this we can conclude that a tetrad is not to be thought of as a tensor, but instead as four orthonormal vector fields. Still working with the coordinate systems, it is possible for us to take the Lorentz transform of the local system for every point in space-time [?]: α 0α α β zX −! zX = Λβ (X)zX (10) Plugging this into the definition of the tetrad: @ V α(X) = Λαzβ −! (11) µ @xµ β x @zβ V α(X) = Λα X (12) µ β @xµ Thus: α α β Vµ (X) ! Λβ Vµ (X) (13) α From this we see Vµ (X) transforms as a Lorentz contravariant vector. The usefulness of tetrads comes as follows: if we take a generally covariant µ vector Bµ and contract it into a tetrad Vα , we obtain the following [?]: µ Vα Bµ = Bα (14) 2 This particular object Bα has two useful properties: (1) Under a local Lorentz transformation it will behave as a vector. (2) Under a general coordinate transformation it will transform as four scalars. The reason tetrads are so useful is because now we can take any tensor field and contract it into a group of scalars, allowing us to use the substitution process discussed in the Introduction. This contraction method can even be used for a spinor field, which eliminates the problem of how spinors act under an infinitesimal Lorentz transformation. Covariant Derivatives It is not enough to focus only on the field; we must also look into the derivative of the field. This is because the derivative of a tensor field will itself be a tensor field in Minkowski space. To be useful within the Lagrangian density, the derivatives must be Lorentz vectors and coordinate scalars. We choose the following covariant derivative rα to act on a field where [?]: β rα ! Λα(x)D Λ(x) rβ (x) (15) where D(Λ) is the matrix representation of the infinitesimal Lorentz group. This expression is chosen since it is a Lorentz vector, coordinate scalar and transforms easily under a Lorentz transformation. To make the last property here possible, we define [?]: µ rα = Vα (@µ + Γµ) (16) where 1 Γ (x) = ΣαβV ν r V (x) (17) µ 2 α µ βν and Σαβ is the group generator for the Lorentz group. Also [?]: µ Vβν = gµν Vβ (18) We are now in a position to use the method described in the Introduction to generalize a Lagrangian density containing spinor fields into curved space. All it will involve is contracting all spinors using tetrads and replacing all partial derivatives with covariant ones. 1 3 Example: The Spin 2 Field The Lagrangian density for this particular case will be [?]: 1 L(x) = i( γα@ − @ γα ) − m (19) 2 α α Using what was developed in the previous section, we wish to generalize this to curved space-time. First, for the Dirac spinor field: 1 Σ = [γ ; γ ] (20) αβ 4 α β 3 where the γ terms are the Dirac matrices. Plugging this into our value of Γµ: 1 Γ (x) = [γ ; γ ]V ν (x)r g (x)V µ(x) (21) µ 8 α β α µ µν β Putting this into the definition rα and putting it into the Lagrangian density in place of @α, we get: n1 o L(x) = det V i γαV µr − V µ(r )γα − m (22) 2 α µ α µ Here we multiplied by det V to keep the Lagrangian density a scalar density. µ α Let γµ = Vα γ . Now we get [?]: n1 o L(x) = det V i γµr − (r )γµ − m (23) 2 µ µ We renamed the γ matrices because they are now the Dirac matrices in curved space. Taking their anti-commutator: fγµ; γν g = γµγν + γν γµ ! (24) µ α ν β ν β µ α Vα γ Vβ γ + Vβ γ Vα γ ! (25) µ ν α β Vα Vβ fγ ; γ g (26) By definition: α β fγ ; γ g = 2ηαβ (27) So: µ ν µ ν fγ ; γ g = 2Vα Vβ ηαβ (28) Thus we see: µ ν fγ ; γ g = 2gµν (29) This is the anti-commutator relation of the Dirac matrices generalized to curved space. 4 Conclusion With the ability to describe spinors in curved space, it becomes possible to see the effects of the gravitational action on systems of fermions. Of course, everything discussed here is still just an approximation, for we neglected to include quantum effects of gravity which are not completely understood at this time. References [1] G.B. Arfken and H.J Weber. Mathematical Methods for Physicists. Elsevier Adademic Press, sixth edition, 2005. [2] N.D. Birrell and P.C.W Davies. Quantum Fields in Curved Space. Cam- bridge University Press, 1982. [3] Steven Weinberg. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. John Wiley and Sons, 1972. 4.