Penrose's spinor calculus of 4-dimensional Lorentzian geometry is extended to the case of 5-dimensional Lorentzian geometry. Such fruitful ideas in Penrose's spinor calculus as the spin covariant derivative, the curvature spinors or the definition of the spin coefficients on a spin frame can be carried over to the spinor calculus in 5-dimensional Lorentzian geometry. The algebraic and differential properties of the curvature spinors are studied in detail and as an application we extend the well-known 4-dimensional Newman-Penrose formalism to a 5-dimensional spacetime.
In this paper we present a space-time calculus for symmetric spinors, including a product with a number of index contractions followed by symmetrization. As all operations stay within the class of symmetric spinors, no involved index manipulations are needed. In fact spinor indices are not needed in the formalism. It is also general because any covariant tensor expression in a 4-dimensional Lorentzian spacetime can be translated to this formalism. The computer algebra implementation SymSpin as part of xAct for Mathematica is also presented.
The spinor-helicity formalism is an essential technique of the amplitudes community. We draw on this method to construct a scheme for classifying higher-dimensional spacetimes in the style of the four-dimensional Petrov classification and the Newman-Penrose formalism. We focus on the five-dimensional case for concreteness. Our spinorial scheme naturally reproduces the full structure previously seen in both the CMPP and de Smet classifications, and resolves longstanding questions concerning the relationship between the two classifications.
Benefiting from the index spinorial formalism, the Killing spinor equation is integrated in six-dimensional spacetimes. The integrability conditions for the existence of a Killing spinor are worked out and the Killing spinors are classified in two algebraic types, in the first type the scalar curvature of the spacetime must be negative, while in the second type the spacetime must be an Einstein manifold. In addition, the equations that define Killing-Yano (KY) and closed conformal Killing-Yano (CCKY) tensors are expressed in the index notation and, as consequence, all non-vanishing KY and CCKY…
In this work, we study quantum fluctuations in the stress energy tensor of spinor fields evolving in general FRW spacetimes. We quantify these fluctuations by the noise kernel of spinor fields. For the particular case of de Sitter spacetime, we place the spinor field in what is called the fermionic Bunch Davies vacuum and study the variation of the noise kernel with the mass of the field. We then make use of the conformal invariance of massless spinor fields and employ an equivalence that relates a massless spinor field in any given FRW spacetime with a corresponding massless spinor field in d…
We investigate the implications of the existence of Killing spinors in a spacetime. In particular, we show that in vacuum and electrovacuum a Killing spinor, along with some assumptions on the associated Killing vector in an asymptotic region, guarantees that the spacetime is locally isometric to the Kerr or Kerr-Newman solutions. We show that the characterisation of these spacetimes in terms of Killing spinors is an alternative expression of characterisation results of Mars (Kerr) and Wong (Kerr-Newman) involving restrictions on the Weyl curvature and matter content.
In spinor formalism, since any massless free-field spinor with spin higher than $1/2$ can be constructed with spin-1/2 spinors (Dirac-Weyl spinors) and scalars, we introduce a map between Weyl fields and Dirac-Weyl fields. We determine the corresponding Dirac-Weyl spinors in a given empty spacetime. Regarding them as basic units, other higher spin massless free-field spinors are then identified. Along this way, we find some hidden fundamental features related to these fields. In particular, for non-twisting vacuum Petrov type N solutions, we show that all higher spin massless free-field spinor…
Flagpole and flag-dipole spinors are particular classes of spinor fields that has been recently used in different branches of theoretical physics. In this paper, we study the possibility and consequences of these spinor fields to induce an underlying fluid flow structure in the background of Kerr spacetimes. We show that flag-dipole spinor fields are solutions of the equations of motion in this context. To our knowledge, this is the second time that this class of spinor field appears as a physical solution, the first one occurring as a solution of the Dirac equation in ESK gravities.
Exact solutions to an Einstein-Maxwell(E-D-M) system with an electric current are simple models of global cosmic magnetic phenomena in the universe. In this paper, we consider an E-D-M system with two chiral spinors $ ψ^\pm$ coupled with a gauge field $ A$ and an electromagnetic field $F=dA$. We construct a family of exact solutions to the E-D-M system on four-dimensional static Sasakian spacetimes. The electromagnetic field in the solutions is a contact magnetic field, and the chiral spinors are induced from Sasakian quasi-Killing spinors. We find closed universe models whose weak energy cond…
On Cosmological Spacetime Structure and Symmetry: Manifold as a Lie Group, Spinor Structure and Symmetry Group, Minkowski Metric, and Unnecessariness of Double-Valued Representations
Within the scope of Bianchi type-III spacetime we study the role of spinor field on the evolution of the Universe as well as the influence of gravity on the spinor field. In doing so we have considered a polynomial type of nonlinearity. In this case the spacetime remains locally rotationally symmetric and anisotropic all the time. It is found that depending on the sign of nonlinearity the models allows both accelerated and oscillatory modes of expansion. The non-diagonal components of energy-momentum tensor though impose some restrictions on metric functions and components of spinor field, unl…
The Nontriviality of Trivial General Covariance: How Electrons Restrict 'Time' Coordinates, Spinors (Almost) Fit into Tensor Calculus, and 7/16 of a Tetrad Is Surplus Structure
A self-consistent system of nonlinear spinor and Bianchi type-V anisotropic gravitational fields are investigated. It is found that the presence of nontrivial non-diagonal components of the energy-momentum tensor of the spinor field imposes some severe restrictions to the system. As a result two different solutions are found. In one case the metric functions are similar to each other, i.e., $a_1 \sim a_2 \sim a_3$ and the spinor mass and spinor field nonlinearity do not disappear from the system. In this case the spacetime expands with acceleration in case of a positive self-coupling constant …
In this article we construct and discuss several aspects of the two-component spinorial formalism for six-dimensional spacetimes, in which chiral spinors are represented by objects with two quaternionic components and the spin group is identified with $SL(2;\mathbb{H})$, which is a double covering for the Lorentz group in six dimensions. We present the fundamental representations of this group and show how vectors, bivectors, and 3-vectors are represented in such spinorial formalism. We also complexify the spacetime, so that other signatures can be tackled. We argue that, in general, objects b…
Within the scope of Bianchi type-VI cosmological model we study the role of spinor field in the evolution of the Universe. It is found that due to the spinor affine connections the energy-momentum tensor of the spinor field possesses non-diagonal components. The non-triviality of non-diagonal components of the energy-momentum tensor imposes some severe restrictions either on the spinor field or on the metric functions or on both of them. But unlike in cases of Bianchi type-I or $VI_0$, in case of Bianchi type-VI model it does not lead to the elimination of spinor field nonlinearity or mass ter…
We show how to transform a Dirac equation in curved spacetime into a Dirac equation in flat spacetime. In particular, we show that any solution of the free massless Dirac equation in a 1+1 dimensional flat spacetime can be transformed via a local phase transformation into a solution of the corresponding Dirac equation in a curved background, where the spacetime metric is encoded into the phase. In this way, the existing quantum simulators of the Dirac equation can naturally incorporate curved spacetimes. As a first example we use our technique to obtain solutions of the Dirac equation in a par…
We describe the construction of a geometric invariant characterising initial data for the Kerr-Newman spacetime. This geometric invariant vanishes if and only if the initial data set corresponds to exact Kerr-Newman initial data, and so characterises this type of data. We first illustrate the characterisation of the Kerr-Newman spacetime in terms of Killing spinors. The space spinor formalism is then used to obtain a set of four independent conditions on an initial Cauchy hypersurface that guarantee the existence of a Killing spinor on the development of the initial data. Following a similar a…
We study the interaction, in general curved spacetime, between a spinor and a scalar field describing dark energy; the so-called DE$_ν$ model in curved space. The dominant term is the dimension 5 operator, which results in different energy shifts for the neutrino states: an Aharonov-Bohm-like effect. We study the phenomenology of this term and make observational predictions to detect dark energy interactions in the laboratory due to its effect on neutrino oscillation experiments, which opens up the possibility of designing underground experiments to detect dark energy. This dimension 5 operato…
In flat spacetime, the Dirac equation is the "square root" of the Klein-Gordon equation in the sense that by applying the square of the Dirac operator to the Dirac spinor, one recovers the Klein-Gordon equation duplicated for each component of the spinor. In the presence of gravity, applying the square of the curved-spacetime Dirac operator to the Dirac spinor does not yield the curved-spacetime Klein-Gordon equation, but yields, instead, the Schrödinger-Dirac covariant equation. First, we show that the latter equation gives rise to a generalization to spinors of the covariant Gross-Pitaevskii…
