[Paper Review] Nonlinear Spinor Fields in Bianchi type-V spacetime

Bijan Saha|arXiv (Cornell University)|Feb 18, 2016

Cosmology and Gravitation Theories3 citations

TL;DR

This paper investigates self-consistent nonlinear spinor fields in Bianchi type-V anisotropic spacetime, showing that non-diagonal components of the spinor field's energy-momentum tensor impose strong constraints. Two distinct solutions emerge: one with massive, nonlinear spinor fields leading to accelerated expansion for positive self-coupling (λ > 0), and another with massless, linear spinor fields resulting in linear expansion, analogous to FRW-like evolution under similar constraints in other Bianchi models.

ABSTRACT

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 $λ$. A negative $λ$ gives rise to a cyclic or periodical mode of expansion. In the second case the spinor mass and the spinor field nonlinearity vanish and the Universe expands linearly with time.

Motivation & Objective

• To investigate the self-consistent interaction between nonlinear spinor fields and Bianchi type-V anisotropic gravitational fields.
• To determine how non-diagonal components of the spinor field's energy-momentum tensor constrain the spacetime geometry and field dynamics.
• To classify possible cosmological solutions under different symmetry and field constraints.
• To analyze the role of the self-coupling constant λ in determining the expansion mode—accelerated, cyclic, or linear.

Proposed method

• Formulates the action principle with gravitational and spinor field Lagrangians, using the Einstein-Hilbert action and a nonlinear spinor Lagrangian with self-interaction term F(F(K)).
• Applies the Fierz identity to express the nonlinear term F as a function of invariants K ∈ {I, J, I+J, I−J}, where I = (ψ̄ψ)² and J = (iψ̄γ⁵ψ)².
• Uses the Bianchi type-V metric with time-dependent scale factors a₁(t), a₂(t), a₃(t), and exponential spatial dependence e²ᵐˣ³, introducing a parameter m to model anisotropy.
• Derives the Einstein field equations from the metric, computing nontrivial Christoffel symbols and the Einstein tensor components Gᵢⱼ.
• Imposes constraints on the energy-momentum tensor to eliminate non-diagonal components, leading to two distinct solution branches.
• Solves the resulting system of ODEs for the scale factors and spinor field components under two symmetry-restricted cases.

Experimental results

Research questions

• RQ1 How do non-diagonal components of the spinor field's energy-momentum tensor affect the dynamics of a Bianchi type-V spacetime?
• RQ2 Can massive, nonlinear spinor fields coexist with anisotropic Bianchi type-V geometry under physical constraints?
• RQ3 What cosmological expansion modes (accelerated, cyclic, linear) emerge from the self-consistent system of spinor and gravitational fields?
• RQ4 How does the sign and magnitude of the self-coupling constant λ influence the long-term evolution of the Universe in this model?

Key findings

• For a positive self-coupling constant λ > 0, the model exhibits accelerated expansion, with the volume scale V(t) increasing over time and the deceleration parameter q < 0.
• For a negative self-coupling constant λ < 0, the volume scale V(t) exhibits cyclic or periodic behavior, with the Universe expanding to a maximum and then contracting.
• In the first solution branch, the metric functions satisfy a₁ ∼ a₂ ∼ a₃, and both spinor mass and nonlinearity remain non-zero, leading to accelerated expansion.
• In the second solution branch, the spinor mass and nonlinearity vanish, resulting in a linear expansion of the Universe with V(t) = b₀t + b₁, where b₀ and b₁ are constants.
• The spinor field components in the second case are ψᵢ = cᵢ / √V, with constants cᵢ satisfying c₁* c₁ + c₂* c₂ − c₃* c₃ − c₄* c₄ = 0 and c₁* c₃ + c₂* c₄ − c₃* c₁ − c₄* c₂ = 0.
• The results are consistent with analogous findings in Bianchi type-I and type-VI₀ models, indicating a universal behavior under similar constraints.