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Torsional four-fermion interaction and the Raychaudhuri equation

Shibendu Gupta Choudhury, Sagar Kumar Maity, Amitabha Lahiri

TL;DR

The paper investigates whether torsion arising from fermion spin in the Einstein–Cartan–Sciama–Kibble framework can produce a repulsive contribution to gravitational collapse and avert singularities. It reformulates ECSK gravity as General Relativity with an effective energy-momentum tensor $T_{\mu\nu}=T_{\mu\nu}^{free}+T_{\mu\nu}^{int}$, where $T_{\mu\nu}^{int}$ is a four-fermion term sourced by vector and axial currents with couplings $\lambda_{V}$ and $\lambda_{A}$, and analyzes collapse dynamics using the Raychaudhuri equation for timelike congruences. The authors show that at high temperature the interaction energy density scales as $\mathcal{T}^6$ and can violate the strong energy condition depending on couplings and species, offering a mechanism to hinder focusing and potentially avoid final singularities in certain cases; a single species with maximal chirality may nullify the effect, while axial-only coupling tends to be attractive. Overall, the work highlights a possible route to singularity avoidance in fermionic collapse within a GR framework augmented by torsion-induced four-fermion interactions, while noting the need for curved-background analyses and nonlinear Dirac dynamics for a definitive conclusion.

Abstract

Intrinsic spin of fermions can generate torsion in spacetime. This torsion is a non-propagating field that can be integrated out, leaving an effective non-universal four-fermion interaction. This geometrical interaction affects fermions inside a matter distribution and can be expected to become stronger as the density grows. In this work, we investigate the role of this interaction in a gravitationally collapsing fermionic distribution. Our specific aim is to explore if this interaction can provide a repulsive contribution and prevent the final singularity formation. We consider a collapsing distribution of massive fermions, ignoring other interactions. Using simplified yet reasonable assumptions, we establish that a repulsive contribution can arise depending on how torsion couples with different chiralities. Also, the interaction starts to dominate as the collapse proceeds, accelerating or decelerating the collapse depending on the relative signs of the geometrical interaction between different species of fermions.

Torsional four-fermion interaction and the Raychaudhuri equation

TL;DR

The paper investigates whether torsion arising from fermion spin in the Einstein–Cartan–Sciama–Kibble framework can produce a repulsive contribution to gravitational collapse and avert singularities. It reformulates ECSK gravity as General Relativity with an effective energy-momentum tensor , where is a four-fermion term sourced by vector and axial currents with couplings and , and analyzes collapse dynamics using the Raychaudhuri equation for timelike congruences. The authors show that at high temperature the interaction energy density scales as and can violate the strong energy condition depending on couplings and species, offering a mechanism to hinder focusing and potentially avoid final singularities in certain cases; a single species with maximal chirality may nullify the effect, while axial-only coupling tends to be attractive. Overall, the work highlights a possible route to singularity avoidance in fermionic collapse within a GR framework augmented by torsion-induced four-fermion interactions, while noting the need for curved-background analyses and nonlinear Dirac dynamics for a definitive conclusion.

Abstract

Intrinsic spin of fermions can generate torsion in spacetime. This torsion is a non-propagating field that can be integrated out, leaving an effective non-universal four-fermion interaction. This geometrical interaction affects fermions inside a matter distribution and can be expected to become stronger as the density grows. In this work, we investigate the role of this interaction in a gravitationally collapsing fermionic distribution. Our specific aim is to explore if this interaction can provide a repulsive contribution and prevent the final singularity formation. We consider a collapsing distribution of massive fermions, ignoring other interactions. Using simplified yet reasonable assumptions, we establish that a repulsive contribution can arise depending on how torsion couples with different chiralities. Also, the interaction starts to dominate as the collapse proceeds, accelerating or decelerating the collapse depending on the relative signs of the geometrical interaction between different species of fermions.
Paper Structure (10 sections, 97 equations, 1 figure)

This paper contains 10 sections, 97 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Energy-Momentum tensor of the collapsing fermionic distribution
  3. Raychaudhuri equation and Focusing
  4. Energy conditions and Fate of the focusing condition
  5. Conclusion and final remarks
  6. Derivation of the effective Energy-Momentum tensor
  7. Calculation of $\left\langle J^I J_I\right\rangle$
  8. Calculation of $\left\langle J^I_5 J_{I5}\right\rangle$
  9. Calculation of $\left\langle J^{I5}J_{I}\right\rangle$
  10. Calculation of $\left\langle T_{\mu\nu}^\mathrm{free}\right\rangle$

Figures (1)

  • Figure 1: Plot of $A$ and $B$ with $\beta$ for $m=1$.