[Paper Review] 2+1 dimensional solution of Einstein Cartan equations

M. Hortaçsu, H. T. Özçelik+1 more|ArXiv.org|Jul 28, 2008

Advanced Differential Geometry Research1 references3 citations

TL;DR

This paper presents an exact static solution to the 2+1 dimensional Einstein-Cartan equations by treating a classical spin-1/2 field as a non-dynamical source for spacetime torsion. The solution features a metric with non-zero cosmological constant, where torsion couples to the spinor field, and the Dirac equation is solved consistently in the curved, torsionful background, yielding a complete solution to both the Einstein-Cartan and Dirac equations with a specific logarithmic dependence on radial coordinate.

ABSTRACT

In this work a static solution of Einstein-Cartan (EC) equations in 2+1 dimensional space-time is given by considering classical spin-1/2 field as external source for torsion of the space-time. Here, the torsion tensor is obtained from metricity condition for the connection and the static spinor field is determined as the solution of Dirac equation in 2+1 spacetime with non-zero cosmological constant and torsion. The torsion itself is considered as a non-dynamical field.

Motivation & Objective

• To construct an exact solution of the Einstein-Cartan equations in 2+1 dimensions with non-zero cosmological constant.
• To investigate the role of a classical, static, radial spin-1/2 field as a source for spacetime torsion.
• To solve the Dirac equation consistently in a curved, torsionful spacetime background.
• To determine whether the torsion tensor restricts the spinor field to a specific type (e.g., Dirac vs. Majorana).
• To verify that the full system of Einstein-Cartan and Dirac equations is satisfied simultaneously.

Proposed method

• The spacetime is assumed static and spherically symmetric, with a metric ansatz in 2+1 dimensions including a non-zero cosmological constant.
• Torsion is introduced via the contorsion tensor derived from the spinor field's axial current, treating torsion as non-dynamical.
• The Dirac equation is solved in the presence of the spin connection and metric, using a specific spinor ansatz with radial and angular dependence.
• The energy-momentum tensor from the spinor field is used as the source in the Einstein equations, while the torsion source term comes from the spinor's spin density.
• The full system is solved by matching the metric functions and spinor components to satisfy both the Einstein-Cartan and Dirac equations simultaneously.
• The solution is derived under the constraint that the cosmological constant must satisfy $ c > ilde{c} $, with $ 4m(c - m) > c^2 - \Lambda $ for real logarithmic solutions.

Experimental results

Research questions

• RQ1 Can a static, 2+1 dimensional spacetime with non-zero torsion be constructed using a classical spin-1/2 field as the torsion source?
• RQ2 How does the presence of torsion affect the solution of the Dirac equation in a curved spacetime background?
• RQ3 Does the torsion tensor force the spinor field to be of Dirac type rather than Majorana type in this configuration?
• RQ4 What is the explicit form of the metric and spinor field when both Einstein-Cartan and Dirac equations are solved consistently?
• RQ5 How does the cosmological constant influence the existence and structure of the solution?

Key findings

• The solution yields a metric of the form $ ds^2 = -r^2 dt^2 + \frac{1}{(c^2 - \Lambda)r^2} dr^2 + r^2 d\phi^2 $, with non-zero scalar curvature $ \tilde{R} = 6\Lambda $.
• The non-zero curvature components are $ \tilde{R}_{tt} = -2r^2\Lambda $, $ \tilde{R}_{rr} = \frac{2\Lambda}{r^2(c^2 - \Lambda)} $, and $ \tilde{R}_{\phi\phi} = 2r^2\Lambda $, confirming non-trivial geometry.
• The spinor field components $ \psi_1, \psi_2, \bar{\psi}_2 $ are expressed as logarithmic functions of $ r $, with coefficients depending on mass $ m $, cosmological constant $ \Lambda $, and parameter $ c $.
• The solution requires the constraint $ 4m(c - m) > c^2 - \Lambda $ to ensure real-valued logarithmic arguments in the trigonometric functions.
• The Laplace operator in the torsionful spacetime reduces to the Riemannian form because the contorsion tensor vanishes for this solution.
• The spinor field is uniquely determined to be of Dirac type; Majorana-type fields do not satisfy the coupled system of equations.