[Paper Review] On the stress-energy tensor of a rotating wormhole

Santiago Esteban Perez Bergliaffa, Katrina Hibberd|ArXiv.org|Jun 12, 2000

Methane Hydrates and Related Phenomena3 citations

TL;DR

This paper derives geometric constraints on the stress-energy tensor required to generate a rotating wormhole in a stationary, axisymmetric spacetime. Using tetrad formalism and Einstein's equations, it shows that the Teo-type rotating wormhole metric cannot be sourced by either a perfect fluid or a fluid with anisotropic stresses due to violations of energy condition constraints, implying more complex matter sources are needed.

ABSTRACT

We analyze the stress-energy tensor necessary to generate a general stationary and axisymmetric spacetime. The constraints on the geometry arising from considering a perfect fluid as a source are derived. For a fluid with a nonzero stress tensor, we obtain two necessary conditions on the metric. As an example, we show that the rotating wormhole presented in the literature can not be described by either a perfect fluid or by a fluid with anisotropic stresses.

Motivation & Objective

• To determine the constraints on the stress-energy tensor for a stationary, axisymmetric spacetime that could generate a rotating wormhole.
• To investigate whether the Teo rotating wormhole metric can be sourced by a perfect fluid or a fluid with anisotropic stresses.
• To identify the necessary geometric and physical conditions on the metric coefficients for such matter sources to be viable.
• To demonstrate that the specific rotating wormhole solution proposed by Teo (2000) violates the required energy condition constraints for both perfect and anisotropic fluids.

Proposed method

• Derives the Einstein tensor components in an orthonormal tetrad basis for a general stationary and axisymmetric metric.
• Expresses the stress-energy tensor of a fluid with anisotropic stresses using the four-velocity and the traceless, transverse stress tensor Πμν.
• Imposes the constraints uμΠμν = 0 and Πμμ = 0 to preserve fluid properties under symmetry.
• Uses the tetrad components of Einstein's equations to relate Gμν to ρ, p, and Πμν, forming a system of ten equations in ten unknowns.
• Introduces the variable x = u₀/u₃ to algebraically solve the system and derive expressions for fluid components and stress tensor components.
• Applies the derived constraints (G₀₀ + G₃₃ ≥ 2G₀₃ and x² > 1) to test the viability of the Teo wormhole metric.

Experimental results

Research questions

• RQ1 Can the Teo rotating wormhole metric be generated by a perfect fluid source in a stationary, axisymmetric spacetime?
• RQ2 What geometric and physical constraints must the metric coefficients satisfy for a fluid with anisotropic stresses to act as a source?
• RQ3 Does the Teo metric satisfy the necessary energy condition constraints derived for a fluid with anisotropic stresses?
• RQ4 Are there coordinate-independent constraints on the matter content required to generate a rotating wormhole?
• RQ5 What implications do the derived constraints have for the physical realizability of rotating wormhole solutions?

Key findings

• The Teo rotating wormhole metric violates the constraint G₁₂ = 0 required for a perfect fluid source, proving it cannot be generated by such a fluid.
• The constraint G₀₀ + G₃₃ ≥ 2G₀₃ is violated for the Teo metric across all θ, ruling out a fluid with anisotropic stresses as a source.
• The condition x² > 1, required for a real and physical fluid four-velocity, is not satisfied for the Teo metric, invalidating the fluid solution.
• The system of equations for the fluid components cannot be solved with real, positive-definite energy density and pressure for the Teo metric.
• The analysis confirms that the Teo rotating wormhole cannot be sourced by either a perfect fluid or a fluid with anisotropic stresses, indicating the need for more complex matter fields such as heat flux.