[Paper Review] Purely magnetic spacetimes

Alan Barnes|ArXiv.org|Jan 15, 2004

Cosmology and Gravitation Theories5 references3 citations

TL;DR

This paper investigates the existence of purely magnetic vacuum spacetimes—where the electric part of the Weyl tensor vanishes—using the Newman-Penrose formalism with a canonical null tetrad. It proves that constant argument spacetimes of Petrov type II and D cannot exist in vacuum, extending prior results and supporting the broader conjecture that purely magnetic vacuum spacetimes do not exist, even with a cosmological constant.

ABSTRACT

Spacetimes in which the electric part of the Weyl tensor vanishes (relative to some timelike unit vector field) are said to be purely magnetic. Examples of purely magnetic spacetimes are known and are relatively easy to construct, if no restrictions are placed on the energy-momentum tensor. However it has long been conjectured that purely magnetic vacuum spacetimes (with or without a cosmological constant) do not exist. The history of this conjecture is reviewed and some advances made in the last year are described briefly. A generalisation of this conjecture first suggested for type D vacuum spacetimes by Ferrando and Saez is stated and proved in a number of special cases. Finally an approach to a general proof of the conjecture is described using the Newman-Penrose formalism based on a canonical null tetrad of the Weyl tensor.

Motivation & Objective

• To investigate the longstanding conjecture that purely magnetic vacuum spacetimes do not exist, particularly in the absence of a cosmological constant.
• To generalize previous results on purely magnetic spacetimes by extending them to the case of constant argument fields, where the eigenvalues of the complex Weyl tensor are proportional with fixed phase.
• To provide a unified and more direct proof of non-existence for constant argument type II and D vacuum spacetimes using the Newman-Penrose formalism.
• To explore the integrability of the resulting algebraic and differential constraints in the type I case, aiming to either confirm the non-existence of such spacetimes or find counterexamples.

Proposed method

• The Newman-Penrose formalism is employed with a canonical null tetrad adapted to the Weyl tensor's principal null directions, enabling a systematic analysis of the Weyl spin coefficients.
• The complex Weyl tensor components are decomposed into real and imaginary parts via the eigenvalues of $ Q_{ab} = E_{ab} + iH_{ab} $, with the constant argument condition implying all eigenvalues share the same phase.
• Vacuum Bianchi identities are applied in the NP formalism, reducing to a system of differential equations involving spin coefficients $ ho, au, u, ho, ar{ ho}, ext{etc.} $, which are then analyzed under the constant argument assumption.
• Algebraic constraints are derived from the vanishing of $ heta_1 = heta_3 = 0 $ and the proportionality of $ \Psi_0, \Psi_4, \Psi_2 $, leading to conditions like $ \kappa = \sigma = \tau + \bar{\pi} = 0 $.
• Commutator identities and Ricci identities are used to derive a contradiction unless $ A = 0 $ or $ B = 0, \pi $, thus ruling out non-trivial constant argument fields.
• The method is extended to type I fields, where the system of algebraic and differential equations is analyzed for integrability, though a full solution remains open.

Experimental results

Research questions

• RQ1 Do purely magnetic vacuum spacetimes exist in general relativity, even with a non-zero cosmological constant?
• RQ2 Can constant argument spacetimes of Petrov type II or D exist in vacuum, given that they generalize purely magnetic and purely electric cases?
• RQ3 What constraints do the Newman-Penrose equations impose on the spin coefficients when the Weyl tensor components have constant argument?
• RQ4 Is the non-existence of such spacetimes provable via the NP formalism without assuming additional symmetries like shear-free or geodesic flow?
• RQ5 Can the algebraic and differential constraints in the type I case be integrated to either confirm non-existence or construct counterexamples?

Key findings

• The paper proves that vacuum spacetimes of Petrov type II with constant argument (i.e., all eigenvalues of $ Q_{ab} $ have the same phase) cannot exist, as the Bianchi identities lead to a contradiction unless the amplitude $ A = 0 $ or the phase $ B = 0, \pi $.
• The result extends Hall’s earlier proof for purely magnetic type D spacetimes to the broader class of constant argument fields, providing a more direct and general argument.
• For type D and II fields, the conditions $ \kappa = \sigma = \tau + \bar{\pi} = 0 $, $ \rho = \bar{\rho} $, and $ \mu = \bar{\mu} $ are derived, which are inconsistent with non-trivial constant argument solutions.
• The proof is valid even when the cosmological constant $ \Lambda \neq 0 $, showing the non-existence result is robust under such modifications.
• In the type I case, the system reduces to a set of algebraic and differential equations in the spin coefficients, which are currently under investigation for integrability.
• The work supports the broader conjecture that purely magnetic vacuum spacetimes do not exist, and provides a framework for a general proof using the NP formalism.