[Paper Review] Killing-Yano symmetry for a class of spacetimes admitting parallel null 1-planes

Dumitru Bǎleanu, Sibel Başkal|ArXiv.org|Jun 17, 2002

Advanced Differential Geometry Research3 citations

TL;DR

This paper investigates Killing-Yano (KY) tensors of rank two and three in a generalized pp-wave metric admitting parallel null 1-planes, demonstrating their existence under specific curvature and functional constraints. It derives corresponding Killing tensors and establishes compatibility between geometric duality and non-generic symmetries, particularly in a subclass with zero scalar curvature and non-invertible Killing tensors.

ABSTRACT

A possible generalization of plane fronted waves with parallel rays (gpp-wave) fall into a more general class of metrics admitting parallel null 1-planes. For gpp-wave metric, the zero-curvature condition is given, the Killing-Yano tensors of order two and three are found and the corresponding Killing tensors are constructed. Henceforth, the compatibility between geometric duality and non-generic symmetries is presented.

Motivation & Objective

• To generalize the pp-wave metric to a class of spacetimes admitting parallel null 1-planes, extending beyond standard gpp-wave metrics.
• To investigate the existence of Killing-Yano tensors of rank two and three in this generalized class, particularly under zero and non-zero scalar curvature conditions.
• To construct associated Killing tensors via contraction of KY tensors and analyze their invertibility and geometric duality properties.
• To explore the compatibility between geometric duality and non-generic symmetries in spacetimes with higher-rank KY tensors.
• To determine Petrov types and pure radiation conditions for the metric using the Weyl tensor and Ricci tensor analysis.

Proposed method

• Derives the generalized pp-wave metric in the form $ ds^2 = 2dvdu + A(x,y,u)(dx^2 + dy^2) + H(v,x,y,u)du^2 $, where $ A $ and $ H $ are arbitrary functions.
• Imposes the condition of a parallel null 1-plane field via $ \nabla_\nu l_\mu = \kappa_\nu l_\mu $, with $ l_\mu $ null and non-vanishing.
• Solves the Killing-Yano equation $ f_{\nu_1\cdots\nu_n;\lambda} = 0 $ for rank-2 and rank-3 tensors, yielding explicit solutions $ f_{123} = uA(x,y,u) $, $ f_{234} = r(u)A(x,y,u) $.
• Constructs associated symmetric Killing tensors via $ K_{\mu\nu} = g^{\alpha\beta} f_{\mu\alpha} f_{\beta\nu} $ (rank-2) and $ K_{\mu\nu} = g^{\alpha\delta}g^{\beta\gamma} f_{\mu\alpha\beta} f_{\gamma\delta\nu} $ (rank-3).
• Analyzes geometric duality by testing whether the constructed Killing tensors are invertible and can generate dual metrics via $ K^{\mu\alpha}k_{\alpha\nu} = \delta^\mu_\nu $.
• Performs Petrov classification using the Weyl tensor and principal null vector $ l_\mu = \delta^\mu_1 $, identifying types N, III, II/D based on curvature and function dependence.

Experimental results

Research questions

• RQ1 Under what conditions do Killing-Yano tensors of rank two and three exist in generalized pp-wave metrics with parallel null 1-planes?
• RQ2 How do the functional forms of $ A(x,y,u) $, $ H(v,x,y,u) $, and $ r(u) $ constrain the existence of higher-rank KY tensors?
• RQ3 Are the Killing tensors derived from rank-3 KY tensors invertible, and what does this imply for geometric duality?
• RQ4 What is the Petrov type of the spacetime under different curvature and function dependence assumptions?
• RQ5 Does the pure radiation condition hold, and how is it related to the vanishing of the Ricci tensor components and energy density?

Key findings

• A subclass of the generalized pp-wave metric admits rank-2 and rank-3 Killing-Yano tensors when $ A(x,y,u) = u s(x,y) $, $ H = \frac{v}{u} + h_2(u) $, and $ r(u) $ satisfies $ 2u r(u)_{,u} + r(u) - u^2 h_2(u)_{,u} - u h_2(u) = 0 $.
• The Killing tensors derived from rank-3 KY tensors are non-invertible, indicating that geometric duality via dual metrics does not hold in the standard sense for this subclass.
• Despite non-invertibility, geometric duality and non-generic symmetries are compatible in a specific subclass, as shown by consistent solutions to the KY and Killing tensor equations.
• The spacetime is not conformally flat, as evidenced by non-vanishing Weyl tensor components.
• Petrov classification shows type N for $ R=0 $, $ h_1 $ depending only on $ u $, and $ H = v h_1(u) + h_2(x,y,u) $; type III for $ H = v h_1(x,y,u) + h_2(x,y,u) $; and type II/D for $ R \neq 0 $.
• Pure radiation is realized when $ R=0 $, $ h_1 $ is independent of $ x,y $, and the Einstein tensor takes the form $ G_{\mu\nu} = \rho(x^\sigma) l_\mu l_\nu $, indicating a null fluid source.