The failure of the Ehlers--Kundt conjecture in the impulsive case

2025/09/29 by Frauenberger, Moriz L., Grant, James D. E., Steinbauer, Roland
#34A36 #46F30 #83C10 #83C15 #83C35 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2509.24989

Abstract

In 1962, Ehlers and Kundt conjectured that plane waves are the only class of complete Ricci-flat~pp-waves, i.e. metrics on \mathbb R⁴ of the form ds²=2du dv+dx²+dy²+H(x,y,u)du² . Recently, Flores and Sánchez gave a proof of the conjecture in the fundamental case of spatially polynomially bounded profile functions H. However, impulsive pp-waves, i.e., waves with concentrated profile functions of the form H(x,y,u)=f(x,y) δ(u) (δ, the Dirac measure) have been found to be complete. We summarise completeness results for several classes of impulsive wave spacetimes achieved during the last years and discuss them in the context of the Ehlers--Kundt conjecture.

Citations

• The essential regularity of singular connections in geometry
• On the Optimal Regularity Implied by the Assumptions of Geometry I: Connections on Tangent Bundles
• The use of generalized functions and distributions in general relativity
• Causality and Conjugate Points in General Plane Waves
• Generalized pseudo-Riemannian geometry
• Foundations of a nonlinear distributional geometry
• On the geometry of impulsive gravitational waves

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