In 1962, Ehlers and Kundt conjectured that plane waves are the only class of complete Ricci-flat pp-waves, i.e. metrics on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathbb {R}}^4$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>4</mn> </msup> </math></EquationSource> </InlineEquation> of the form <Equation ID="Equ30"> <EquationSource Format="TEX">$ ds^2=2du\,dv+dx^2+dy^2+H(x,y,u)du^2\,. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>d</mi> <msup> <mi>s</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>2</mn> <mi>d</mi> <mi>u</mi> <mspace width="0.166667em" /> <mi>d</mi> <mi>v</mi> <mo>+</mo> <mi>d</mi> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>H</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <msup> <mi>u</mi> <mn>2</mn> </msup> <mspace width="0.166667em" /> <mo>.</mo> </mrow> </math></EquationSource> </Equation>Recently, Flores and Sánchez gave a proof of the conjecture in the fundamental case of spatially polynomially bounded profile functions H. However, impulsive pp-waves, i.e. waves with concentrated profile functions of the form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$H(x,y,u)=f(x,y)\,\delta (u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="0.166667em" /> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>, the Dirac measure) have been found to be complete for arbitrary (smooth) spatial profile functions f. 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.

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

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2026年6月3日

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Moriz L. Frauenberger,
James D. E. Grant,
Roland Steinbauer

摘要

In 1962, Ehlers and Kundt conjectured that plane waves are the only class of complete Ricci-flat pp-waves, i.e. metrics on ${\mathbb {R}}^4$ $R^{4}$ of the form $ ds^2=2du\,dv+dx^2+dy^2+H(x,y,u)du^2\,. $ $d s^{2} = 2 d u d v + d x^{2} + d y^{2} + H (x, y, u) d u^{2} .$ 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 $H(x,y,u)=f(x,y)\,\delta (u)$ $H (x, y, u) = f (x, y) δ (u)$ ( $\delta $ $δ$ , the Dirac measure) have been found to be complete for arbitrary (smooth) spatial profile functions f. 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.