We study the ray transform L over null (light) rays in the pseudo-Euclidean space with signature <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2086_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$(n',n'')$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>n</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2086_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$n'\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>n</mi> <mo>′</mo> </msup> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2086_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$n''\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>n</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We analyze the normal operator <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2086_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$L'L$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mo>′</mo> </msup> <mi>L</mi> </mrow> </math></EquationSource> </InlineEquation>, derive an inversion formula, and prove stability estimates. We show that the symbol <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2086_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$p(\xi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is elliptic but singular at the light cone <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2086_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation> with conormal singularities there. We analyze L as a Fourier Integral Operator as well. Finally, we compare this to the Minkowski case.

The Light Ray Transform for Pseudo-Euclidean Metrics

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2025年7月10日

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Divyansh Agrawal,
Plamen Stefanov

摘要

We study the ray transform L over null (light) rays in the pseudo-Euclidean space with signature $(n',n'')$ $(n^{'}, n^{''})$ , $n'\ge 2$ $n^{'} \geq 2$ , $n''\ge 2$ $n^{''} \geq 2$ . We analyze the normal operator $L'L$ $L^{'} L$ , derive an inversion formula, and prove stability estimates. We show that the symbol $p(\xi )$ $p (ξ)$ is elliptic but singular at the light cone $\mathcal {L}$ $L$ with conormal singularities there. We analyze L as a Fourier Integral Operator as well. Finally, we compare this to the Minkowski case.