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• W4319984920 abstract "Abstract Let H be a (past directed) horizon in a time-oriented Lorentz manifold and $$gamma :[left( alpha ,beta right) rightarrow H$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>:</mml:mo> <mml:mo>[</mml:mo> <mml:mfenced> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> </mml:mfenced> <mml:mo>→</mml:mo> <mml:mi>H</mml:mi> </mml:mrow> </mml:math> a past directed generator of the horizon, where $$[left( alpha ,beta right) $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>[</mml:mo> <mml:mfenced> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> </mml:mfenced> </mml:mrow> </mml:math> is $$[alpha ,beta )$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> or $$left( alpha ,beta right) $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mfenced> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> </mml:mfenced> </mml:math> . It is proved that either at every point of $$gamma left( tright) ,~tin left( alpha ,beta right) $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mfenced> <mml:mi>t</mml:mi> </mml:mfenced> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mfenced> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> </mml:mfenced> </mml:mrow> </mml:math> the differentiability order of H is the same, or there is a so-called differentiability jumping point $$gamma left( t_{0}right) ,~t_{0}in left( alpha ,beta right) $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mfenced> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mfenced> <mml:mo>,</mml:mo> <mml:mspace /> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>∈</mml:mo> <mml:mfenced> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> </mml:mfenced> </mml:mrow> </mml:math> such that H is only differentiable at every point $$gamma left( tright) ,~tin left( alpha ,t_{0}right) $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mfenced> <mml:mi>t</mml:mi> </mml:mfenced> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mfenced> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mfenced> </mml:mrow> </mml:math> but not of class $$C^{1}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> and H is exactly of class $$C^{1}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> at every point $$gamma left( tright) ,~tin left( t_{0},beta right) $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mfenced> <mml:mi>t</mml:mi> </mml:mfenced> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mfenced> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> </mml:mfenced> </mml:mrow> </mml:math> . We will use in the proof a result which shows that every mathematical horizon in the sense of P. T. Chruściel locally coincides with a Cauchy horizon." @default.
• W4319984920 created "2023-02-11" @default.
• W4319984920 creator A5065391101 @default.
• W4319984920 date "2023-02-09" @default.
• W4319984920 modified "2023-10-05" @default.
• W4319984920 title "The Differentiability of Horizons Along Their Generators" @default.
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