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Two new conditions for the occurrence of Krolak strong curvature ...

Two new conditions for the occurrence of Krolak strong curvature ...

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<strong>Twostrong> <strong>newstrong> <strong>conditionsstrong> <strong>forstrong> <strong>thestrong> <strong>occurrencestrong><strong>ofstrong> <strong>Krolakstrong> strong curvature singularitiesB. E. Whale, S. M. ScottCentre <strong>forstrong> Gravitational PhysicsCollege <strong>ofstrong> Physical SciencesAshley, M. J. S. L. and S. M. Scott, Curvature singularities and abstract boundary singularity <strong>thestrong>orems <strong>forstrong> space-time. Contem.Math. Vol. 337, pages 9-19.Clarke, C.J.S. and A. Królak, Conditions <strong>forstrong> <strong>thestrong> <strong>occurrencestrong> <strong>ofstrong> strong curvature singularities. J. Geom. Phys., 1985. 2(2): p. 127-143.Hawking, S.W., Comments on cosmic censorship. General Relativity and Gravitation, 1979. 10: p. 1047 – 1049.


Why more <strong>conditionsstrong>?TimeHawking, S. W., The Occurrence <strong>ofstrong> Singularities in Cosmology. III. Causality and Singularities. Royal Society <strong>ofstrong> LondonProceedings Series A, 1967. 300: p. 187-201.


Why more <strong>conditionsstrong>?TimeClarke, C.J.S. and A. Królak, Conditions <strong>forstrong> <strong>thestrong> <strong>occurrencestrong> <strong>ofstrong> strong curvature singularities. J. Geom. Phys., 1985. 2(2): p. 127-143.


Why more <strong>conditionsstrong>?The integralgeodesicdoes not converge along <strong>thestrong>The integraldiverges to infinityClarke, C.J.S. and A. Królak, Conditions <strong>forstrong> <strong>thestrong> <strong>occurrencestrong> <strong>ofstrong> strong curvature singularities. J. Geom. Phys., 1985. 2(2): p. 127-143.


Jacobi tensorsJacobi Tensor along a timelike geodesicBeem, J.K., P.E. Ehrlich, and K.L. Easley, Global Lorentzian Geometry. Vol. 202. 1996: Marcel Dekker, Inc.


Divergence from a pointGeodesic divergence atwhere,Beem, J.K., P.E. Ehrlich, and K.L. Easley, Global Lorentzian Geometry. Vol. 202. 1996: Marcel Dekker, Inc.


Illustration <strong>ofstrong> divergenceAssuming <strong>thestrong> strong energy condition


Illustration <strong>ofstrong> divergenceAssuming <strong>thestrong> strong energy condition


Illustration <strong>ofstrong> divergenceAssuming <strong>thestrong> strong energy condition


Point collapse conditionThe point collapse condition <strong>ofstrong> <strong>thestrong> singularity <strong>thestrong>oremsspecifies that <strong>forstrong> <strong>thestrong> geodesic<strong>thestrong>re existsso that


<strong>Krolakstrong> strong curvature conditionThe geodesic<strong>forstrong> allsatisfies <strong>thestrong> <strong>Krolakstrong> condition if<strong>thestrong>re existsso that


An important equationBeem, J.K., P.E. Ehrlich, and K.L. Easley, Global Lorentzian Geometry. Vol. 202. 1996: Marcel Dekker, Inc.


An important equationGivenwe know that, <strong>forstrong> anyBeem, J.K., P.E. Ehrlich, and K.L. Easley, Global Lorentzian Geometry. Vol. 202. 1996: Marcel Dekker, Inc.


New condition 1+Implicit function <strong>thestrong>orem.Beem, J.K., P.E. Ehrlich, and K.L. Easley, Global Lorentzian Geometry. Vol. 202. 1996: Marcel Dekker, Inc.


New condition 1The timelike geodesicsatisfies <strong>thestrong> <strong>Krolakstrong>condition if and only if <strong>thestrong>re exists a unique continuouslydifferentiable functionthat satisfies<strong>forstrong> alland


New condition 2+positivity <strong>ofstrong>Beem, J.K., P.E. Ehrlich, and K.L. Easley, Global Lorentzian Geometry. Vol. 202. 1996: Marcel Dekker, Inc.


New condition 2The timelike geodesiccondition if and only if <strong>forstrong> allso that,satisfies <strong>thestrong> <strong>Krolakstrong><strong>thestrong>re existsand<strong>forstrong> some


Links to singularity <strong>thestrong>oremsFor singularity <strong>thestrong>orems with point collapse <strong>conditionsstrong> we cansee that,)?


What else is needed?•Can <strong>thestrong> <strong>conditionsstrong> be given <strong>forstrong> null geodesics?•Can <strong>thestrong> <strong>conditionsstrong> be given <strong>forstrong> surface collapse?•How doesbehave in <strong>thestrong> limit?•Lack <strong>ofstrong> metric extension equivalent to certain behaviour?


Thank you•Ashley, M. J. S. L. and S. M. Scott, Curvature singularities andabstract boundary singularity <strong>thestrong>orems <strong>forstrong> space-time. Contem.Math. Vol. 337, pages 9-19.•Beem, J.K., P.E. Ehrlich, and K.L. Easley, Global LorentzianGeometry. Vol. 202. 1996: Marcel Dekker, Inc.•Clarke, C.J.S. and A. Królak, Conditions <strong>forstrong> <strong>thestrong> <strong>occurrencestrong> <strong>ofstrong>strong curvature singularities. J. Geom. Phys., 1985. 2(2): p. 127-143.•Hawking, S.W., Comments on cosmic censorship. GeneralRelativity and Gravitation, 1979. 10: p. 1047 – 1049.•Hawking, S. W., The Occurrence <strong>ofstrong> Singularities in Cosmology. III.Causality and Singularities. Royal Society <strong>ofstrong> London ProceedingsSeries A, 1967. 300: p. 187-201.


The difference <strong>ofstrong> divergencesLet with <strong>thestrong>n may bedefined aswhereis <strong>thestrong> Jacobi tensor with initial <strong>conditionsstrong>


The all important resultGivenwe know that, <strong>forstrong> anyThis gives us that,Beem, J.K., P.E. Ehrlich, and K.L. Easley, Global Lorentzian Geometry. Vol. 202. 1996: Marcel Dekker, Inc.


Links to singularity <strong>thestrong>oremsFor singularity <strong>thestrong>orems with point collapse <strong>conditionsstrong> we cansee that,•<strong>thestrong>re exists so that satisfies <strong>thestrong>differential equation,•<strong>thestrong>re exists so that <strong>forstrong> all <strong>thestrong>re existsso thatand<strong>forstrong> some

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