GIULIO MAGLI DIPARTIMENTO DI MATEMATICA. The final state of spherically symmetric perfect fluids' collapse: new insights.

Transcription

1 GIULIO MAGLI DIPARTIMENTO DI MATEMATICA The final state of spherically symmetric perfect fluids' collapse: new insights.

2 Dedicated to the memory of our friend Mauro Francaviglia Austin - Center for Relativity-1978

3 The final fate of gravitational collapse - M< 1.4 M White dwarfs (electron degeneracy pressure) - M< 3 M Neutron stars (neutron degeneracy pressure) - No stable state exists for greater M - A singularity forms at the endstate of continued gravitational collapse - The nature of such a singularity is a priori unknown

4 We are thus presented with what is perhaps the most fundamental unanswered question of generalrelativistic collapse theory, namely: does there exist a cosmic censor who forbids the appearance of naked singularities, clothing each one in an absolute event horizon?... If in fact naked singularities do arise, then there is a whole new realm opened up for wild speculations! BLACKHOLE NAKED SINGULARITY R. Penrose, (1969)

5 THE MATHEMATICAL FRAMEWORK FOR SPHERICAL COLLAPSE 1- THE METRIC 2- THE MATTER SOURCE 3- THE EQUATION OF STATE

6 THE FIELD EQUATIONS CAN BE WRITTEN AS - REGULAR INITIAL DATA: INITIAL DISTRIBUTIONS OF MASS AND VELOCITY - MATCHING CONDITIONS: WITH SCHWARZSCHILD VACUUM, SATISFIED AT ANY COMOVING TIME - WEAK ENERGY CONDITION: SATISFIED AT ANY COMOVING TIME

7 THE NATURE OF SINGULARITIES - SHELL CROSSING ( WEAK? ) - SHELL FOCUSSING SINGULARITY CURVE: APPARENT HORIZON CURVE: A NECESSARY CONDITION FOR NAKEDNESS IS THAT THE SHELL DOES NOT BECOME TRAPPED BEFORE BECOMING SINGULAR:

8 THE FIRST EXAMPLES OF NS: THE DUST SOLUTIONS THE ENDSTATE OF THE COLLAPSE CAN BE STUDIED DIRECTLY USING THE EXACT SOLUTION AND DEPENDS ON THE EXPANSION OF THE INITIAL DENSITY PROFILE - IF n=1 OR n=2 THE SINGULARITY IS NAKED - IF n>3 THE SINGULARITY IS COVERED - IF n=3 A TRANSITION OCCURS AT A SORT OF GOLDEN NUMBER OF SPHERICAL COLLAPSE Christodoulou, CMP 93, 1984, Dwivedi & Joshi CMP 166 (1994); Singh & Joshi CQG 13 (1996)

9 A SINGULARITY IS VISIBLE IF THERE ARE OUTGOING NULL GEODESICS MEETING THE SINGULARITY IN THE PAST FEW RESULTS OF EXISTENCE ARE KNOWN FOR CAUCHY PROBLEMS AT SINGULAR POINTS

10 Giambo', Giannoni, Magli, Piccione CQG 20 (2003) Giambo', Giannoni, Magli, Piccione Comm. Math. Phys. 235 (2003)

11 SO, THE PROBLEM IS HOW TO USE THESE TECNIQUES WITHOUT KNOWING THE SOLUTION TO INVESTIGATE THE ROLE OF PRESSURE WE OBSERVE THAT THE DUST SOLUTIONS CAN BE WRITTEN AS FOLLOWS WHERE

12 SO THAT IF THE DATA k(r) ARE TAYLOR EXPANDABLE, THE SOLUTION IS TAYLOR-EXPANDABLE WITH RESPECT TO r. THIS LEADS US TO THE FOLLOWING Giambo', Magli, CQG 31 (2014)

13 THE EFE TO LOWEST ORDER THE MATTER EQUATIONS TO LOWEST ORDER

14 THE CASE OF LINEAR EQUATIONS OF STATE

15 Sketch of the proof: for r-exp. solutions the EFE imply A special role is thus played by 1+3β: - If 1+3β<0, the apparent horizon does not form: the singularity is naked - If 1+3β>0, the behaviour of the geodesics must be studied, and it turns out that no escaping light ray exists: a blackhole forms

16 REMARK: IF WE REQUIRE ALSO THE STRONG ENERGY CONDITION THEN : ONLY EOS WITH 1+3β >0 ARE ALLOWED THIS BECOMES A COSMIC CENSORSHIP THEOREM FOR r- exp PERFECT FLUIDS

17 OF COURSE FOR LINEAR EOS THE PRESSURE DIVERGES WHENEVER THE DENSITY DOES. QUALITATIVELY, WE EXPECT THAT SEC ENFORCES BLACKHOLE FORMATION ALSO WITH GENERAL EOS HAVING THIS PROPERTY THE SITUATION CAN, HOWEVER, CHANGE COMPLETELY IF THE PRESSURE REMAINS BOUNDED AT THE SINGULARITY

18 CONCLUSIONS - A CONDITIONED COSMIC CENSORSHIP THEOREM HOLDS FOR A CLASS OF PERFECT FLUID SOLUTIONS SATISFYING A REGULARITY REQUIREMENT - THE CLASS IS OF COURSE NOT EMPTY BUT IT IS DIFFICULT TO ASSESS ITS DIMENSION - THE ROLE OF A DIVERGING PRESSURE TO ASSURE BLACKHOLE FORMATION IS SINGLED OUT AND DESERVES FUTURE INVESTIGATIONS