Univ. of Alberta, Edmonton Mainz, MITP Workshop Quantum Vacuum and - - PowerPoint PPT Presentation

Valeri P. Frolov,

• Univ. of Alberta, Edmonton

Mainz, MITP Workshop “Quantum Vacuum and Gravitation, 22-26 June, 2015

Based on: "Spherical collapse of small masses in the ghost-free gravity V.F, A. Zelnikov, T. Netto, e-Print: arXiv:1504.00412 (2015); (to appear in JHEP) "Mass-gap for black hole formation in higher derivative and ghost free gravity", V. F. ,arXiv:1505.00492 (2015); "Information loss problem and a 'black hole` model with a closed apparent horizon", V.F., JHEP 1405 (2014) 049, arXiv:1402.5446

Outline of the talk:

1.Brief Introduction 2.Higher Derivative (HD) and Ghost Free (GH) Gravity 3.Weak Gravity: Gravitational Field of a Point Mass 4.HD and GF Gyratons 5.Null Shell Collapse in HD and GH Gravity 6.Mass Gap for Mini-BH formation 7.Strong Gravity: Models with Closed Apparent Horizon 8.New Universe Formation inside a Black Hole? 9.Summary and Discussions

(Remark: Everything in four dimensions, however 3 and 4 have been done for arbitrary D)

Black hole is a spacetime domain from where no information carrying signals can escape to infinity. The black hole boundary is an event horizon. Can we prove that an object in the center of our Galaxy is a black hole (according to this definition) ? Yes, only if you expect to live forever. This definition is very useful for proof of theorems, but certainly is not very practical.

Event horizon vs.apparent horizon

7

`Quasi-local definition’ of BH: Apparent horizon

A compact smooth surface is called a trapped surface if both, in- and out-going null surfaces,

• rthogonal to , are non-expanding .

A trapped region is a region inside . A boundary of all trapped r B B B egions is called an apparent horizon.

Trapped surface + N Null energy condition: EC =Event horizo n existence T l l

  



According to GR: Singularity exists inside a black hole. Theorems on singularities: Penrose and Hawkin Penrose theorem: Assume 1.The null energy condition holds 0;

• 2. There exists a no

g. nc T l l

  



• mpact connected Cauchy surface.
• 3. There exist a closed trapped null surface .

Then, we either have null geodesic incompleteness, or closed timelike curves.

Schwarzschild ST has a spacelike singularity. RN and Kerr ST have a timelike singularity. In both cases this is a curvature singularity.

Expectation 1: When curvature becomes high (e.g. reaches the Planckian value) the classical GR must be modified (quantum corrections, it is an emergent theory, etc.). Expectation 2: Singularities of GR would be resolved.

Regularity at r=0 and AH

2 2 2 2 2 2 2 1 1 2 2 2 2 4 2 2

( , ) , ( , ) ( , ) ( ) ( ) , ( , ) ( ) ( ) , 4( 1) [( ) 1] 4 . Apparent horizon: g=( )

• 0. If an AH crosses

0, then before this the curvature singularity dr ds F t r dt r d g t r F t r F t F t r g t r g t g t r g r r r r r                      is developed at 0. r 

2 2 2 2 2 2 2 6 2 2 2 2 2

Schwarzschild metric: / . , 1 2 . Apparent (event) horizon at 0, 2 . 48 ( ) Kretschmann scalar . Linearized version (1 2 ) (1 2 )( . ) GM r dr ds Fdt r d F F F r GM GM r ds dt dr r d                       

Three connected problems:

• 1. Regularity of potential

at 0;

• 2. Finiteness of the self-energy of a point charge;
• 3. Existence of AH: |

| . For 2, . / r CM M C F      

2

Regularization : 4 ( ) , ( ) 4 ( ) , (1 ) ( ) ( ) ( ) , (0) Pauli-Villars regularizat n io

r r reg reg

GM GM r r GMe GM r r GM e r r r r GM

 

             

 

              

2 2 2 2 2 1

(1 / ) 4 ( ) Higher-derivative theor Source- , ( ) , 1 1 1 ; (1 smearing vs non-locality: 4 , ( / ) 1 ) / . . / y

r reg r reg eg

G I G I G G G G r Me r M G



            

 

                             

Quadratic in Curvature Action

ˆ / 2 , ˆ is an operator constructed from and g. Biswas, Gerwick, Koivisto, Mazumdar (2012): The number of arbitrary functions of operator (after using the Bianchi identiti S dx g R R O R O

   

       



es) is 6. For metric perturbations over the flat background

• nly 2 arbitrary functions survive.

4

1 ( ) ( ) 2 S d x h a h h b h

           

    



1 ( ) ( ) 2 hc h hd h

  

    ( ) f h h

          

     

a b c d b c f          

IR GR limit: (0) (0) (0) (0) 1 a c b d      

2 2 2 2 1 3

• 1. General Relativity (GR):

, 1;

• 2. ( ) gravity: ( )

(0) (0) 1 2 (0) ; 1, 1 (0);

• 3. Weyl gravity:

, 1 , 1 ;

• 4. Higher derivative (HD) gravi

L R a c L R L R L L R L R … a c L L R C C a c

 

  

  

                   

2 1 2 1 2

ty: (1 ) , (1 ).

• 5. Ghost free (GF) gravity:

exp( ).

c

n i i n k k

a c a c   

   

       

 

2 2 2

(1 2 ) (1 2 2 ) ds dt d          

Static solutions of linearized gravity equations in the Newtonian limit

( ) r

  

     Stress-energy tensor:

Biswas, Gerwick, Koivisto, Mazumdar (2 Modesto, Netto, Shapiro (2014 012) )

( ) 8 ( ( ) 3 ( ))( 2 ) : 8 a G a c G                 

2 3 1 2 2 3 1 2

finite For a point mass ( ) the solution is spherically symmetric. We call it if near 0 it is of the form 1 ( ) ( ), 2 1 ( ) ( ). 2 m r r r r r O r r r r O r                  

2 2 1 2 2 2 2 1 1 1 1 1 1 2 2 1 2 2 1 1 1

A finite solution is not necessary regular one. (1), 8(4 5 3 ), 16[ (5 4 ) 4 ( )]. The solut regular ion is if 0. The solution is if

• r

0. I egular f A A R R R O r r A A

 

                           

• regu

, la = , r and is regular a c    

The gravitational collapse is regular in linearized regular HD and in GF theories of gravity. Mass gap for mini black hole production. 

1 1

ˆ ( ) , ˆ ( ) ( ) [ ( )] , ( ) ( ) , 1 ( ) ( ) 2

s i s i

O a Q a O Q ds f s e f s d Q e i

   

       

       

           

 

( ) ( ) Q f s  

• image and -image of the field equa

. tion Q f

ˆ ˆ ˆ Green function: . OG I  

1

ˆ ˆ ( ) ,

s

G O ds f s e

  

   

2 (4 )

3 2

Heat kernel: ( ) (4 )

x x s s

e < x e x > K x x s s 

     

        

( ) 8 ( ) ( ), ( )

i r i

Gm r Gm ds f s K d e r s Q ir

  

    

      

  

 

2 1 1 1 2 2 1 1 1 1 1

HD gravity: ( ) [ (1 )] , The Heaviside expansion theor General Relativity: ( ) 1 em: ( ) (1 ), (1 ( ) 2 (1 , ( ) ). . ) 2

i i

n i i n n s i i j i j n r i j i i i

Q f s P e P f r Gmr P e s r

 

       

           

              

   

( ) 2 . r Gm r    

1 1 1 1 1 2 1 2

2 Soluti The solution i

• n ne

s

• regular

a if r 0: . . 1.

n n i k i i i k i

GmS G r P mS S S P    

   

        

 

2

( ) 2 ( For the ) 2 er GF gravity ( ) ( ), The solution i f ( 2) s regula . r at 0. f s r s r Gm r r r               

[V.F., Fursa

In 4 Einstein gravity -- Aichelburg-Sexl solution. Higher dimensional generalization for a W e obtain now a solution for an ultra-r particle with spin -- gyraton metric elativistic particle. D

ev, PRD 71 104034 (2005); V.F., Israel, Zelnikov, PRD 72, 084031]

2 2 2 2 2 2 2 2 2 2 1 2

Boost transformation: , (1 ) 2, ( , , 1 ) . . ds ds dh ds dt d d dy d t v u y v u          



      

                

2 2 2 2 2 2 2 2 2 2 2 2

Penrose limit: const; li In the limit

• ne has

, , , , 2lim( m exp( (4 )) 4 ( ) ) . . dh du y u t u u ds dudv d d M m u s u h s

 

            

 

 

                  

2

2 (4 ) 2

4 ( ) ( ), ( ) ) . (

s

GMF u ds F f s e s



  

  

  

    

2

For GR, as well as for GB and ( ) gravity: ( ) ln( ), is IR cut-off parameter. L R F z z    

( ) 8 ( ) ( ), r Gm ds f s K r s  



 



2 (4 )

3 2

( ) (4 )

x x s

e K x x s s 

     

    

2 2 2 2 1 1 2 2 1 1

For HD gravity: 1 1 ( ) (ln 2 ) ( ), 4 4 1 ln 2 , ln( ) . For

• regular

( ) ln( ) 2 ( theo ) . , ry

n i n i i i i i i

F F z C S z z c Sz O z c S P S z z P K z      

   

           

 

2 3

For ghost free gravity ( ) ln Ei(1 ) 1 ( ) 4 F z z z z z O z       

Metric for Thin Null Shell

P

We consider a set of gyratons, passing through a point in Minkowski spacetime. They form a null cone with the vertex at . A section const

• f the cone is a sphere. We take a continuous

limit of this P P t  destibution, assuming that the mass density per a unit solid angle is constant / 4 . M 

n

eY eZ

Y X Z k P  k

X

e

p p

Gyraton frame vs Minkowski f rame

2 2 2 2 2 2 2

The result of averaging: 2 ( ) 2 MF r t t r t < dh > dt dr d r r 

           

            

2 2 2 2 2 2 2 2

For const the metric is flat. can be "gauged away" 0 inside t 2 For he nu ( ) ln , ll c ( )

• nes.

. M F z z ds ds F ds ds < dh > < d d h t F > dr r        

Apparent Horizon

1 1 1 2 2 2

For GR (as well as for GB and ( )-gravity) ( ) 1, so that an apparent horizon exi F 1 2 ( ) (

• r HD gravity ( )

1 sts for any value o ( ) ( ), ) . f . ) (

n i i i

GM g g r z g GMr q z q z zF z L R q z M F z q z z P K r



  

  

             



1 2 2 2 1 1 2

( ), 1 1 ( ) (ln 2 1) ( ) 4 4 ln( ) .

n i i i i i

z q z S z z c S S P z O z   

 

      



2 1 1 1 1 1

Outside the null shell | | / 1, 1 , ( ) ( ) , ( ) 0 4 1 ( ) ( ).

n i i i i i i n i i i

t r t r q z r P Z y y r q z r P Z y K y y     

   

                

 

max

( ) is positive, 0.399 at 1.114 Z y Z y  

In the "regular" HD theory and GF gravity, if 1,then there is no apparent horizo . n GM  

Curvature Invariant

2 2 2 2 2 2 6 2 2 2 2 2 2

The Kretschmann curvature vanishes on the null shells. However, in a general 48 2 2 , 1 [( 4 case 5) 2( 2) ] it is diverg , ent at 16 . 2 . ln G M R F F z zqq q q r F z w w S w SS S z c r w               

1

HD and GF theories have the mass scale parameter . "Physical" null shell should be constructed from

• flields. One can expect that the thickness of the shell

should be larger than . Let us show that      this makes curvature finite.

Metric for Thick Null Shell

R T- T+ N+ N-

• b/2

b/2 r t I v = -b/2 v = b/2 u = b/2 u = -b/2 O

2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 5 2 2

Additional averaging results in the metric in -domain: 2 1 [ ( ) ], 2 ( ), [(6 5) 3 ], ln ln 2, 9 [(30 31) 15 ]. 225 32 [(36 5 27

r k k r

I GM dr < < dh > > c dt c c r c d br r c dxx F r x r c u S S u r c r R G u M c u S S 



                  



2 2 2 2

) 36 9 ]. S uS S S  

2 2 2 4 32 2 3 2

For HD gra In both cases for small G the linear perturbarion is uniformly small and higher

• rder corr

vity is finite for a

• regular

theory, that is whe For ghost free gravity: n 0. . R G M R M S     ections can be neglected. This means that a no-apparent-horizon result is robust.

• 1

2 2

Denote = . Then we have 1 . For . If . Then one can neglect all higher in curvature corrections (weak field regime!)

cr

GM M R T R T M R R          

For regular HD and GF theories there exists the mass gap for mini black hole formation. For small mass the (time-dependent) gravitational field of the collapsing body is regular and no apparent horizon.

Limiting curvature conjecture: Markov, JETP Letters, 36 (1982) 266; Ann.Phys., 155 (1984) 333; Polchinski, Nucl.Phys. B325 (1989) 619.

2

1 || R ||   Spherically symmetric ST: An apparent horizon does not cross r=0. It is either closed or unlimited.

2/3

ST with a geometry satisfying modified gravitational equations; Limiting curvature conjecture ; Geometry differes from the classical one in the domains, where ( / ) ; Hawking radiati

S Pl S

g r r r l r



 

• n to the infinity is accompanied by negative energy

flux through the horizon, which slowly reduces the black hole's mass; Null fluid approximation for incoming and outgoing energy fluxes; This massive shell approximation for the region near the horizon, where massless quanta are created.

Main assumptions of the model

56

• T. A. Roman and P.G. Bergmann (1983);
• P. Bolashenko and V.F. (1986)
• S. N. Solodukhin, (1999);
• S. A. Hayward (2006);
• S. Ansoldi (2008);
• C. Bambi, D. Malafarina, L. Modesto (2013);
• V. N. Lukas and V. N. Strokov (2013);
• V. Frolov (2014);

J.M. Bardeen (2014);

• C. Rovelli and F. Vidotto (2014);
• T. De Lorenzo, C. Pacilio , C. Rovelli, S. Speziale (2015);

D.I. Kazakov, S.N. Solodukhin (1993)

• P. Hajicek (2002);
• D. Grumiller (2003, 2004;
• J. Ziprick and G. Kunstatter (2010)

Other publications on regular BH models with closed apparent horizons

Hawking radiation at far distance is effectively described by a properly chosen null fluid flux. The conservation law requires that this radiation is accompanied by the negative energy flux through the horizon, which we also approximate by the null

• fluid. In order to make a model consistent one needs to

assume that between the two regions with pure outgoing and pure incoming fluxes there exists a transition region, corresponding to the domain where the particle are created. We assume that this region is narrow and approximate it by a massive thin shell. Main conclusion: For slow change of the black hole the back-reaction of the shell is negligible.

Model for Particle Creation Domain

58

2 2 2 2 2 2 2 2 3 2 2

Use `Plankian scale parameter' to transform the metric into its dimensionless form , 2 , 2 ( ) 1 . 2 ( ) In the limit 0 one has 1 and the (curvature) 1 b dS b ds ds fdv dvd d v f v f                   

Apparent horizon: ( ) f    

Modified Vaidya model

[Hayward ‘06, Frolov ‘14]

v

1

v

'  

'  

'  

'  

*



*



'  

'  

* 3 3

3 3 is the minimal mass 4

• f the black hole.

Simplest model: , for 0, (1 / ), for v v v v v v             

62

Quasi-horizon

2 2 2

1 Radial out-going null rays: 2 Quasi-horizon: 2 Another definition: ( )

v

d f dv d f f f dv f



          

63

64

( ) J v



  ( ) J u



 

r  r 

1

3 2

Apparent horizon

“Through A Black Hole Into A New Universe?” V.F., Markov, Mukhanov, Phys.Lett. B216 (1989) 272; “Black Holes As Possible Sources Of Closed and Semiclosed Worlds” ,V. F., Markov, Mukhanov, IC/88/91. May 1988. Phys.Rev. D41 (1990) 383; “How many new worlds are inside a black hole?” Barrabes and V. F. Phys.Rev. D53 (1996) 3215

Smolin, The Life of the Cosmos ‘97: “A collapsing black hole causes the emergence of a new universe on the "other side", whose fundamental constant parameters (speed of light, Planck length and so forth) may differ slightly from those of the universe where the black hole collapsed. Each universe therefore gives rise to as many new universes as it has black holes.” Buonanno, Damour, Veneziano ‘99: “Gravitational instability, leading to the possible formation of many black holes” … each

• f which becomes the place of “birth of a baby Friedmann

universe after a period of dilaton-driven inflation”.

• E. Poisson, W. Israel (1990);

I.G. Dymnikova (1991);

• E. Poisson (1991).
• E. Elizalde, S. R. Hildebrandt (2000);

I.V. Artemova, I. D. Novikov (2002);

• L. Ford (2003);
• S. Conboy, K. Lake (2005);
• S. Ansoldi (2008);
• O. B. Zaslavskii (2009);
• S. Hossenfelder, L. Smolin (2010);
• S. Hossenfelder, L. Modesto, I. Premont-Schwarz (2010);
• J. P.S. Lemos, V.T. Zanchin (2011);

V.N. Lukash, E.V. Mikheeva (2013)

• A. Vilenkin, J. Zhang (2014)

“Black holes in cutoff gravity”, D. Morgan, PRD 43 (1991) 3144

"Extrema of the action are either local extrema, leading to the ordinary equations of motion of general relativity, or extrema on the boundary of field space, with at least one eigenvalue of the curva

2

ture attaining its maximum 1/ ."  "The singularities are replaced by perfectly well-behaved regions, and an infalling observer ends up in an exponentially expanding de Sitter-like core."

2 2 2 2 3 2 2 2 2 2 3

3 ( 2 , 1 , 2 . ). dS b ds f d ds fd d R O t f

   

                 

Vacuum SET for 2D Black Holes

2 2

To describes a formation of the 2D black hole in the collapse of the thin null We take the 2D metric shell at v=0 we shal in the for l put 4 ( 2D curvature is m 2 , 1 2 ( ) . ( ( ) ). M v ds fdv dvdr M v M R f M v r         

3

) . v r

; H H H

Conformal anomaly: plus boundary conditions= no in-fluxes determines . 1 Energy current: , ( ), 2 , 1 1 , . 2 2

l n n l r nn l

T cR T K T K K l K n F T r K K n T T K

              

    



        

  

(in-coming energy flux)

l

K

(out-going energy flux)

n

K

Vacuum stress-energy tensor for 2D black hole

horizon

2 (1 ) r M X   X









null shell flat ST flat ST

domain U

r



r



r 

r  r 

Two type of models: With “closed” and “open” apparent horizon. Common feature is regularity of the BH interior. Difference: Either V or U dominated energy fluxes? In V-model: Solution for information loss paradox; Extended time of the final phase; Large blue shift of out-coming particles (trans-Planckian energy); Anti-Hawking effect, etc