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

`Quasi- local definition’ of BH: Apparent horizon A compact smooth surface is called a trapped B surface if both, in- and out-going null surfaces, orthogonal to , are non-expanding . B A trapped region is a region inside . B A boundary of all trapped r egions is called an apparent horizon. 7

   Null energy condition: 0 T l l  Trapped surface + N EC =Event horizo n existence

According to GR: Singularity exists inside a black hole. Theorems on singularities: Penrose and Hawkin g. Penrose theorem: Assume    1.The null energy condition holds 0; T l l  2. There exists a no nc ompact 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 dr      2 2 2 2 ( , ) , ds F t r dt r d ( , ) g t r   2 2 ( , ) ( ) ( ) , ( , ) ( ) ( ) , F t r F t F t r g t r g t g t r 0 1 0 1 2      2 2 4( 1) [( ) 1] g r   2 0 4 .   4 2   r r   2 Apparent horizon: g=( ) 0. If an AH crosses r  0, then before this the curvature singularity is r r  developed at 0.

   Schwarzschild metric: / . GM r 2 dr         2 2 2 2 , 1 2 . ds Fdt r d F F   Apparent (event) horizon at 0, 2 . F r GM 2 48 ( ) GM  2 Kretschmann scalar . 6 r Linearized version          2 2 2 2 2 (1 2 ) (1 2 )( ) . ds dt dr r d

Three connected problems:   1. Regularity of potential at 0; r 2. Finiteness of the self-energy of a point charge;     3. Existence of AH: | | . For / 2, 0 . CM M C F Regularization : GM         4 ( ) , GM r r   r GMe           2 ( ) 4 ( ) , GM r r    r (1 ) GM e       ( ) ( ) ( ) , r r r reg r     (0) Pauli-Villars regularizat n io GM reg

        2 , ( ) , G I G I 1 1 1       ; G G G         r eg 2 2 (1 / )          2 (1 / ) 4 ( ) G M r reg Higher-derivative theor y . Source- smearing vs non-locality:                2 1 r 4 , ( 1 / ) / . G Me r reg

Quadratic in Curvature Action ˆ        / 2 , S dx g R R O R     ˆ   is an operator constructed from and g. O Biswas, Gerwick, Koivisto, Mazumdar (2012): The number of arbitrary functions of operator (after using the Bianchi identiti es) is 6. For metric perturbations over the flat background only 2 arbitrary functions survive.

  1          4 ( ) ( ) S d x h a h h b h      2   1      ( ) ( ) hc h hd h   2  ( ) f          h h                  0 0 0 a b c d b c f       IR GR limit: (0) (0) (0) (0) 1 a c b d

   1. General Relativity (GR): , 1; L R a c        L R L R L L R L R … 2 2. ( ) gravity: ( ) (0) (0) 1 2 (0) ;     1, 1 (0); a c L          2 2 3. Weyl gravity: , 1 , L R C C a      2 1 1 ; c 3  n     2 4. Higher derivative (HD) gravi ty: (1 ) , a  i 1 i  n     2 c (1 ). c  k 1 k      2 5. Ghost free (GF) gravity: exp( ). a c

Static solutions of linearized gravity equations in the Newtonian limit      0 0 ( ) r Stress-energy tensor:              2 2 2 (1 2 ) (1 2 2 ) ds dt d : Biswas, Gerwick, Koivisto, Mazumdar (2 012)       ( ) 8 a G            ( ( ) 3 ( ))( 2 ) 8 a c G Modesto, Netto, Shapiro (2014 )

   For a point mass ( ) the solution is m r spherically symmetric. We call it finite if  near 0 it is of the form r 1        2 3 ( ) ( ), r r r O r 0 1 2 2 1        2 3 ( ) ( ). r r r O r 0 1 2 2

A finite solution is not necessary regular one. A A      2 2 1 (1), R R R O  2 r r        2 2 8(4 5 3 ), A 2 1 1 1 1           16[ (5 4 ) 4 ( )]. A 1 1 2 2 1 2 2     The solut ion is regular if 0. 1 1    The solution is -r egular if 0. 1     I f , = , and -regu la r 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.

ˆ    ( ) , O a              ˆ 1 1 ( ) ( ) [ ( )] , Q a O       s ( ) ( ) , Q ds f s e 0 1     i     s ( ) ( ) f s d Q e     2 i i   ( ) ( ) Q f s -image and -image of the field equa tion . Q f

ˆ ˆ ˆ   Green function: . OG I  ˆ ˆ        1 s ( ) , G O ds f s e 0 Heat kernel: 2 (4 )      x x s e          s ( ) < x e x > K x x s   3 2 (4 ) s       ( ) 8 ( ) ( ), r Gm ds f s K r s 0    Gm  i       r ( ) d Q e     ir i

n           2 1 HD gravity: ( ) [ (1 )] , Q i  1 i The Heaviside expansion theor em: n      n         1 2 2 s ( ) (1 ), (1 ). f s P e P i    i i j i 1 j j i  1 i n          1 1 r ( ) 2 (1 ) . r Gmr P e i i  1 i        General Relativity: ( ) 1 , ( ) 2 ( ) 2 . f s r r Gm r

n     1 Soluti on ne a r 0: 1. r P i  1 i n            1 k 2 . GmS G mS S P 0 1 1 2 k i i  1 i   The solution i s -regular if 0 . S 2       2 For the GF gravity ( ) ( ), f s s         ( ) 2 ( ) 2 er f ( 2) . r r Gm r r  The solution i s regula r at 0. r