Valeri P. Frolov, Univ. of Alberta, Edmonton GC2018, Yukawa - PowerPoint PPT Presentation

Valeri P. Frolov, Univ. of Alberta, Edmonton GC2018, Yukawa Institute, Kyoto, February 16, 2018

Based on: " Principal Killing strings in higher-dimensional Kerr-NUT-(A)dS spacetimes", Jens Boos and V.F, e-Print: arXiv:1801.00122 (2018); "Stationary black holes with st ringy hair", Jens Boos and V.F., e-Print: arXiv:1711.06357 (2017), (to appear in PRD); "Stationary strings and branes in the higher- dimensional Kerr-NUT-(A)dS spacetimes", David Kubiznak and V.F., JHEP 0802 (2008) 007; e-Print: arXiv:0711.2300.

"Black holes, hidden symmetries, and complete integrability", V.F., Pavel Krtous and David Kubiznak, Living Rev.Rel. 20 (2017) no.1, 6; e-Print: arXiv:1705.05482.

Solving stationary string equations in the Kerr-NUT-(A)dS background

Killing-Yano object s Conformal KY tensor (CKY) of rank in dims: p D 1 1            X ( ) X ( ) 0 ,    X p 1 D p 1     Killing tensor (KYT): 0;      Closed conformal KY tensor (C CKY) : d 0 . 1 1             ( ) X ( ) X ( ),    X p 1 D p 1 * *   p D p *

Properties: •  (KYT)=CCKY •  (CCKY)=KYT •  CCKY CCKY=CCKY

Principal ten so r = non-deg enera te rank 2 CCKY tenso r    D 2 n 1           b h g g h  c ab ca b cb c a ba D 1  n 1         h r l l x e ˆ e      1  n 1              0 0 g l l l l ( e e ) ˆ ˆ ˆ ˆ e e e e       1                0 0 ( l l ) 1 ( e e ) ( ) ( ) 1 ˆ ˆ ˆ ˆ e e e e         0 Darboux basis: ( l l e ). ˆ ˆ e e   Non-degenerate: There are exactly non-vanishing n "eigenvalues" ( r x  , ) that are functi onally independent in some domain. In this domain none f o the gradients of them is a null vector.

A metric which admits a principal tensor is off-shell Kerr-NUT-(A)dS metric. It contains arbitr n ary functi ons of 1 var iable.  On-shell: Einstein equations are satisf ied Kerr- NUT-(A)dS so lution.

•    is a primary Killing vector: L g L h 0;   •    ( ) j j 1 h h is a CCKY 2 j form;  j •  ( ) j ( ) j f h is a KY (D-2j) form  •  ( ) j bc c ab ( ) j a   k 1 f f is a rank 2 Killing tensor; 1 D 2 j 1     ( ) j c c ( D 2 j 1)   1 D 2 j 1 •            k , ( j 0 … n 1 m ) are commuting ( ) j ( ) j (secondary) Killing vectors; •  k g ; (0) •      Frobenius theorem: = , ;   ( ) j j •   ( , r x , , ) are canonical coordinates;  j • are principal null d l irections; their integral lines  are geodesics.

Geodesic equations are completely integrable: There exist integrals of motion for a free particle, D    ( n ) first ord er p and second n or der p k p . ( ) j ( ) j Q: If instead of a particle one has a string: Are Nambu-Goto string equations completely integrable in the Kerr-NUT-( A)dS geometr y? A: In a general case - No. If a string is stat ionary - Yes.

Stationary strings in Kerr-NUT-(A)dS    i Killing vector , coordin ates - ( , t y ) t          2 a b i g p , F , A ,   ab ab i 2 2      2 i 2 i j ds F dt ( Ady ) p d d y y i ij Nambu-Goto action for a string   i j dy dy         I tE E Fdl d F p .   ij d d    i String configuration ( ) is a geodesic in ( y D 1)  dimensional space with metric p Fp . ij ij

If metric g admits the principal tensor it has ab   n Killing vectors and rank 2 Killing tensors. n    This gives D 2 n integrals of motion for a free parti cle. The reduced ref-shifted metric p does not admit ij  a principal tensor, However, when is a primary    Killing vector, it has n 1 Killing vectors and n     rank 2 Killing tensors. This gives D 2 n 1 integrals of motion. Thus, the stationary string equa t io ns a re compl etely integrable . [ V .F . an d David Kubiznak (20 08)]

Principal Killing Strings   l is a tangent vector to a principal   null geodesic in the affine parametrization:     0, [ , ] 0.    Frobenius theorem implies that a ST is foliated   a A i by 2-D (Killing) s urfaces . Coordinates Y ( z , y ).   i Equations of a given are y const .     A z ( v ) are coordinates on , such that           v 

The Killing surface in the off -shell Kerr -NUT -(A)dS metric is minimal.         2 A B 2 2 Induced 2-D metric: d d z d z d 2 d v d .   AB  n are (D-2) m utually orthogonal unit vectors normal ( i )  to . The extrin s c i curvature is:        a c b a b g n Y Y , g n Z 0.   ( ) i AB ab ( ) i A c B ( ) i ab ( ) i                b AB c b a b a b 2 a b Z Y Y .         A c B  a a a          b a b b a b Z 2 F .     a a 1       a b a a 2 F , ( )       b a 2

Incoming principal Killing string:         ( ) j r , P ( ) r x const,  j n    2( n 1 j ) r dr   ( ) j P n X r ( ) n           ˆ ( r x ) ( v r x )   k k  2( n 1) m r        ˆ d d v a d d r j j X  j 1 n 1         1 2( n 1 j ) ˆ d a d r d r j j j X n String equation in the null incoming coordinates        0 0 ˆ ˆ const x x const.   j j

String's stress-energy tensor in in-coming coordinates          ab ( a b ) 2 a b s T 2 q      g        ˆ ˆ 0 0 q q x ( x )   j j  n 1 m          ˆ ˆ 0 0 ( x x ) ( )   j j    1 j 1

Applications to Myers-Perry ST  D 3        a b 16 M d S  D 2 ab B        a b 16 J d S ( ) i ( ) i ab B 2  J Ma .  i ( ) i D 2           a b a b E T d , J T d b a i b i a   D 1 D 1 H H   2  0      J M 0 a s i i ( ) i 2 M 1    0 J J exp( v v / ), v .    ( ) i ( ) i i i 0 2 D 2 ( ) s i

 J Projection( ) τ   ˆ ˆ i i e and are unit vectors along and ; e x y and are dual unit i i i i m m 2 Ma           ˆ δ i i 0 0 i forms. J ; x 0 a e ,  i i i i D 2   i 1 i 1  m          ˆ ˆ 0 0 0 s F [ x e y e (1 ) z e ]    i i i i  z r  i 1 m m                     ˆ τ δ 0 0 i j 0 0 0 i 0 F [ a a 0 (1 ) z ] s i i i i i    i j 1 i 1

n   n 1 2 string segments 2 "infinite captured strings"