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

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

"

Based on:

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

Conformal KY tensor (CKY) of rank in dims: 1 1 Killing tensor (KYT): ( ) ( ) , 1 1 Closed conformal KY tensor (C 0; CKY) : ( ) .

Killing-Yano object s

X X

p D X X p D p d                          

* * *

1 1 ( ) ( ), 1 1 X X p D p p D p              

(KYT)=CCKY (CCKY)=KYT CCKY CCKY=CCKY

Properties:

  

1 1 1 1

2 1 1 ˆ ( ) ˆ ˆ ˆ ˆ ( ) 1 ( ) (

Principal rank 2 CCKY tenso non-deg ten enera so te r r =

b c ab ca b cb c a ba n n

h g D n g h D r x

          

    

           

                           

 

h l l e e g l l l l e e e e e e l l e e Non-degenerate: There are exactly non-vanishing "eigenvalues" ( ) ( ) 1 ˆ ˆ ˆ ˆ Darboux basis: ( , ) that are functi ). ˆ

• nally independent

in some domain. In this domain none f ˆ

• n

r x

     

         e e e e l l e e e the gradients

• f them is a null vector.

A metric which admits a principal tensor is off-shell Kerr-NUT-(A)dS metric. It contains arbitr On-shell: Einstein equations are satisf ary functi ied Kerr-

• ns of 1 var

NUT-(A)dS so iable. lution. n 

 



   

      

• 



• 



• 
• 
• 1

2 1 1 2 1

( ) 1 ( ) ( ) ( ) ( ) 1 ( ) ( 2 1)

is a primary Killing vector: 0; is a CCKY 2 form; is a KY (D-2j) form is a rank 2 Killing tensor;

D j D j

j j j j j j bc c ab j a j c c D j

L L j k f f g h h h f h

  

      



       

• 
• 

 

• ( )

( ) (0) ( )

, ( 1 ) are commuting (secondary) Killing vectors; ; Frobenius theorem: = , ; ( , , , ) are canonical coordinates; are principal null d

j

j j j j

j … n m g r x l k k irections; their integral lines are geodesics.

  

( ) ( )

Geodesic equations are completely integrable: There exist integrals of motion for a free particle, ( Q: If ) first ord instead of er and second a particle one

• r

has a string: der p .

j j

D n p n p k Are Nambu-Goto string equations completely integrable in the Kerr-NUT-( A: In a general case - No. If a string is stat A)dS geometr ionary - y? Yes.

                           

 

2 2 2 2 2

Killing vector , coordin Nambu-Goto action for a ates - ( , ) , , string , ( )

Stationary strings in Kerr-NUT-(A)dS

i t a b i ab ab i i ij i i j i ij

t y dy I tE E Fdl d g p F A ds F dt Ady p y F p d d d y      . String configuration ( ) is a geodesic in ( 1) dimensional space with metric .

j i ij ij

dy d y D p Fp

     If metric admits the principal tensor it has Killing vectors and rank 2 Killing tensors. This gives 2 integrals of motion for a free parti The reduced ref-shifted metric does not cle.

ab ij

g p n n D n         admit a principal tensor, However, when is a primary Killing vector, it has 1 Killing vectors and rank 2 Killing tensors. This gives 2 1 integrals of motion. Thus, the stationary string equa n n D n [ t V io .F ns a . an re compl d David etely integrable Kubiznak (20 . 08)]

is a tangent vector to a principal null geodesic in the affine parametrization: 0, [ , ] Frobenius theorem implies that a ST is foliated by 2-D (Killing) 0. s

Principal Killing Strings

l  



   

    urfaces . Coordinates ( , ). Equations of a given are . ( ) are coordinates on , such that

a A i i A v

Y z y y const z v



 



 

            



    

  

   

( 2 ) 2 2

are (D-2) m Induced 2-D utually orthogonal metric: unit vectors normal to . The extrin d d d 2 s c d . i d

A B AB i

d z z v n

The Killing surface in the off -shell Kerr -NUT -(A)dS metric is minimal.

        

                       

                       

( ) ( ) ( ) ( ) 2 2

curvature . 2 . 1 , ( ) 2 is: , 0.

b AB c b a b a c b a b i AB ab i A c B i ab i a b a b A c B a a a b a b b a b a a a b a a b a

Z Y Y Z F F g n Y Y g n Z

String equation in the null incoming coordinates ˆ ˆ const const.

j j

x x

 

      

( ) 2( 1 ) ( )

Incoming principal Killing string: , ( ) const, ( )

j j n n j j n n

r P r x r dr P X r



 

 

       



2( 1) 1 1 2( 1 )

ˆ ( ) ( ) ˆ d d d d 1 ˆ d d d

k k n m j j j n n j j j j n

r x v r x r v a r X a r r X

 

      

    

             



( ) 2 1 1 1

String's stress-energy tensor in in-coming coordinates 2 ˆ ˆ ( ) ˆ ˆ ( ) ( )

ab a b a b s j j n m j j j

T q g q q x x x x

    

        

        

            

 

 





   

2 ( ) s i i

M a i

J

Applications to Myers-Perry ST

( ) ( ) 0 2

2 1 exp( / ), . 2 ( )

i i i i s i

M J J v v v D      

               

 

( ) ( ) ( )

3 16 d 2 16 d 2 . 2

a b ab B a b ab i i B i i

D M S D J S J Ma D

1 1

d , d

D D

a b a b b a i b i a H H

E T J T  

 

     

 

1 1 1 1 1

ˆ ˆ and are unit vectors along and ; and are dual unit 2 ˆ forms. , ˆ ˆ [ (1 ) ] ; 2 [ ˆ

m i i i i i m i i i i i i m i i m m i j s i i i i j i i i s i i i i z i

x a e x e y e z e e x y Ma D a e r            

           

               

    

τ F δ J δ F

0 (1

) ]

i i i

a z        

Projection( )  J τ

1

2 string segments 2 "infinite captured strings"

n n

