Dilatonic black holes in heterotic string theory: perturbations and stability

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1 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 1/2 Dilatonic black holes in heteotic sting theoy: petubations and stability Filipe Moua Cento de Matemática, Univesidade do Minho, Baga, Potugal axiv: [hep-th]

2 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 2/2 The Tanghelini black hole Metic of the type f) = g) = ds 2 = f)dt 2 + g 1 )d d Ω 2 d 2 ; 1 R H ) d 3 ); d = 4: Schwazschild solution.

3 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 3/2 Leading α coections Effective action in the Einstein fame 1 16πG [ g R 4 ] d 2 µ φ) µ φ + e 2 d 4 φ λ 2 Rµνρσ R µνρσ d d x, λ = α 2, α 4 bosonic, heteotic). Field equations R µν + λe 4 2 d φ R µρστ R ρστ ν ) 1 2d 2) g µνr ρσλτ R ρσλτ = 0; 2 φ λ 4 e 4 2 d φ R ρσλτ R ρσλτ) = 0.

4 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 4/2 α -coections to the dilaton Dilaton field equation in the backgound of a Tanghelini black hole: Fist integation: d 2 R d 3 H )φ ) = λ d 2)2 d 3)d 1) 4 ) d 2 d 3 H φ = λ d 2)2 d 3) 4 2d 6 H Fo each d it is always possible to choose d 2)2 Σ = 4 λ d 5 H, such that φ is egula at the hoizon. R 2d 3) H d d 1 d 3)Σ

5 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 5/2 α -coected dilaton solution Solution: φ) λ = 2 d 1 d 3 + d 3)d 2)2 8d 1) 2 d 2)2 4R 2 H R H ln ) 2 B 1 [ d 1) + 2 RH ) d 3 ; ) ) d 3 RH RH ) d 3 )] 2 d 3, 0 with Bx; a,b) = x 0 t a 1 1 t) b 1 dt.

6 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 6/2 Asymptotic expansion Close to infinity: φ) Σ Σd 3 H + d 3 2 2d 6 + λ 8 H d 2)d 3)2d 6 2d 4. Solution of Boulwae and Dese 1986), but: φ) is of ode λ; seconday hai.

7 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 7/2 The Callan-Myes-Pey black hole The only fee paamete is the hoizon adius R H seconday hai), which is not changed; f) = g) = 1 R H ) d 3 ) [ 1 λ d 3)d 4) 2 R d 5 H d 1 R d 1 d 1 H d 3 R d 3 α = 0: Schwazschild-Tanghelini solution; α -coected ADM black hole mass: ) d 3)d 4) λ d 2)Ad 2 m = κ 2 R 2 H H ] ; R d 3 H dilaton vanishes classically and only gets α -coections 1988).

8 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 8/2 Dilatonic BH and compactified stings Metic in d s = 10 o 26) dimensions of the type ds 2 = f)dt 2 +g 1 )d d Ω 2 d 2 +hφ)g mny)dy m dy n ; Solution: hφ) = 1 2 d s 2 φ)2 ; g) = 1 RH ) d 3 ) 1 d 3)d 4) 2 λ R 2 H RH ) ) d 3 1 RH d 1 ) 1 RH d 3 Callan-Myes-Pey);

9 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 9/2 Dilatonic BH and compactified stings f) = g) + 4 = 1 d 2)2 RH 1 RH ) d 3 ) ) d 3 ) 1 d s d λ d s 2) 2 2 ) d 3 RH 2B ; R 2 H + d 2) 2 d s d d s 2) 2 λ R 2 H [ d s d φ φ ) d s 2) 2 d 3)d 4) 2 d 3) )] 2 d 3,0 ln 1 d 2) 2 d 3) d s d d s 2) 2 λ R 2 H RH RH RH λ R 2 H RH ) d 3 ) d 3 ) d 1 ) ) d 3 1 RH d 1 ) 1 RH d 3 RH ) d 1 ) ) d 3 1 RH d 1 ) 1 RH d 3.

10 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 10/2 Themodynamical popeties I) Wald entopy: S = 2πG Σ L R µνρσ ε µν ε ρσ h dωd 2 ; ε t = f g ; 8πG L R µνρσ ε µν ε ρσ = f g + e 4 d 2 φ λf ) g f ; At ode λ = 0, R tt = 1 2 f = 1 d 3)d 2), φ = 0, 2H 2 f = g; One gets S = A H d 3)d 2) λ 2 H ) like CMP).

11 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 11/2 Themodynamical popeties II) Tempeatue: T = lim H g 2π T = d 3 4 H π [ 1 + d f d. 1 4d 1)d s 2) 2 3d 5 + 2γ 3d s + 6)) d d s d s + γ 13) 10γ) d d 2 s 83d s + 2γ 5d s + 8) + 28 ) d 2 ) 2 2d s 9d s + 8γ 46) + 4γ + 38) d + 2d 2) 2 d 1)d d s ) ψ 0) d 3 ] 28d s + 8d s d s + γ) + 32) λ H 2 α coections decease T fo evey elevant values of d and d s. This suggests that T may each a maximum. Fo d s = 10, T max = α ; fo d s = 26, T max = α. Fo d s = 10, T cit = 0.16 α ; fo d s = 26, T cit = 0.08 α.

12 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 12/2 lack hole inetial and gavitational mass M I = d 2) Ω d 2 16πG = 1 + d 3)d 4) 2 ) lim d 3 1 f ) ) λ d 2) Ωd 2 16πG 2 H d 3 H. M G = d 2) Ω d 2 16πG = M I + λ 2 H ) lim d 3 1 g ) d s d) d 2) 3 d 2) Ω d 2 d s 2) 2 16πG d 3 H.

13 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 13/2 Geneal petubation setup Metic of the type ds 2 = f)dt 2 + g 1 )d d Ω 2 d 2 ; Vaiation of the metic h µν = δg µν ; Vaiation of the Riemann tenso: δr ρσµν = 1 Rµνρ λ h λσ Rµνσ λ h λρ 2 µ ρ h νσ + µ σ h νρ ν σ h µρ + ν ρ h µσ ).

14 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 14/2 Petubations on the d 2)-sphee Geneal tensos of ank at least 2 on the d 2)-sphee can be uniquely decomposed in thei tensoial, vectoial and scala components. One can in geneal conside petubations to the metic and any othe physical field of the system unde consideation.

15 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 15/2 Tensoial petubations of the metic We conside only the tensoial pat of h µν : h ij = 2 2 H T,t)T ij θ i ), h ia = 0,h ab = 0 with γ kl D k D l + k T ) T ij = 0, D i T ij = 0, g ij T ij = 0. D i : d 2)-sphee covaiant deivative, associated to the metic γ ij. T ij ae the eigentensos of D 2 on S d 2 k T = 2 l l + d 3) ae the eigenvalues of D 2 on S d 2, whee l = 2, 3, 4,...

16 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 16/2 Tensoial petubations of R ρσµν δr ijkl = [3g 1) H T + g H T ] g il T jk g ik T jl g jl T ik + g jk T il ) + 2 ) H T Di D l T jk D i D k T jl D j D l T ik + D j D k T il ; δr itjt = [ 2 2t H T + 12 ] ff 2 H T + ff H T T ij ; δr itj = δr ij = 2 12 f ) t H T t H T + 2 f th T T ij ; g g H T 1 ) g 2 2 g H T 2 H T 2 H 2 T T ij. All othe tensoial petubations ae 0.

17 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 17/2 Petubations of the field equations δ 2 φ λ 4 e 2 d 4 φ δ R ρσλτ R ρσλτ) + ) δr ij + λe 2 d [δ 4 φ R iρστ R ρστ j λ d 2 e 4 2 d φ R ρσλτ R ρσλτ δφ = 0, 1 2d 2) R ρσλτ R ρσλτ h ij 1 2d 2) g ijδ R ρσλτ R ρσλτ)] + 4 d 2 R ijδφ = 0. Spheical symmety, k φ = 0, a,b =,t) : δ 2 φ = g ab a b δφ g ab Γ c ab cδφ + g ij i j δφ g ij Γ k ij kδφ g ij Γ a ij aδφ. Using δ R ρσλτ R ρσλτ) = 0, we can set δφ = 0.

18 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 18/2 Petubed gaviton field equation 1 2λ f ) 2 ) f 2 t H T 1 2λ g 2 g 2 H T [ d 2)g f + g ) g 1 g) + 4λd 4) 4λgg λ f 2 + g 2)] H T + [ + ll + d 3) + 2d 2) 2d 3)g f + g ) + + λ 8 1 g d 3) 1 + 4λ ) 1 g) 2 can be witten in the fom 1 g)2 2 2 d 2 [ f f g g f 2 f )] 2 )] 2 t H T F 2 ) 2 H T + P) H T + Q) H T = 0. H T = 0

19 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 19/2 The Maste Equation The petubation equation can be witten as a "maste equation" 2 Φ x 2 2 Φ t 2 =: V TΦ. dx/d = 1/ fg "totoise" coodinate); Φ = k)h T "maste" vaiable); V T : potential fo tenso-type gavitational petubations. In classical EH gavity it is the same as the potential fo scala fields Ishibashi, Kodama); k) = 1 4 fg exp d 2)g f +g )+4λd 4) g1 g) 2fg 4λgg λf 2 +g 2 ) ) d

20 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 20/2 The sting-coected tenso potential V T [f), g)] = 1 4 fg ll + d 3) 2 f 2 g d 2)d 4)2 f 2 g d 6)3 f 2 gf + 3 fg 2 f f 2 f g 2 f d 2)3 f 2 gg fg + f)f g fgg f)f ) + λ 4 fg 4ll + d 3)1 g)gf 2 + 2d 4)d 5)1 g)g 2 f 2 + d 4)f 2 gf + 2ll + d 3)f 2 gf + d 3)d 4)f 2 g 2 f d 6)2 f 2 gf fg 2 f 2 + d 4)f 2 gg 5d 4)f 2 g 2 g + d 7 ) 2 f 2 gf g f 2 f 2 g 1 2 d 1)2 f 2 gg f 2 f g f 2 g 3 + d 2) 2 f 2 g 2 f f 2 gg f 2 2 f 2 g 2 g f 2 gf g 3 f 2 gg g )

21 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 21/2 Study of the stability That was the potential fo tenso type gavitational petubations of any kind of static, spheically symmetic R 2 sting coected black hole in d dimensions. Solutions of the fom Φx,t) = e iωt φx); The maste equation is then witten in the Schödinge fom, ] [ d2 dx 2 + V φx) =: Aφx) = ω 2 φx); A solution to the field equation is then stable if the opeato A has no negative eigenvalues Ishibashi, Kodama; Dotti, Gleise).

22 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 22/2 "S-defomation" appoach Stability means positivity fo evey possible φ) of the following inne poduct: φ, Aφ = = = φx) [ dφ dx [ d2 2 dx 2 + V + V φ 2 ] ] φx) dx dx [ Dφ 2 + Ṽ φ 2] dx with D = d dx + S, Ṽ = V + fg ds d S2.

23 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 23/2 "S-defomation" appoach cont.) Taking S = fg k dk d we ae left with φ,aφ = + Dφ 2 dx + + Q) fg φ 2 dx, with Q = l 3 + d + l)f 2 + 4λ1 g) ) + 3 g f)f 3 2λf ) afte using equations of motion).

24 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 24/2 Stability condition The second tem of φ,aφ can be witten as + R H Q) φ 2 d. fg Fo > R H, f),g) > 0. This condition keeps valid with α coections as long as the black hole in consideation is lage, i.e. R H λ, which is tue in sting petubation theoy. This way the petubative stability of a given black hole solution, with espect to tenso type gavitational petubations, follows if and only if one has Q) > 0 fo R H.

25 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 25/2 Stability of solutions with seconday hai Fo any sting theoy coected, spheically symmetic, static solution, which has no dilaton field at the classical level, one has Q) + 2λ l 3 + d + l)f + g f)f 2 l 3 + d + l)f 21 g) + f ) + g f)f 2 ) 3. One will have Q) 0 fo R H, in any spacetime dimension, as long as g f)f > 0, 2 1 g) + f ) > 0. λ=0

26 Typical behavio of g f and f g f f Dilatonic black holesin heteotic sting theoy:petubations and stability p. 26/2

27 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 27/2 Stability of dilaton-coupled BH At the classical level, the solution is unique Tanghelini, Myes, Pey) and one has 2 1 g) which is positive fo any > R H. + g ) = d 1) Rd 3 H λ=0 d 2, This poves stability unde tenso type gavitational petubations of any spheically symmetic static solution with no dilaton at λ = 0 fo any d > 4.

28 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 28/2 Scatteing Theoy The equation descibing gavitational petubations to a black hole solution allows fo a study of scatteing in this spacetime geomety. Classical esult in EH gavity: fo any spheically symmetic black hole in abitay dimension, the absoption coss section of minimally coupled massless scala fields equals the aea of the black hole hoizon Das, Gibbons, Mathu, 1997). Univesality of the low fequency absoption coss sections of geneic black holes in EH gavity Halmak, Natáio, Schiappa, 2007)? Wok that needs to be done: tying to extend such esult with the inclusion of highe deivative coections.

29 Dilatonic black holesin heteotic sting theoy:petubations and stability p. 29/2 Conclusions We found out the dilatonic black hole solution with R 2 coections in d dimensions; We extended the petubation theoy to R 2 stingy gavity; We studied the stability of black hole solutions unde tenso type gavitational petubations, and poved the petubative stability of the dilatonic R 2 black hole fo any space-time dimension.