Noncommutative Scalar Quasinormal modes of RN Black Hole Nikola - PowerPoint PPT Presentation

Noncommutative Scalar Quasinormal modes of RN Black Hole Nikola Konjik (University of Belgrade) 9 - 14 September 2019 Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 1 / 21

Content 1 Introduction 2 Noncommutative geometry 3 Angular noncommutativity 4 Scalar U ( 1 ) gauge theory in RN background 5 Continued fractions 6 Outlook Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 2 / 21

Introduction Physics between LHC and Planck scale → problem of modern theoretical physics Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 3 / 21

Introduction Physics between LHC and Planck scale → problem of modern theoretical physics Possible solutions • String Theory Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 3 / 21

Introduction Physics between LHC and Planck scale → problem of modern theoretical physics Possible solutions • String Theory • Quantum loop gravity Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 3 / 21

Introduction Physics between LHC and Planck scale → problem of modern theoretical physics Possible solutions • String Theory • Quantum loop gravity • Noncommutative geometry Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 3 / 21

Introduction Physics between LHC and Planck scale → problem of modern theoretical physics Possible solutions • String Theory • Quantum loop gravity • Noncommutative geometry • . . . Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 3 / 21

Introduction Physics between LHC and Planck scale → problem of modern theoretical physics Possible solutions • String Theory • Quantum loop gravity • Noncommutative geometry • . . . Detection of the gravitational waves can help better understanding of structure of space-time Dominant stage of the perturbed BH are dumped oscillations of the geometry or matter fields (Quasinormal modes) Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 3 / 21

Noncommutative geometry • Local coordinates x µ are changed with hermitian operators ˆ x µ Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 4 / 21

Noncommutative geometry • Local coordinates x µ are changed with hermitian operators ˆ x µ x µ , ˆ x ν ] = i θ µν • Algebra of operators is [ˆ Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 4 / 21

Noncommutative geometry • Local coordinates x µ are changed with hermitian operators ˆ x µ x µ , ˆ x ν ] = i θ µν • Algebra of operators is [ˆ x ν ≥ 1 x µ ∆ˆ 2 | θ µν | • For θ = const ⇒ ∆ˆ Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 4 / 21

Noncommutative geometry • Local coordinates x µ are changed with hermitian operators ˆ x µ x µ , ˆ x ν ] = i θ µν • Algebra of operators is [ˆ x ν ≥ 1 x µ ∆ˆ 2 | θ µν | • For θ = const ⇒ ∆ˆ • The notion of a point loses its meaning ⇒ we describe NC space with algebra of functions (theorems of Gelfand and Naimark) Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 4 / 21

Noncommutative geometry • Local coordinates x µ are changed with hermitian operators ˆ x µ x µ , ˆ x ν ] = i θ µν • Algebra of operators is [ˆ x ν ≥ 1 x µ ∆ˆ 2 | θ µν | • For θ = const ⇒ ∆ˆ • The notion of a point loses its meaning ⇒ we describe NC space with algebra of functions (theorems of Gelfand and Naimark) Approaches to NC geometry ⋆ -product, NC spectral triple, NC vierbein formalism, matrix models, . . . Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 4 / 21

NC space-time from the angular twist Twist is used to deform a symmetry Hopf algebra Twist F is invertible bidifferential operator from the universal enveloping algebra of the symmetry algebra We work in 4D and deform the space-time by the following twist 2 θ ab X a � X b F = e − i [ X a , X b ] = 0, a,b=1,2 X 1 = ∂ 0 and X 2 = x ∂ y − y ∂ x − ia 2 ( ∂ 0 ⊗ ( x ∂ y − y ∂ x ) − ( x ∂ y − y ∂ x ) ⊗ ∂ 0 ) F = e Bilinear maps are deformed by twist! Bilinear map µ µ : X × Y → Z µ ⋆ = µ F − 1 Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 5 / 21

Commutation relations between coordinates are: x 0 , ˆ [ˆ x ] = ia ˆ y , All other commutation relations are zero x 0 , ˆ [ˆ y ] = − ia ˆ x Our approach: deform space-time by an Abelian twist to obtain commutation relations Angular twist in curved coordinates X 1 = ∂ 0 and X 2 = ∂ ϕ -supose that metric tensor g µν does not depend on t and ϕ coordinates -Hodge dual becomes same as in commutative case Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 6 / 21

Angular noncommutativity • Product of two plane waves is e − ip · x ⋆ e − iq · x = e − i ( p + ⋆ q ) · x where is p + ⋆ q = R ( q 3 ) p + R ( − p 3 ) q and   1 0 0 0 � at � at � � 0 cos sin 0   R ( t ) ≡ 2 2 � at � at   � � 0 − sin cos 0   2 2 0 0 0 1 Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 7 / 21

Angular noncommutativity • e − ip · x ⋆ e − iq · x ⋆ e − ir · x = e − i ( p + ⋆ q + ⋆ r ) · x gives p + ⋆ q + ⋆ r = R ( r 3 + q 3 ) p + R ( − p 3 + r 3 ) q + R ( − p 3 − q 3 ) r Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 8 / 21

Angular noncommutativity • e − ip · x ⋆ e − iq · x ⋆ e − ir · x = e − i ( p + ⋆ q + ⋆ r ) · x gives p + ⋆ q + ⋆ r = R ( r 3 + q 3 ) p + R ( − p 3 + r 3 ) q + R ( − p 3 − q 3 ) r • General case   N p ( 1 ) + ⋆ ... + ⋆ p ( N ) = p ( k ) p ( k ) � � �  p ( j ) R  − + 3 3 j = 1 1 ≤ k < j j < k ≤ N • Conservation of momentum is broken! Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 8 / 21

Scalar U ( 1 ) ⋆ gauge theory A µ ⋆ dx µ is introduced to the model If a one-form gauge field ˆ A = ˆ through a minimal coupling, the relevant action becomes � � � + � � S [ˆ φ, ˆ d ˆ φ − i ˆ A ⋆ ˆ d ˆ φ − i ˆ A ⋆ ˆ A ] = φ ∧ ⋆ ∗ H φ � µ 2 φ + ⋆ ˆ φǫ abcd e a ∧ ⋆ e b ∧ ⋆ e c ∧ ⋆ e d ˆ − 4 ! � d 4 x √− g ⋆ � φ − µ 2 ˆ � g µν ⋆ D µ ˆ φ + ⋆ D ν ˆ φ + ⋆ ˆ = φ Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 9 / 21

After expanding action and varying with respect to Φ + EOM is � � � − 1 g µν D µ D ν φ − Γ λ 4 θ αβ g µν D µ ( F αβ D ν φ ) − Γ λ µν D λ φ µν F αβ D λ φ � − 2 D µ ( F αν D β φ ) + 2 Γ λ µν F αλ D β φ − 2 D β ( F αµ D ν φ ) = 0 Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 10 / 21

Scalar field in the Reissner–Nordström background RN metric tensor is  0 0 0  f − 1 0 0 0   f g µν =  − r 2  0 0 0   − r 2 sin 2 θ 0 0 0 + Q 2 G with f = 1 − 2 MG which gives two horizons ( r + and r − ) r r 2 Q-charge of RN BH M-mass of RN BH Non-zero components of gauge fields are A 0 = − qQ i.e. F r 0 = qQ r 2 r q-charge of scalar field Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 11 / 21

EOM for scalar field in RN space-time rf ∂ t − q 2 Q 2 � 1 r + 2 MG r 2 ∂ r + 2 iqQ 1 � f ∂ 2 t − ∆ + ( 1 − f ) ∂ 2 φ r 2 f − GQ 2 + aqQ ( MG � � r 2 ) ∂ ϕ + rf ∂ r ∂ ϕ φ = 0 r 3 r where a is θ t ϕ Assuming ansatz φ lm ( t , r , θ, ϕ ) = R lm ( r ) e − i ω t Y m l ( θ, ϕ ) we got equation for radial part lm + 2 1 − MG � l ( l + 1 ) − 1 f ( ω − qQ r ) 2 � fR ′′ � � R ′ lm − R lm r 2 r r − GQ 2 − imaqQ ( MG � � r 2 ) R lm + rfR ′ = 0 (1) lm r 3 r Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 12 / 21

NC QNM solutions QNM -special solution of equation -damped oscillations of a perturbed black hole A set of the boudary condition which leads to this solution is the following: at the horizon, the QNMs are purely incoming, while in the infinity the QNMs are purely outgoing Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 13 / 21

Continued fraction method To get form d 2 ψ dy 2 + V ψ = 0 y must be r + r − � � � � y = r + r + − iamqQ ln( r − r + ) − r − − iamqQ ln( r − r − ) r + − r − r + − r − y is modified Tortoise RN coordinate Asymptotic form of the eq. (1) Z out e i Ω y y − 1 − i ω qQ − µ 2 M  − amqQ Ω za y → ∞ Ω     R ( r ) → � �� �  ω − qQ 1 + iam qQ − i y   Z in e r + r + za y → −∞  Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 14 / 21

Combining assymptotic forms, we get general solution in the form ∞ � n + δ � r − r + R ( r ) = e i Ω r ( r − r − ) ǫ � a n (2) r − r − n = 0 Nikola Konjik (University of Belgrade) Belgrade, Serbia 9 - 14 September 2019 15 / 21