Higher order black holes of scalar tensor theories E Babichev and CC - PowerPoint PPT Presentation

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Gravity modification:issues and guidelines Hairy black hole Adding matter Conclusions Since GR is unique we need to introduce new and genuine gravitational degrees of freedom! They must not lead to higher derivative equations of motion. For then additional degrees of freedom are ghosts and vacuum is unstable (Ostrogradski theorem 1850 [ Woodard 2006, Rubakov 2014 ] ) Matter must not directly couple to novel gravity degrees of freedom. Matter sees only the metric and evolves in metric geodesics. As such EEP is preserved and space-time can be put locally in an inertial frame. Novel degrees of freedom need to be screened from local gravity experiments. Need a well defined GR local limit (Chameleon [ Khoury 2013 ] , Vainshtein [ Babichev and Deffayet 2013 ] ). Exact solutions essential in modified gravity in order to understand strong gravity regimes and novel characteristics. Need to deal with no hair paradigm. A modified gravity theory should tell us something about the cosmological constant problem and in particular how to screen an a priori enormous cosmological constant. C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Gravity modification:issues and guidelines Hairy black hole Adding matter Conclusions Possible modified gravity theories Assume extra dimensions : Extension of GR to Lovelock theory with modified yet second order field equations [ Deruelle et.al ’03, Garraffo ’08, CC ’09 ] . Braneworlds DGP model RS models, Kaluza-Klein et.al. compactification Graviton is not massless but massive! dRGT theory and bigravity theory. Theories are unique. [ C DeRham, 2014 ] 4-dimensional modification of GR: Scalar-tensor theories, f ( R ) , Galileon/Hornedski theories [ Sotiriou 2014, CC 2014 ] . Lorentz breaking theories: Horava gravity, Einstein Aether theories [ Audren, Blas, Lesgourgues and Sibiryakov ] Theories modifying geometry: inclusion of torsion, choice of geometric connection [ Zanelli ’08, Olmo 2012 ] C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Hairy black hole Adding matter Conclusions Introduction/Motivation 1 Gravity modification:issues and guidelines Scalar-tensor theories and no hair 2 Scalar-tensor black holes and the no hair paradigm 3 Conformal secondary hair? Building higher order scalar-tensor black holes 4 Resolution step by step Example solutions Hairy black hole 5 Adding matter 6 Conclusions 7 C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Hairy black hole Adding matter Conclusions Scalar-tensor theories are the simplest modification of gravity with one additional degree of freedom Admit a uniqueness theorem due to Horndeski 1973 contain or are limits of other modified gravity theories. F ( R ) is a scalar tensor theory in disguise (Can) have insightful screening mechanisms (Chameleon, Vainshtein) Include terms that can screen classically a big cosmological constant (Fab 4 [ CC, Copeland, Padilla and Saffin 2012 ] ) C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Hairy black hole Adding matter Conclusions What is the most general scalar-tensor theory with second order field equations [ Horndeski 1973 ] , [ Deffayet et.al. ] ? Horndeski has shown that the most general action with this property is � d 4 x √− g ( L 2 + L 3 + L 4 + L 5 ) = S H L 2 = K ( φ, X ) , L 3 = − G 3 ( φ, X ) � φ, ( � φ ) 2 − ( ∇ µ ∇ ν φ ) 2 � � L 4 = G 4 ( φ, X ) R + G 4 X , L 5 = G 5 ( φ, X ) G µν ∇ µ ∇ ν φ − G 5 X ( � φ ) 3 − 3 � φ ( ∇ µ ∇ ν φ ) 2 + 2 ( ∇ µ ∇ ν φ ) 3 � � 6 the G i are unspecified functions of φ and X ≡ − 1 2 ∇ µ φ ∇ µ φ and G iX ≡ ∂ G i /∂ X . In fact same action as covariant Galileons [ Deffayet, Esposito-Farese, Vikman ] Theory screens generically scalar mode locally by the Vainshtein mechanism. C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Conformal secondary hair? Hairy black hole Adding matter Conclusions Introduction/Motivation 1 Gravity modification:issues and guidelines Scalar-tensor theories and no hair 2 Scalar-tensor black holes and the no hair paradigm 3 Conformal secondary hair? Building higher order scalar-tensor black holes 4 Resolution step by step Example solutions Hairy black hole 5 Adding matter 6 Conclusions 7 C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Conformal secondary hair? Hairy black hole Adding matter Conclusions Black holes have no hair During gravitational collapse... Black holes eat or expel surrounding matter their stationary phase is characterized by a limited number of charges and no details black holes are bald... No hair arguments/theorems dictate that adding degrees of freedom lead to singular solutions... For example in vanilla scalar-tensor theories black hole solutions are GR black holes with constant scalar. except C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Conformal secondary hair? Hairy black hole Adding matter Conclusions Conformally coupled scalar field Consider a conformally coupled scalar field φ : � √− g 16 π G − 1 2 ∂ α φ∂ α φ − 1 R � 12 R φ 2 � d 4 x + S m [ g µν , ψ ] S [ g µν , φ, ψ ] = M Invariance of the EOM of φ under the conformal transformation � g αβ = Ω 2 g αβ g αβ �→ ˜ φ �→ ˜ φ = Ω − 1 φ There exists a black hole geometry with non-trivial scalar field and secondary black hole hair. The BBMB solution [N. Bocharova et al.-70 , J. Bekenstein-74 ] C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Conformal secondary hair? Hairy black hole Adding matter Conclusions The BBMB solution [N. Bocharova et al.-70 , J. Bekenstein-74 ] Static and spherically symmetric solution d r 2 � 2 1 − m d s 2 = − � d t 2 + d θ 2 + sin 2 θ d ϕ 2 � � 2 + r 2 � r � 1 − m r with secondary scalar hair � 3 m φ = 4 π G r − m Geometry is that of an extremal RN. Problem:The scalar field is unbounded at ( r = m ). Controversy on the stability [Bronnikov et al.-78, McFadden et al.-05] Not clear that the solution is a black hole. C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Conformal secondary hair? Hairy black hole Adding matter Conclusions Scalar-tensor theories and black holes In scalar tensor theories "regular" black hole solutions are GR black holes with a constant scalar field Is it possible to have non-trivial and regular scalar-tensor black holes for an asymptotically flat space-time? How can we evade no-hair theorems? C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Resolution step by step Building higher order scalar-tensor black holes Example solutions Hairy black hole Adding matter Conclusions Introduction/Motivation 1 Gravity modification:issues and guidelines Scalar-tensor theories and no hair 2 Scalar-tensor black holes and the no hair paradigm 3 Conformal secondary hair? Building higher order scalar-tensor black holes 4 Resolution step by step Example solutions Hairy black hole 5 Adding matter 6 Conclusions 7 C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Resolution step by step Building higher order scalar-tensor black holes Example solutions Hairy black hole Adding matter Conclusions Higher order scalar-tensor theory Construct black hole solutions for, Higher order scalar tensor theory: Horndeski/Galileon theory (Lovelock/Lanczos theory) Shift symmetry for the scalar Spherically symmetric and static space-time. C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Resolution step by step Building higher order scalar-tensor black holes Example solutions Hairy black hole Adding matter Conclusions Example theory Consider the action, � d 4 x √− g � ζ R − 2 Λ − η ( ∂φ ) 2 + β G µν ∂ µ φ∂ ν φ � S = , Metric field equations read, ∂ µ φ∂ ν φ − 1 � 2 g µν ( ∂φ ) 2 � ζ G µν − η + g µν Λ + β ( ∂φ ) 2 G µν + 2 P µανβ ∇ α φ ∇ β φ � 2 + g µα δ αρσ νγδ ∇ γ ∇ ρ φ ∇ δ ∇ σ φ � = 0 , Scalar field has translational invariance : φ → φ + const., Scalar field equation can be written in terms of a current ∇ µ J µ = 0 , J µ = ( η g µν − β G µν ) ∂ ν φ. Take ds 2 = − h ( r ) dt 2 + dr 2 f ( r ) + r 2 d Ω 2 , φ = φ ( r ) then scalar equation is integrable... C. Charmousis Higher order black holes of scalar tensor theories G rr √ g g rr ′ c

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Resolution step by step Building higher order scalar-tensor black holes Example solutions Hairy black hole Adding matter Conclusions Example theory Consider the action, � d 4 x √− g � ζ R − 2 Λ − η ( ∂φ ) 2 + β G µν ∂ µ φ∂ ν φ � S = , Scalar field has translational invariance : φ → φ + const., Scalar field equation can be written in terms of a current ∇ µ J µ = 0 , J µ = ( η g µν − β G µν ) ∂ ν φ. Take ds 2 = − h ( r ) dt 2 + dr 2 f ( r ) + r 2 d Ω 2 , φ = φ ( r ) then scalar equation is integrable... ( η g rr − β G rr ) √ g φ ′ = c but current is singular J 2 = J µ J ν g µν = ( J r ) 2 g rr unless J r = 0 at the horizon... Generically φ = constant everywhere [ Hui and Nicolis ] and we have again the appearance of a no-hair theorem... unless.... C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Resolution step by step Building higher order scalar-tensor black holes Example solutions Hairy black hole Adding matter Conclusions Example theory Consider the action, � d 4 x √− g � ζ R − 2 Λ − η ( ∂φ ) 2 + β G µν ∂ µ φ∂ ν φ � S = , Scalar field has translational invariance : φ → φ + const., Scalar field equation can be written in terms of a current ∇ µ J µ = 0 , J µ = ( η g µν − β G µν ) ∂ ν φ. In scalar equation, η g µν − β G µν → metric EoM R → G µν ∂ µ φ∂ ν φ , Λ → g µν ∂ µ φ∂ ν φ Take ds 2 = − h ( r ) dt 2 + dr 2 f ( r ) + r 2 d Ω 2 , φ = φ ( r ) then scalar equation is integrable... ( η g rr − β G rr ) √ g φ ′ = c but current is singular J 2 = J µ J ν g µν = ( J r ) 2 g rr unless J r = 0 at the horizon... Generically φ = constant everywhere [ Hui and Nicolis ] and we have again the C. Charmousis Higher order black holes of scalar tensor theories appearance of a no-hair theorem...

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Resolution step by step Building higher order scalar-tensor black holes Example solutions Hairy black hole Adding matter Conclusions Example theory Consider the action, � d 4 x √− g � ζ R − 2 Λ − η ( ∂φ ) 2 + β G µν ∂ µ φ∂ ν φ � S = , Scalar field has translational invariance : φ → φ + const., Scalar field equation can be written in terms of a current ∇ µ J µ = 0 , J µ = ( η g µν − β G µν ) ∂ ν φ. Take ds 2 = − h ( r ) dt 2 + dr 2 f ( r ) + r 2 d Ω 2 , φ = φ ( r ) then scalar equation is integrable... ( η g rr − β G rr ) √ g φ ′ = c but current is singular J 2 = J µ J ν g µν = ( J r ) 2 g rr unless J r = 0 at the horizon... Generically φ = constant everywhere [ Hui and Nicolis ] and we have again the appearance of a no-hair theorem... unless.... C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Resolution step by step Building higher order scalar-tensor black holes Example solutions Hairy black hole Adding matter Conclusions Time dependent scalar field Set β G rr − η g rr = 0 rendering the scalar equation "redundant"... Consider φ = φ ( t , r ) with static space-time, ds 2 = − h ( r ) dt 2 + dr 2 f ( r ) + r 2 d Ω 2 (tr)-component of EoM is non trivial and reads, � � � φ ′ � f − 1 − η r 2 βφ ′ rfh ′ φ − 2 rf ˙ ˙ h + = 0 r 2 β General solution, φ ( t , r ) = ψ ( r ) + q 1 ( t ) e X ( r ) with � � rf − η r β f + h ′ X ( r ) = 1 � 1 r − 1 dr and ¨ q 1 ( t ) = C 1 q 1 ( t ) + C 2 2 h Simplest solution softly breaking translational invariance q 1 ( t ) = q t and thus φ ( t , r ) = q t + ψ ( r ) No time derivatives present in the field equations C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Resolution step by step Building higher order scalar-tensor black holes Example solutions Hairy black hole Adding matter Conclusions Scalar field equation Hypotheses: β G rr − η g rr = 0 and φ ( t , r ) = q t + ψ ( r ) , − ∂ r [( β G rr − η g rr ) ∂ r ψ ] − ∂ t [( β G tt − η g tt ) ∂ t ( qt )] = 0 no scalar charge, current ok, φ � = 0, and ( tr ) -eq satisfied Geometric constraint, f = ( β + η r 2 ) h β ( rh ) ′ , fixing spherically symmetric gauge. ds 2 = − h ( r ) dt 2 + dr 2 f ( r ) + r 2 d Ω 2 We need to find ψ ( r ) and h ( r ) and have two ODE’s to solve, the ( rr ) and ( tt ) . Hence hypotheses are consistent. C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Resolution step by step Building higher order scalar-tensor black holes Example solutions Hairy black hole Adding matter Conclusions Solving the remaining EoM From (rr)-component get ψ ′ √ r 2 ( h 2 r 2 ) ′ � 1 / 2 q 2 β ( β + η r 2 ) h ′ − λ � ψ ′ = ± . h ( β + η r 2 ) with λ ≡ ζη + β Λ . For η = Λ = 0 time dependence is essential!! and finally (tt)-component gives h ( r ) via, � h ( r ) = − µ r + 1 k ( r ) β + η r 2 dr , r with k + C 0 k 3 / 2 = 0 , q 2 β ( β + η r 2 ) 2 − � 2 ζβ + ( 2 ζη − λ ) r 2 � Any solution to the algebraic eq for k = k ( r ) gives full solution to the system! C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Resolution step by step Building higher order scalar-tensor black holes Example solutions Hairy black hole Adding matter Conclusions Fab 4 limit: Λ = 0, η = 0 d 4 x √− g [ ζ R + β G µν ∂ µ φ∂ ν φ ] Consider S = � G µν √ g ∂ ν φ � G µν ∇ µ ∇ ν φ = ∇ µ ( G µν ∇ ν φ ) = 1 � = 0 √ g in Eq of scalar β G µν → Einstein equation G rr = 0 → f = h ( rh ) ′ and φ ( t , r ) = q t + ψ ( r ) √ r −√ µ 2 � r � � ( rr ) -EOM gives φ ± = qt ± q µ µ + log + φ 0 √ r + √ µ ( tt ) -EOM q 2 β 3 − 2 ζβ k + C 0 k 3 / 2 = 0 → k = constant ! f ( r ) = h ( r ) = 1 − µ/ r Schwarzschild geometry with a non-trivial scalar field. But is the scalar regular on the horizon? C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Resolution step by step Building higher order scalar-tensor black holes Example solutions Hairy black hole Adding matter Conclusions Scalar-tensor Schwarzschild black hole √ r −√ µ � 2 � r � φ ± = qt ± q µ µ + log + φ 0 √ r + √ µ Scalar looks singular for r → r h but t h → ∞ ! ( fh ) − 1 / 2 dr then ds 2 = − hdv 2 + 2 Consider v = t + � � h / f dvdr + r 2 d Ω 2 Regular chart for horizon, EF coordinates ( [ Jacobson ] , [ Ayon-Beato, Martinez & Zanelli ] ) v − r + 2 √ µ r − 2 µ log �� r � �� φ + = q µ + 1 + const Scalar regular at future black hole horizon! Metric is Schwarzschild, scalar is regular and non-trivial Scalar linearly diverges at past and future null infinity but not its derivatives, current is constant. C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Resolution step by step Building higher order scalar-tensor black holes Example solutions Hairy black hole Adding matter Conclusions All solutions are not "GR like" (but need η � = 0 or Λ � = 0) Need to solve: q 2 β ( β + η r 2 ) 2 − � k + C 0 k 3 / 2 = 0 2 ζβ + ( 2 ζη − λ ) r 2 � with � h ( r ) = − µ r + 1 k ( r ) β + η r 2 dr r Example: Black hole in an Einstein static universe ( ζη + β Λ = 0) � � � 1 + η r 2 h = 1 − µ r , f = � 1 − µ , r β ψ ′ = ± q � µ β r 2 ) and φ = qt + ψ ( r ) . r ( 1 + η h Solution is not asymptotically flat or de Sitter. Can we get de Sitter asymptotics? C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Resolution step by step Building higher order scalar-tensor black holes Example solutions Hairy black hole Adding matter Conclusions de Sitter black hole d 4 x √− g � ζ R − 2 Λ − η ( ∂φ ) 2 + β G µν ∂ µ φ∂ ν φ � Consider S = � k ( r ) has to verify q 2 β ( β + η r 2 ) 2 − � k + C 0 k 3 / 2 = 0 2 ζβ + ( 2 ζη − λ ) r 2 � Infinite number of solutions with differing asymptotics, but are there de Sitter asymptotics? Particular solution reads k ( r ) = ( β + η r 2 ) 2 β with q 2 = ( ζη + β Λ) / ( βη ) and C 0 = ( ζη − β Λ) √ β/η 3 β r 2 de Sitter Schwarzschild! with f = h = 1 − µ η r + √ ψ ′ = ± q 1 − h and φ ( t , r ) = q t + ψ ( r ) h Solution is regular at the horizon for de Sitter asymptotics C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Resolution step by step Building higher order scalar-tensor black holes Example solutions Hairy black hole Adding matter Conclusions Self tuned de Sitter Schwarzschild 3 β r 2 with Λ eff = − η/β We have f = h = 1 − µ η r + d 4 x √− g � R − 2 Λ − η ( ∂φ ) 2 + β G µν ∂ µ φ∂ ν φ � S = � The effective cosmological constant is not the vacuum cosmological constant. In fact, q 2 η = Λ − Λ eff > 0 Hence for any arbitrary Λ > Λ eff fixes q , integration constant. where Λ eff is a geometric acceleration Solution self tunes vacuum cosmological constant but has "action induced" effective cosmological constant C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Hairy black hole Adding matter Conclusions Conformally coupled scalar field Consider a conformally coupled scalar field φ : � √− g 16 π G − 1 2 ∂ α φ∂ α φ − 1 R � 12 R φ 2 � d 4 x + S m [ g µν , ψ ] S [ g µν , φ, ψ ] = M Invariance of the EOM of φ under the conformal transformation � g αβ = Ω 2 g αβ g αβ �→ ˜ φ �→ ˜ φ = Ω − 1 φ There exists a black hole geometry with non-trivial scalar field and secondary black hole hair. The BBMB solution [N. Bocharova et al.-70 , J. Bekenstein-74 ] C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Hairy black hole Adding matter Conclusions BBMB completion [ CC, Kolyvaris, Papantonopoulos and Tsoukalas ] We would like to combine the above properties in order to obtain a hairy black hole. Consider the following action, S ( g µν , φ, ψ ) = S 0 + S 1 where � dx 4 √− g − 1 2 ( ∂φ ) 2 − 1 � � �� 12 φ 2 R S 0 = ζ R + η and � dx 4 √− g � β G µν ∇ µ Ψ ∇ ν Ψ − γ T BBMB ∇ µ Ψ ∇ ν Ψ � S 1 = , µν where = 1 2 ∇ µ φ ∇ ν φ − 1 4 g µν ∇ α φ ∇ α φ + 1 12 ( g µν � − ∇ µ ∇ ν + G µν ) φ 2 . T BBMB µν Scalar field equation of S 1 contains metric equation of S 0 . ∇ µ J µ = 0 , J µ = � β G µν − γ T BBMB � ∇ ν Ψ . µν C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Hairy black hole Adding matter Conclusions Black hole with primary hair Solve as before assuming linear time dependence for Ψ Scalar φ has an additional branch regular at the "horizon" A second solution reads, � γ c 2 � h ( r ) = 1 − m f ( r ) = ( 1 − m 0 r , r ) 1 − 12 β r 2 φ ( r ) = c 0 r , � dr ψ = qv − q . �� γ c 2 � ( 1 ∓ � m 1 − 0 r ) 12 β r 2 C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Hairy black hole Adding matter Conclusions Black hole with primary hair Solve as before assuming linear time dependence for Ψ Scalar φ has an additional branch regular at the "horizon" Solution reads, γ c 2 f ( r ) = h ( r ) = 1 − m 0 r + 12 β r 2 , φ ( r ) = c 0 r , � γ c 2 mr − 0 12 β ψ ′ ( r ) = ± q , r h ( r ) βη + γ ( q 2 β − ζ ) = 0 . A second solution reads, � γ c 2 � h ( r ) = 1 − m f ( r ) = ( 1 − m 0 r , r ) 1 − 12 β r 2 C. Charmousis Higher order black holes of scalar tensor theories c 0 r

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Hairy black hole Adding matter Conclusions Black hole with primary hair Solve as before assuming linear time dependence for Ψ Scalar φ has an additional branch regular at the "horizon" Solution reads, γ c 2 f ( r ) = h ( r ) = 1 − m 0 r + 12 β r 2 , φ ( r ) = c 0 r , � γ c 2 mr − 0 12 β ψ ′ ( r ) = ± q , r h ( r ) βη + γ ( q 2 β − ζ ) = 0 . Scalar charge c 0 playing similar role to EM charge in RN A second solution reads, � γ c 2 � h ( r ) = 1 − m f ( r ) = ( 1 − m 0 r , r ) 1 − 12 β r 2 C. Charmousis Higher order black holes of scalar tensor theories c 0 r

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Hairy black hole Adding matter Conclusions Black hole with primary hair Solve as before assuming linear time dependence for Ψ Scalar φ has an additional branch regular at the "horizon" Solution reads, γ c 2 f ( r ) = h ( r ) = 1 − m 0 r + 12 β r 2 , φ ( r ) = c 0 r , � γ c 2 mr − 0 12 β ψ ′ ( r ) = ± q , r h ( r ) βη + γ ( q 2 β − ζ ) = 0 . Scalar charge c 0 playing similar role to EM charge in RN Galileon Ψ regular on the future horizon � dr ψ = qv − q � 1 ± 1 − h ( r ) C. Charmousis Higher order black holes of scalar tensor theories A second solution reads,

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Hairy black hole Adding matter Conclusions Black hole with primary hair Solve as before assuming linear time dependence for Ψ Scalar φ has an additional branch regular at the "horizon" A second solution reads, � γ c 2 � h ( r ) = 1 − m f ( r ) = ( 1 − m 0 r , r ) 1 − 12 β r 2 φ ( r ) = c 0 r , � dr ψ = qv − q . �� γ c 2 � ( 1 ∓ � m 1 − 0 r ) 12 β r 2 C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Hairy black hole Adding matter Conclusions Adding electromagnetic charge Following the same idea we can add an EM field � √− gd 4 x � R − η ( ∂φ ) 2 − 2 Λ + β G µν ∇ µ φ ∇ ν φ I [ g µν , φ, A µ ] = − 1 4 F µν F µν − γ T µν ∇ µ φ ∇ ν φ � , where we have defined T µν := 1 − 1 � 4 g µν F αβ F αβ � σ F µσ F . ν 2 Note that the coupling of the EM field is not trivial. But the scalar field equations defines a current as before ∇ µ J µ = ∇ µ [( β G µν − η g µν − γ T µν ) ∇ ν φ ] = 0 , C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Hairy black hole Adding matter Conclusions Adding electromagnetic charge We consider, dr 2 r 2 ds 2 = − h ( r ) dt 2 + 1 − θ 2 d θ 2 + r 2 θ 2 d χ 2 , A µ dx µ = A ( r ) dt . + φ ( t , r ) = ψ ( r ) + q t , (1) f ( r ) We define � η r 2 + β �� r 2 B ( r ) 2 γ + 4 ( r h ( r )) ′ β � B ( r ) = A ′ ( r ) , S ( r ) = , (2) 4 β and the EOM reduce to, η r 2 + β � 2 r 2 � ( β − γ ) B ( r ) 2 η r 2 + β � 2 q 2 β � − S ( r ) � ( η − β Λ) r 2 + 2 β � + C 0 S ( r ) 3 / 2 = 0 , + 4 β β ( β − γ )( η r 2 + β ) � � βγ C 0 2 Q + B ( r ) = S ( r ) 1 / 2 r 2 2 C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Hairy black hole Adding matter Conclusions RN like solution F rt = B ( r ) = 2 Q r 2 . (3) The metric functions take the form h ( r ) = f ( r ) = 1 + η r 2 r + Q 2 ψ ′ 2 = − ( f ( r ) − 1 ) q 2 3 β − µ r 2 , , (4) f ( r ) 2 while the coupling constants are, √ β q 2 = η + Λ β β = γ, C 0 = ( η − β Λ) ηβ η C. Charmousis Higher order black holes of scalar tensor theories

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Hairy black hole Adding matter Conclusions Conclusions Have found GR black holes with a non-trivial and regular scalar field Shift symmetry and higher order essential!! Rendered scalar field eq redundant and allowed for linear time dependence Time dependence essential for regularity on the event horizon Solutions are hairy(charge q ) and non-hairy (time dependent), hence fake. Method can be applied in differing Gallileon context [ Kobayashi and Tanahashi ] , in higher dimensions, including EM and other matter fields. Is there a way to find observable for q ? Is there a distinction possible? Thermodynamics and stability. Can we go beyond spherical symmetry? C. Charmousis Higher order black holes of scalar tensor theories

Higher order black holes of scalar tensor theories E Babichev and CC - PowerPoint PPT Presentation

Introduction/Motivation Scalar-tensor theories and no hair Scalar-tensor black holes and the no hair paradigm Building higher order scalar-tensor black holes Hairy black hole Adding matter Conclusions Higher order black holes of scalar tensor

Hairy black holes in scalar tensor theories E Babichev and CC gr-qc/1312.3204 CC, T Kolyvaris, E

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