Tidal deformation and dynamics of black holes Eric Poisson - PowerPoint PPT Presentation

Introduction Newtonian tides Relativistic tides Conclusion Tidal deformation and dynamics of black holes Eric Poisson Department of Physics, University of Guelph IH´ ES, Bures-sur-Yvette, 30 October 2014 Eric Poisson Tidal deformation and dynamics of black holes

Introduction Newtonian tides Relativistic tides Conclusion Context The tidal dynamics of compact bodies in general relativity is now the subject of vigourous development. The tidal deformation of neutron stars could have measurable effects on gravitational waves produced during inspirals, well before merger occurs. [Flanagan, Hinderer (2008); Postnikov, Prakash, Lattimer (2010); Pannarale et al (2011), Lackey el al (2012), Damour, Nagar, Villain (2012); Read et al (2013); Vines, Flanagan (2013)] Tidal interactions are important in extreme mass-ratio inspirals: tidal torquing of the large black hole leads to a significant gain of orbital angular momentum. [Hughes (2001); Martel (2004); Yunes et al (2010, 2011)] Relativistic theory of Love numbers. [Damour, Nagar (2009); Binnington, Poisson (2009); Damour, Lecian (2009); Landry, Poisson (2014)] I -Love- Q relations. [Yagi, Yunes (2013); Doneva, Yazadjiev, Stergioulas, Kokkotas (2013); Maselli et al (2013); Haskell et al (2014)] Tidal invariants for point-particle actions [Bini, Damour, Faye (2012); Dolan, Nolan, Ottewill (2014); Bini, Damour (2014)] Eric Poisson Tidal deformation and dynamics of black holes

Introduction Newtonian tides Relativistic tides Conclusion Goal The main, long-term goal of this work is to develop a relativistic theory of tidal deformations and interactions that is as complete and elegant as the Newtonian theory. In this talk I shall focus on black holes. Eric Poisson Tidal deformation and dynamics of black holes

Introduction Newtonian tides Relativistic tides Conclusion Outline Tides on Newtonian bodies, in four easy steps Tides on black holes, in the same four easy steps Conclusion Eric Poisson Tidal deformation and dynamics of black holes

Introduction Newtonian tides Relativistic tides Conclusion Newtonian tides: Setting and assumptions We consider a self-gravitating body (“the body”) in a generic tidal environment created by remote external matter. The body has a mass M and radius R . It is spherical in isolation. The body may be rotating, but we ignore the rotational deformation. The tidal forces are weak and the deformation is small . The external time scales are long compared with the time scales associated with internal processes in the body; the tides are slow . We work in the noninertial frame of the moving body, with its origin at the centre-of-mass. We focus our attention on a domain N that does not extend far beyond the body. Eric Poisson Tidal deformation and dynamics of black holes

Introduction Newtonian tides Relativistic tides Conclusion The domain N The local domain N excludes all external matter. N Eric Poisson Tidal deformation and dynamics of black holes

Introduction Newtonian tides Relativistic tides Conclusion 1. Characterize the tidal environment The Newtonian potential in N is decomposed as U = U body + U ext ∇ 2 U body = − 4 πρ, ∇ 2 U ext = 0 Because the external matter is remote, the external potential can be Taylor-expanded about the body’s centre-of-mass, U ext ( t, x ) = U ext ( t, 0 ) + g a ( t ) x a − 1 2 E ab ( t ) x a x b + · · · g a ( t ) = ∂ a U ext ( t, 0 ) = body’s CM acceleration E ab ( t ) = − ∂ ab U ext ( t, 0 ) = tidal tensor The tidal tensor is not determined by the field equations restricted to N ; it provides a characterization of a generic tidal environment. Eric Poisson Tidal deformation and dynamics of black holes

Introduction Newtonian tides Relativistic tides Conclusion 2. Describe the body’s deformation The deformation of the body is measured by its quadrupole-moment tensor, x a x b U body = M r + 3 2 Q ab + · · · r 5 To relate Q ab to E ab requires formulating a model for the body and solving the structure equations (eg, equations of hydrostatic equilibrium) for the perturbed configuration. Generically, Q ab ( t ) = − 2 3 k 2 R 5 E ab ( t − τ ) k 2 = gravitational Love number τ = viscous delay The Love number k 2 and viscous delay τ depend on the details of internal structure, composition, dissipation mechanism, etc. Eric Poisson Tidal deformation and dynamics of black holes

Introduction Newtonian tides Relativistic tides Conclusion 3. Deduce dynamical consequences The tidal interaction leads to an exchange of angular momentum between the body and the external matter. For a tidal environment in a state of rigid rotation of angular frequency Ω tide around the body’s rotation axis, E 11 = E 0 + E 2 cos(2Ω tide t ) , E 12 = E 2 sin(2Ω tide t ) , E 13 = 0 E 22 = E 0 − E 2 cos(2Ω tide t ) , E 23 = 0 , E 33 = − 2 E 0 Tidal torquing dS dt = 8 3( k 2 τ ) R 5 ( E 2 ) 2 (Ω tide − Ω body ) Ω body = body’s intrinsic angular velocity The body spins down when Ω tide < Ω body ; it spins up when Ω tide > Ω body . Eric Poisson Tidal deformation and dynamics of black holes

Introduction Newtonian tides Relativistic tides Conclusion 4. Specify the tidal environment In order to apply the general theory, E ab must be specified. This requires leaving the domain N and identifying the source of the tidal environment. When the body is a member of a two-body system with a companion of mass M ′ at position r ( t ) , M ′ U ext ( t, x ) = | x − r ( t ) | and E ab = − ∂ ab U ext ( t, 0 ) is easily computed. For a system in circular motion with orbital radius r , E 0 = − M ′ E 2 = − 3 M ′ � M + M ′ 2 r 3 , 2 r 3 , Ω tide = = Ω orbital r 3 This can then be substituted into the general formulae. Eric Poisson Tidal deformation and dynamics of black holes

Introduction Newtonian tides Relativistic tides Conclusion Relativistic tides: Setting and assumptions The same as in the Newtonian theory. N Eric Poisson Tidal deformation and dynamics of black holes

Introduction Newtonian tides Relativistic tides Conclusion 1. Characterize the tidal environment The metric of a slowly rotating, nearly spherical body is perturbed by a remote distribution of matter, external to the domain N , g αβ = g unpert + p body + p ext αβ αβ αβ p body � � δG αβ = 8 πδT αβ p ext � � δG αβ = 0 The asymptotic behaviour of p ext αβ is specified by two gauge invariant tidal tensors , E ab ( t ) and B ab ( t ) . These can be related to the (electric and magnetic) components of the Weyl tensor evaluated at the edge of N . The tidal tensors are not determined by the field equations restricted to N . Eric Poisson Tidal deformation and dynamics of black holes

Introduction Newtonian tides Relativistic tides Conclusion 2. Determine the body’s deformation The perturbation p body must be continuous across the surface of a αβ material body; this condition determines the relativistic Love numbers 2 and k mag k el , which are gauge invariant. 2 In the case of a black hole, regularity at the event horizon requires p body 2 = k mag = 0 , so that k el = 0 ; the gravitational Love numbers of a αβ 2 black hole are zero. Metric of a tidally deformed, slowly rotating black hole g 00 = − 1 + 2 M r � 2 � E ab x a x b − � � M 2 r 2 − 4 M 3 r 3 + 2 M 4 1 − 2 M χ∂ φ E ab x a x b − r r 4 � � χ p B pa x a − � � M 2 M 3 2 M 2 M 3 2 M − 34 r + 32 r 3 − 8 χ � a B bc � x a x b x c + 5 5 r 2 3 r 4 where χ a = S a /M 2 ≪ 1 is the black hole’s dimensionless spin. Eric Poisson Tidal deformation and dynamics of black holes

Introduction Newtonian tides Relativistic tides Conclusion 3. Deduce dynamical consequences The tidal torquing of a black hole can be calculated on the basis of well-known horizon flux formulae. [Teukolsky, Press (1974); Poisson (2004)] For a tidal environment in a state of rigid rotation of angular frequency Ω tide around the black hole’s rotation axis, E 11 = E 0 + E 2 cos(2Ω tide v ) , E 12 = E 2 sin(2Ω tide v ) , E 13 = 0 E 22 = E 0 − E 2 cos(2Ω tide v ) , E 23 = 0 , E 33 = − 2 E 0 B 11 = 0 , B 12 = 0 , B 13 = B 1 cos(Ω tide v ) , B 22 = 0 , B 23 = B 1 sin(Ω tide v ) , B 33 = 0 . Eric Poisson Tidal deformation and dynamics of black holes

Introduction Newtonian tides Relativistic tides Conclusion 3. Deduce dynamical consequences: continued Tidal torquing of a black hole dS dv = 128 � 2 + 1 1 + 9 �� 45 M 6 E 2 4 B 2 16 E 2 2 + B 2 � � � χM Ω tide Ω tide − Ω H 1 2 Comparison with the Newtonian expression dS dt = 8 3( k 2 τ ) R 5 ( E 2 ) 2 (Ω tide − Ω body ) reveals that ( k 2 τ ) R 5 = 16 15 M 6 for a black hole. With R ∼ M , this implies that ( k 2 τ ) ∼ M . Eric Poisson Tidal deformation and dynamics of black holes

Introduction Newtonian tides Relativistic tides Conclusion 4. Specify the tidal environment The determination of E ab ( t ) and B ab ( t ) requires leaving the domain N and incorporating the external matter that sources the tidal field. This must be done in general relativity, taking into account the nonlinearity of the field equations. To make progress it is useful to assume that the mutual gravity between the black hole and the external matter is weak , so that the metric can be expressed as a post-Newtonian expansion. Gravity is still strong near the black hole, but at a safe distance the metric becomes post-Newtonian. Eric Poisson Tidal deformation and dynamics of black holes