Wave extraction using Weyl scalars: an application

Transcription

1 Wave extraction using Weyl scalars: an application In collaboration with: Chris Beetle, Marco Bruni, Lior Burko, Denis Pollney, Virginia Re Weyl scalars as wave extraction tools The quasi Kinnersley frame The characteristic formulation of Einstein's equations Numerical results Future Developments

2 Weyl scalars as wave extraction tools (I) Weyl scalars can be computed once a set of two real null vectors l, n and two complex conjugate null vectors is chosen, such that l n = 1 and m m =1. The expression for Weyl scalars are 0 = C abcd l a m b l c m d 1 = C abcd l a n b l c m d 2 = C abcd l a m b m c n d 3 = C abcd l a n b m c n d = C abcd n a m b n c m d Ingoing transverse contribution Ingoing longitudinal contribution Coulombian contribution Outgoing longitudinal contribution Outgoing transverse contribution They reflect all the degrees of freedom of the Weyl tensor They are coordinate independent As a general statement, the scalars are given the physical meaning written on the right hand side, however... Weyl scalars do depend on the null vector choice m, m

3 Weyl scalars as wave extraction tools (II) The four null vectors can be rotated, keeping the normalization conditions, in three possible way: 1) Null rotation of l (Type I) n unchanged I =a a a a 3 2) Null rotation of n (Type II) l unchanged II = 3) Spin/Boost (Type III) l Al n A 1 n III =A 2 e 2i We define : Tetrad : A specific choice of the four null vectors. Frame : A set of tetrads connected by a spin/boost (type III) transformation

4 Weyl scalars as wave extraction tools (III) The algebraic properties of space time under consideration can be obtained by looking at principal null directions: I =a a a a 3 =0 Two typologies are of interest in numerical simulations: 1) Type I : All four principal null directions are distinct 2) Type D : The principal null directions coincide in couples Schwarzschild and Kerr are type D space times In numerical simulations we normally deal with type I space times

5 Weyl scalars as wave extraction tools (IV) How do we find principal null directions? I =a a a a 3 =0 Introducing the reduced variable z= 0 a 1 I 4 = z4 6 Hz 2 4 Gz K =0 H, G and K are functions 3 0 of the scalars The solutions can be written as z 1 = 2 ; z 2 = 2 ; z 3 = 2 And the additional variables are given by: ; z 4 = 2 2 = H ; 2 = H ; 2 = H ; ; The value of I = z z 1 z z 2 z z 3 z z after a rotation can now be written as

6 Weyl scalars as wave extraction tools (V) By letting l and n coincide with the two repeated principal null directions (Kinnersley frame), Teukolsky found a perturbative expression for 0 and. Ce i t r Ce i t r 0 = r 5 0 = For outgoing r For ingoing t r t r Ce i radiation Ce i radiation = r 4 = r 5 These expressions are not only coordinate independent, but also tetrad independent at first order This means that, in every tetrad infinitesimally close to the Kinnersley tetrad, is really related to the outgoing gravitational wave contribution Moreover, using an infinitesimal tetrad transformation, 1 and 3 can be set to zero, thus corresponding to gauge contributions

7 The quasi Kinnersley frame (I) In principle, in order to calculate the Weyl scalars, we can use the expression of the Kinnersley tetrad, given by: m =[ l =[ r 2 a 2 n =[ r 2 a 2,1,0, a ] 2, 2, 0, a 2 ] ia sin 2 r ia cos, 0, 1 2 r ia cos, i 2 sin r ia cos ] where =r 2 2 Mr a 2 and =r 2 a 2 cos 2 This tetrad expression depends on the parameters of the black hole, which, in a numerical simulation, are unknown. We need a way to identify the right frame which relies on more general properties of type D space times, and not on the particular parameters of our specific space time.

8 The quasi Kinnersley frame (II) What is a general property of a type D space time, independent of the specific parameters? Principal null directions coinciding in pairs! a 1 Type I a 3 Type D a 3 =a 4 a a 1 =a 2 2 a 4 If we manage to orient the l and n vector in such a way that they are squeezed by the principal null directions... l a 1 a 3 n a 2 a 4

9 The quasi Kinnersley frame (III) Transverse frame : a frame where and vanish 1 3 How do transverse frames see principal null directions? I =a a 2 2 =0 a 1 = a 2 = a 1 a 3 = a 4 = a 3 In the limit of type D space time and a 2 will coincide, and the same a 3 and a 4. But because in a transverse frame we always see the minus sign, it means that the modulus of the parameter must tend to zero in the D limit. This means that the l and n vector of a transverse frame will converge to the repeated principal null directions in the type D limit. a 1

10 The quasi Kinnersley frame (IV) Our task is then to find transverse frames Type I space time : there are three transverse frames. Only one of them is the quasi Kinnersley frame. The geometrical interpretation is clear. How do we find transverse frames? Two methodologies have been developed 1) Transverse frames are found as non null self dual eigenforms of the Weyl tensor. ab 2) We calculate Weyl scalars in a fiducial tetrad, and then perform two tetrad rotations to get to the transverse frames. The parameters for such rotations are then calculated algebraically.

11 The quasi Kinnersley frame (V) I will follow here the second approach: starting from a fiducial tetrad, we need to perform a type I rotation and a type II rotation in order to set 1 = 3 = Type I rotation with parameter a 0 I 1 I 2 I 3 I I Type II rotation with parameter b II II II II II II = 3 II =0 We set the final values thus obtaining two equations for the two parameters a and b.

12 The quasi Kinnersley frame (VI) Some mathematical calculations lead to the final simple equation for the first parameter a: P 1 a 6 P 2 a 5 P 3 a 4 P 4 a 3 P 5 a 2 P 6 a P 7 =0 where: P 1 = P 2 = P 3 = P 4 = P 5 = P 6 = P 7 = After the parameter a for the type I rotation is found, and the Weyl scalars are computed, the parameter for the type II rotation is simply given by b= 3 I I

13 The quasi Kinnersley frame (VII) The equation for the parameter a looks difficult to solve algebraically, however, using some physical properties, we have been able to get the six solutions by using again the reduced variable z= 0 a 1 and the functions of the scalars,, which we introduced for the first time to compute the principal null directions. The solutions are: z 1,2 = ± z 3,4 = ± z 5,6 = ± We obtain six solutions instead of three because of the l n degeneracy. l Tr Only one couple corresponds to the quasi Kinnersley frame. The parameter of the second rotation is then trivially computed. l n Tr

14 The characteristic formulation of Einstein's equations (I) Motivations: We want to test our results on the quasi Kinnersley frame in a numerical application where the determination of such a frame is easier. Moreover, we want to identify easily the quasi Kinnersley tetrad. Once the weyl scalars are computed in the right frame, we would like to compare our results about the gravitational radiation emitted with another well tested wave extraction tool to confirm the validity of our approach. The characteristic formulation of Einstein's equation (Bondi formalism) fullfils both motivations, thus representing an optimal test bed for our analytical results.

15 The characteristic formulation of Einstein's equations (II) Outline of the formalism: The space time is foliated with null hypersurfaces A time like curve parametrized by a variable v is chosen as origin of coordinates Each null foliation is given by setting v = const On each null hypersurface, every null ray is parametrized by a radial variable r (luminosity distance) Two angular variables label each ray on the null hypersurface.,

16 The characteristic formulation of Einstein's equations (III) With this choice of coordinates, the metric looks like: ds 2 = [ 1 2 M r e 2 U 2 r 2 e 2 ]dv 2 2 e 2 dv dr Ur 2 e 2 dv d r 2 e 2 d 2 e 2 sin 2 d 2 The system is in axisymmetry and non rotating v,r, U v,r, M v,r, v,r, are the four unknown functions which we want to find solving Einstein's equations. Einstein's equations translate into the set of equations:,r =H U,r =H U, M,r =H M,,U,rv =H,,U, M Advantages Only one evolution equation Unconstrained initial data for Natural hierarchy for numerical integration Disadvantages Code crashes if caustics are formed

17 The characteristic formulation of Einstein's equations (IV) Wave extraction in the Bondi formalism From a linear study of Einstein's equations in the Bondi formalism it is possible to understand the radial fall offs of all the unknown metric functions. It turns out that: 1 r ; 1 r 2 U 1 r ; M M SCH 1 r This means that the metric, at large distances from the source, becomes ds 2 dv 2 2 dv dr 2 r 2 d 2 sin 2 d 2 By rescaling the angular basis vectors, it can be shown that we are already in the TT gauge, once we set: h =2 ; h =0 We have found that in the linear regime, the function directly represents the wave contribution in the TT gauge

18 The characteristic formulation of Einstein's equations (V) How do we pick up a quasi Kinnersley tetrad? The Kinnersley tetrad for a Schwarzschild space time looks, in our coordinates v,r,, l 2 r =[ r 2 M,1,0,0] Using the well known asymptotic limit of the Bondi functions it is possible to construct a general tetrad which converges to the Kinnersley one. n =[0, r 2 M, 0,0] r m 1 =[0,0, 2 r, i 2 r sin ] The final tetrad looks like: l 2 =[ [ 1 2 M /r e 4 U 2 r 2 e 2 ],e 4,0,0] n 1 =[0, 2[ 1 2 M /r e 2 U 2 r 2 e 2 ],0,0] m =[0, 1 2 r e, rue 2 2, i 2 r sin e ]

19 Numerical results (I) We perform numerical simulations by integrating the set of four equations. We first set M=M SCH of the black hole The initial value for is then chosen to be 0,r, = 2 e r r c 2 2 Y 2 lm where is the amplitude of the perturbation, is the center of the wave packet, and its variance The hypersurfaces equations are integrated from the outside boundary (where all the quantities are zero) inwards, until the excision region is hit. In our simulations we set: =0.15 r c =3.0 =1.0 By choosing the expression for the function Y, we can impose the angular structure of the initial perturbation

20 Numerical results (II) Convergence and radial fall offs We show convergence for and at time v=80. Radial fall offs are shown by plotting the variables and at two different times. 0 2 =r 3 2 =r

21 Numerical results (III) News vs Weyl scalars We have seen that is directly related to the news function. We also know that in the quasi Kinnersley tetrad must be related to the wave contribution. So we expect that there must be a relationship between these two quantities. In the linearized regime QKT can be written as: QKT = R v v ir v v And using the expression which holds in the linearized regime: R v v = 1 2 h 2 v 2 we get : QKT = h v 2 i 2 h v 2 Comparing with the result which relates h in the TT gauge we get : QKT = 2 v 2 to the function

22 Numerical results (IV) News functions

23 Future developments (I) We are trying to find the final value of 0 and in the quasi Kinnersley tetrad by performing tetrad rotations whose parameters are known functions of the scalars we start with. We have determined the parameters required to perform the type I and type II rotations. In this sense, and following our notation, we have identified the quasi Kinnersley frame. However, there are still two degrees of freedom related to the type III (spin/boost) rotation. This means that we still have to identify the particular transformation which lets us pick up the right quasi Kinnersley tetrad out of the quasi Kinnersley frame.

24 Future developments (II) We want to get an algebraic expression for the parameter related to this transformation. Having obtained this parameter, we will be able to perform the three rotations numerically, and come out with an expression for which is the real outgoing gravitational contribution. We will also be able to compute analytically the value of in the final quasi Kinnersley tetrad at least in the linearized regime.