Gauge-invariant quantity. Monday, June 23, 2014

Transcription

1 Gauge-invariant quantity U

2 Topics that will be covered Gauge-invariant quantity, U, (reciprocal of the red-shift invariant, z), the 1 st order (in mass-ratio) change in u t. For eccentric orbits it can be generalized to U. How to calculate it in a radiation gauge? What has already been accomplished & what is in progress? What can be done with it? Dissipative part (fluxes) and its overlap with pn.

3 U 1. Useful quantity to check the computations done in different gauges. 2. Relations between coefficients in pn-expansion of U, and those of binding energy and angular momentum for the binary system. 3. Used in EOB calibration. 4. ISCO shift (from U and its derivatives) is used as a reference point for comparison with analytical and numerical methods.

4 U 1. Useful quantity to check the computations done in different gauges. 2. Relations between coefficients in pn-expansion of U, and those of binding energy and angular momentum for the binary system. 3. Used in EOB calibration. 4. ISCO shift (from U and its derivatives) is used as a reference point for comparison with analytical and numerical methods.

5 U 1. Useful quantity to check the computations done in different gauges. 2. Relations between coefficients in pn-expansion of U, and those of binding energy and angular momentum for the binary system. 3. Used in EOB calibration. 4. ISCO shift (from U and its derivatives) is used as a reference point for comparison with analytical and numerical methods.

6 U 1. Useful quantity to check the computations done in different gauges. 2. Relations between coefficients in pn-expansion of U, and those of binding energy and angular momentum for the binary system. 3. Used in EOB calibration. 4. ISCO shift (from U and its derivatives) is used as a reference point for comparison with analytical and numerical methods.

7 Topics that will be covered Gauge-invariant quantity, U, (reciprocal of the red-shift invariant, z), the 1 st order (in mass-ratio) change in u t. For eccentric orbits it can be generalized to U. How to calculate it in a radiation gauge? What has already been accomplished & what is in progress? What can be done with it? Dissipative part (fluxes) and its overlap with pn.

8 U for circular orbits or U for eccentric orbits uses the renormalized metric perturbation, h R αβ, dotted with background geodesic 4-velocity. u α u β (g αβ + h αβ )=1 u α =[u t 0 + u t 1 + O(µ 2 )]k α U = u t 1 = u t 0 H H := 1 2 hr αβu α 0 u β 0 U is the ratio of radial periods with respect to t and τ and involves H.

9 Metric perturbation (h αβ ) in a modified radiation gauge System of 10 coupled, 2 nd order PDEs in Lorenz gauge Here we only need to solve one, separable, 2 nd order PDE. The PDE for ψ 0 or ψ 4. h αβ n α =0 h =0 ORG h αβ α =0 h =0 IRG

10 Metric perturbation (h αβ ) in a modified radiation gauge Radiative part of the full h αβ is extracted from the perturbed Weyl scalars, ψ 0 / ψ 4. ψ 0 = Ṙαβγδ α m β γ m δ ψ 4 = Ṙαβγδn α m β n γ m δ ( α,n α,m α, m α ) make a null-tetrad. ( α,n α ) are real, outgoing- and ingoing-null vectors. m α is a complex null vector orthogonal to ( α,n α ). These perturbed Weyl scalars, ψ 0 / ψ 4 are invariant under gauge transformations and infinitesimal tetrad rotations.

11 Metric perturbation (h αβ ) in a modified radiation gauge Radiative part of the full h αβ is extracted from the perturbed Weyl scalars, ψ 0 / ψ 4. ψ 0 = Ṙαβγδ α m β γ m δ ψ 4 = Ṙαβγδn α m β n γ m δ And the non-radiative part, which corresponds to the change in mass (δm) and angular-momentum (δj) of spacetime, is then added in a convenient gauge.

12 Teukolsky equation The equation that describes the dynamical perturbation, the Teukolsky equation, has the form OT(h) =SE(h). And h αβ is what we want to extract.

13 Teukolsky equation OT(h) =SE(h) ψ 0 or ψ 4 ( ψ for brevity)

14 Teukolsky equation OT(h) =SE(h) Einstein operator acting on h =8π T µν

15 Teukolsky equation OT(h) =SE(h) 2 nd order derivative operator acting on T µν

16 Teukolsky equation OT(h) =SE(h) 2 nd order derivative operator acting on ψ

17 Teukolsky equation Newman-Penrose equations (Bianchi identities): Derivative operators acting on Weyl scalars = Derivative operators acting on Ricci tensor A combination of them gives us: Derivative operators acting on ψ = Derivative operators acting on R µν T µν OT(h) =SE(h)

18 Separability of the Teukolsky equation ψ = R(r) e iωt S(θ) e imφ s ψ =,m,ω sr,m,ω (r) e iωt ss aω,m(θ) e imφ Ordinary 2 nd order ODE for the radial part and the angular part The solution to both these equations can be written as a sum over known analytical functions.

19 Finding h from ψ Theorem: Suppose SE = OT holds, and suppose Ψ satisfies O Ψ = 0. If E is self-adjoint, then S Ψ satisfies E(f) = 0.

20 Finding h from ψ Theorem: Proof: Suppose SE = OT holds, and suppose Ψ satisfies O Ψ = 0. If E is self-adjoint, then S Ψ satisfies E(f) = 0. Taking adjoint of SE = OT, gives us E S = T O ES = T O If O Ψ = 0, then E(S Ψ) = 0, i.e., h = S Ψ

21 Finding h from ψ How do we connect the solution to the Teukolsky equation, ψ or T (h), to this Ψ?

22 Finding h from ψ How do we connect the solution to the Teukolsky equation, ψ or T (h), to this Ψ? Lets substitute S Ψ back into the Teukolsky equation, SE(S Ψ)=OT (S Ψ) 0=O[T S Ψ]

23 Finding h from ψ How do we connect the solution to the Teukolsky equation, ψ or T (h), to this Ψ? Lets substitute S Ψ back into the Teukolsky equation, SE(S Ψ)=OT (S Ψ) 0=O[T S Ψ] TS maps solutions of O Ψ =0toOψ ψ =0.

24 Finding h from ψ How do we connect the solution to the Teukolsky equation, ψ or T (h), to this Ψ? Lets substitute S Ψ back into the Teukolsky equation, SE(S Ψ)=OT (S Ψ) 0=O[T S Ψ] TS maps solutions of O Ψ =0toO ψ =0. ψ = TS Ψ h = S Ψ

25 Finding h from ψ How do we connect the solution to the Teukolsky equation, ψ or T (h), to this Ψ? Lets substitute S Ψ back into the Teukolsky equation, SE(S Ψ)=OT (S Ψ) 0=O[T S Ψ] TS maps solutions of O Ψ =0toO ψ =0. ψ = TS Ψ h = S Ψ Intermediate Hertz potential

26 Finding h from ψ How do we connect the solution to the Teukolsky equation, ψ or T (h), to this Ψ? Lets substitute S Ψ back into the Teukolsky equation, SE(S Ψ)=OT (S Ψ) 0=O[T S Ψ] TS maps solutions of O Ψ =0toO ψ =0. ψ = TS Ψ h = S Ψ Intermediate Hertz potential More in Cesar s talk

27 Finding h from ψ How do we connect the solution to the Teukolsky equation, ψ or T (h), to this Ψ? Lets substitute S Ψ back into the Teukolsky equation, SE(S Ψ)=OT (S Ψ) 0=O[T S Ψ] TS maps solutions of O Ψ =0toO ψ =0. ψ = TS Ψ h = S Ψ Intermediate Hertz potential First step: Solve for ψ (Teukolsky equation).

28 Second step: Invert to find Ψ ψ = TS Ψ h = S Ψ Intermediate Hertz potential In frequency-domain the operator TS are almost the same as the ones in Teukolsky-Starobinsky identities For circular/spherical orbits, the inversion is algebraic. Radial mode of Ψ = constant radial mode of ψ

29 Second step: Invert to find Ψ ψ = TS Ψ h = S Ψ Intermediate Hertz potential In frequency-domain the operator TS are almost the same as the ones in Teukolsky-Starobinsky identities For generic orbits, the formalism for inversion in frequency-domain has been developed by A. Ori. And its application is in progress for eccentric, equatorial orbits in Kerr (MVD Meent and AG Shah)

30 Third step: Apply the operator S on Ψ to recover the radiative h αβ ψ = TS Ψ h = S Ψ Intermediate Hertz potential

31 ψ = TS Ψ h = S Ψ Intermediate Hertz potential Fourth step: Add to this the h αβ that corresponds to the change in mass and angular momentum of the spacetime.

32 ψ = TS Ψ h = S Ψ Intermediate Hertz potential Fourth step: Add to this the h αβ that corresponds to the change in mass and angular momentum of the spacetime. Fifth step: Subtract the singular piece and sum over all the multipoles.

33 U or U Schwarzschild spacetime Regge- Wheeler- Zerilli Lorenz Gauge Radiation Gauge Circular Eccentric

34 U or U Schwarzschild spacetime Regge- Wheeler- Zerilli Lorenz Gauge Radiation Gauge Circular Eccentric

35 U or U Schwarzschild spacetime Regge- Wheeler- Zerilli Lorenz Gauge Radiation Gauge Circular Eccentric

36 U or U Schwarzschild spacetime Regge- Wheeler- Zerilli Lorenz Gauge Radiation Gauge Circular Eccentric

37 U or U Kerr spacetime Lorenz Gauge Radiation Gauge Circular Eccentric

38 U or U Kerr spacetime Lorenz Gauge Radiation Gauge Circular Eccentric

39 U or U Kerr spacetime Lorenz Gauge Radiation Gauge Circular Eccentric

40 Extracting pn coefficients of U or U Status Circular Schwarzschild Eccentric Schwarzschild Circular Kerr Eccentric Kerr

41 Extracting pn coefficients of U or U Status Circular Schwarzschild Eccentric Schwarzschild Circular Kerr Eccentric Kerr

42 Extracting pn coefficients of U or U Status Circular Schwarzschild Eccentric Schwarzschild Circular Kerr Eccentric Kerr

43 Extracting pn coefficients of U or U Status Circular Schwarzschild Eccentric Schwarzschild Circular Kerr Eccentric Kerr

44 ISCEO shift z =( U) 1 and its derivatives are used to calculate the conservative O(µ/M) shift of the innermost stable circular, equatorial orbits (more in Takahiro s and Soichiro s talks).

45 Ω U is the gauge-invariant shift in u t for a fixed Ω. One can also calculate the gauge-invariant shift in Ω, Ω, for a fixed U. It looks like we know the conservative shift in phase (at least for circular orbits).

46 Ω Question: Can one develop a formalism where the knowledge of H (and its derivatives) is enough to do the orbital evolution? (work in progress by Kyoto-group) If possible, this avoids a lot of lengthy and complex calculations that are involved in going from h to SF. (coupling, different extensions, changing harmonics...)

47 Fluxes A lot of work has been done on calculating very high-order pn expansion of fluxes Case Progress Schwarzschild circular Schwarzschild eccentric Kerr circular Kerr eccentric/inclined/ generic