The Quadrupole Moment of Rotating Fluid Balls

Transcription

1 The Quadrupole Moment of Rotating Fluid Balls Michael Bradley, Umeå University, Sweden Gyula Fodor, KFKI, Budapest, Hungary Current topics in Exact Solutions, Gent, 8- April 04 Phys. Rev. D 79, (009)

2 Outline of talk. Preliminaries. Hartle s formalism for slowly rotating objects 3. The field equations 4. Multipole moments 5. The matching procedure 6. Numerical integration 7. Some numerical results 8. Post-Newtonian limit 9. Conclusions

3 . Preliminaries Ω I p = 0 II Axisymmetric stationary metric: dr ds = A ( r, θ) dt C ( r, θ) r dθ + sin θ dϕ ω( r, θ) dt B ( r, θ) I Rigid rotation: Perfect fluid: II Vacuum ( ) µ t t µ ν σ τσ ( µν ; ) τ σ u = ( u,0,0, Ωu ) u h h = 0 (shear) T = ( ρ+ p) u u pg µν µ ν µν (For review, see e.g. Perjés)

4 . Hartle s scheme for slow rotation (Ap. J. 50 (967) 005) To second order in Ω, the rotational parameter, the metric can be written as: (+ m) dr ds = (+ h) A dt (+ k) r [ dθ + sin θ( dϕ ωdt ) ] B 0:th order functions: A( r ), B ( r ) :st order function:ω. Regularity at origin + asymptotic flatness :nd order functions: h( r, θ ), m( r, θ ), k( r, θ ) ω= ω( r)

5 (i) Equations for h, k and m separate with (ii) For l > no inhomogeneous terms with (iii) Reflection symmetry = l= 0 ω h h ( r) P (cos θ ) l l etc. h( r, θ ) = h ( r) + h ( r) P (cos θ ) 0 m( r, θ ) = m ( r) + m ( r) P (cos θ ) 0 k( r, θ ) = k ( r) P (cos θ ) See also MacCallum, Mars & Vera: Phys. Rev. D. 75 Mars: Class. Quant. Grav. 005 Reina & Vera: 04 (Hartle & Thorne, Chandrasekhar & Miller, Berti et. al., Review on rotating relativistic stars: Stergioulas lrr-003-3)

6 3. Field equations a) Fluid region Perfect fluid: T = ( ρ+ p) u u pg µν µ ν µν Convenient to redefine dependent variables as: ν z dν A = e, = r +, B d r h = hɶ e, m = mɶ e, k = kɶ e ν ν ν

7 0:th order: dz dr = 4 Br ( zb z B ) ( G ) = G 0:th order density and pressure: d rb ( ) ρ0 = r dr p0 = zb B r ( ) If equation of state, ρ = f ( p ) 0 0, given: db dr = + Br ( ρ ) 0r B

8 Equation of state: ρ / γ p = d p + d + d 3 pc Including: Newtonian polytropes: Relativistic polytropes: Linear: Incompressible fluids: d d = d3 = 0 =, d = 0 γ 3 d = 0 d = d = 0 ( p= Cρ γ ) ( p= Cn γ )

9 :st order: d B dω 4 d B r 4r 3 + ω = 0 G 0 = 0 dr A dr dr A ( ) 3

10 :nd order: z, B, ω, h ɶ, k ɶ, mɶ Closed system for

11 b) Vacuum metric With p= ρ = 0 : h 0 = q = 0 q c 0 = = M c Asymptotically flat spacetime Petrov type D, Kerr Mass to second order

12 4. Multipole Moments Quadrupole moment Newtonian : Vacuum region Kerr g q = = c 0 c 6 4 M M a + M c P (cos θ ) M 5 = r r 00 3 Quadrupole moment: 6 Q = Q = Q = 4M a + M c Q Q 6 M c 4 = Kerr 5 a

13 Relativistic Multipole Moments (i) Mɶ = M Λ where Λ is a single point ( ) (ii) hɶ =Ωɶ h ij ij (iii) Ω ɶ =Ω ɶ = 0, Dɶ Dɶ Ω ɶ = hɶ Λ, i Λ i j Λ ij Λ (Geroch, Hansen, Thorne) Asymptotically flatness: A 3-space (M,h) is said to be asymptotically flat if it can be conformally mapped to a 3-space ( M ɶ, h ɶ ) with the following properties: Ω ɶ r

14 Ex: flat space: Change coordinates: x y z x =, y =, z = x + y + z x + y + z x + y + z ds = dx + dy + dz = ( x + y + z ) ( dx + dy + dz ) Conformal metric: ds =Ω ɶ ds = ( dx + dy + dz ) with ɶ Ω= x + y + z = ( x + y + z ) = Ω ɶ =Ω ɶ = 0, D ɶ D ɶ Ω ɶ = h ɶ Λ, i Λ i j Λ ij Λ r

15 Stationary case: a From timelike Killing vector K define the a f = K K a scalars and ψ b : ψ = ε K K, a abcd c; d (gradient due to field equations). E Gravitational potential: ξ = where Ernst potential E= f + iψ + E ɶ ɶ / / ξ =Ω ξ ( φ, Ω ɶ r) r Metric on 3-space: hab = fgab+ KaKb

16 Multipole tensors: i Pɶ ( x ) = ɶ ξ, Pɶ ( x ) = ɶ ξ (0) i () j Pɶ ( x ) = Dɶ Pɶ n(n ) Rɶ Pɶ n+ n+ n 3 n For axisymmetric spacetimes the multipole moments are given in terms of the scalar moments (Hansen, Beige): = ɶ n! ( n) k Pn Pk k... k n n n, j ( n+ ) i ( n) ( n ) k k... k < k k k... k > < k k k... k > kn... where n is the unit vector along the axis of symmetry. Λ

17 Algorithms for calculating the multipole moments developed by, e.g., Fodor & Perjés and Bäckdahl & Herberthson. Fodor-Perjés method: Transform metric to ds = f ( dt ɶ ωdϕ) f e γ ( dρ + dz ) + ρ dϕ with 3 = Ar + h + h + k h + k ( ) ρ sinθ 0 sin θ z ( r M) h ( h k m ) = cosθ sin θ + + r A h0, r ( h, r + k, r) sin θ a am f = KaK = (+ h) A r ω sin θ, ψ = cosθ r

18 Define: Suitable conformal factor: ρ z ρ =, z = ρ + z ρ + z Ω= ɶ r = ρ + z ɶ ξ = ξ r First moments (up to n=3) given by ξ ( ρ = 0) = n 0 (Fodor, Hoenselaers & Perjés): = ɶ m z n n In this case: ɶ 6 ξ = M c + imaz M a + M c z 5 4 Hence: 6 M = M c, J = Ma, Q= M a + M c 5 4

19 5. Matching procedure Matching surface S Fluid region: n p = 0 To zeroth order: p0( r ) = 0 To second order: p( r) = p ( r) + p ( r) + p ( r) P (cos θ ) = r = r + ξ + ξ P (cos θ ) 0

20 Vacuum region: ( v) n r= r + η + η P (cos θ ) 0 where η and η determined 0 from matching conditions

21 Matching conditions On matching surface S: (Darmois, Israel) Projected metrics: ds = ds S ( v) S a) Proj. :nd fund. forms: K S = K ( v) S b) where K K dx dx h h n dx dx h n n +δ µ ν τ σ µ ν µν µ ν ( τ ; σ ) b) No shells of matter on matching surface Pick out zero pressure surface ν ν ν µ µ µ

22 Coordinate changes in fluid region: ϕ ϕ+ω t (to adjust to asymptotically non-rotating observer) t c ( + c ) t 4 3 Solution of matching conditions:

23

24 The vacuum metric above is general enough for the matching to rigidly rotating perfect fluids, but is in general not asymptotically flat. (See Mars & Senovilla, Mod. Phys. Lett. A 3, 509 (998) on overdetermined matching conditions) Note that equation of state not used.

25 6. Numerical integration Regularity conditions: Smoothness at centre and mirror and axial symmetry B= + b r + b r z= + z r + z r ω= ω0+ ωr + ωr +... (0) () () 4 hɶ = h + h r + h r +... mɶ = m + m r +..., kɶ = k + k r +... (0) () (0) () h h () Field equations (without equation of state) give four independent constants = Boundary conditions at r = 0 : b = ρ, z = p, ω, h 6 0c 0c 0

26 New dependent variables: β, ζ, ωɶ, yɶ, hˆ, kˆ, mˆ dω B= + βr, z= + ζ r, ω= ω0+ rɶ ω, = ɶ ω+ yɶ, dr hɶ = hr ˆ, kɶ = r( kr ˆ hˆ ), mɶ = r( mr ˆ hˆ ) Defined to avoid singular terms ˆm can be solved for algebraically Closed first order system for β, ζ, ωɶ, yɶ, hˆ, kˆ Numerical method: 4:th order Runge-Kutta

27 Scaling invariance: ω0 γω0 ˆ ˆ ˆ ˆ ω, yɶ γω, γ yɶ ; h, k, mˆ γ h, γ k, γ mˆ ; β, ζ, r β, ζ, r Rescaling of r: r αr induces a change of the parameters d d p c d d, d, d3, p0c,, α α α of the equation of state: 3 0 Only the central pressure need be varied when scanning the solution space for given equation of state. One of the constants in equation of state can be fixed

28 Requirement of asymptotic flatness, corresponding to: q = 0 h = h ( ρ ( p ), p, ω ) 0 0c 0c 0 Since both the differential equations for k and h and the expressions for q and c are linear in k and h, the correct value for h can easily be found by considering linear combinations of particular and homogeneous solutions: q = q + Cq p h

29 7. Some numerical results -: Incompressible case (Chandrasekhar & Miller) 3: Linear 4: Newtonian polytropes 5: Relativistic polytropes Units: c =, 8π G= If r in units of α :meter d ρ SI = α ρ kg/m 5 3 = α kg/(ms wh = ρ 4 xsi x ) ere x c, p, d, d3 = d SI

30 Incompressible case

31 Incompressible case

32 ρ = d p+

33 The ρ = d p + case: For metrics with d < these metrics approaches anti-de Sitter ρ = p 0c 0c in the central region when 3 = d Q and r M and, but they become infinitely large. Q For the Whittaker metric, with : p 0c ρ = 3p+ µ r 4κ κ M π / µ 0 and + r where κ is a parameter of the Whittaker metric. 0 d ( )

34 Newtonian polytropes p ρ = p c / γ

35 Relativistic polytropes p p ρ = + γ pc / γ

36 7. Post-Minkowskian limit In J.A. Cabezas, J. Martín, A. Molina & E. Ruiz, Gen. Rel. Grav. 39, 707 (007) and J. Martín, A. Molina & E. Ruiz, Class. Quantum Grav. 5, 0509 (008) the deviations from the Kerr metric are investigated for incompressible fluids and Newtonian polytropes respectively in the post-minkowskian limit.. Global harmonic coordinates. Lichnerowicz matching conditions Equivalence of Lichnerowicz and Darmois-Israel: Cuchí et. al.

37 Expansion in 7. Post-Minkowskian limit M GM λ= = r rc A= + A + A +..., B= + B + B +... h = h + h +..., ω= ω + ω +... / 3/ ρ = ρ + ρ +..., p= p + p +... Virial theorem: < p> λ< ρ > Balance between gravitational and centrifugal forces: ω< Also field equations give p = 0 and ω λ and also ω / = Constant λ r

38 Key equations: dp( ρ ) dρ da = ρ dρ dr dr with (i) d da p p r r dr dr = ( ρ) and ρ = ρ d h dh d / d + + h = dr r dr da r 6 da ω r ρ (ii) Particular solution: h p ω r = 3 /

39 Asymptotically flat solutions q = 0 c d ln h / r da 4 5 h 5 dr dr ω / r r= r = 5 48M d da ln rhh / dr dr r= r Q = Q 6 5a 4 M c Condition for being oblate: k ξ + < 0 h > 0 r= r r= r r (M. Bradley, D. Eriksson, G. Fodor & I. Rácz, Phys. Rev. D. 75, 0403 (007))

40 . Incompressible caseρ = Constant: Equations can be completely integrated Q 5ρ = Q 6 p c λ γ ρ. Newtonian polytropes p= pc : ρc In J. Martín, A. Molina & E. Ruiz, Class. Quantum Grav. 5, 0509 (008) it is shown that the properties of the solutions of their equations corresponding to (i) and (ii) are such that. It also follows that c 0 c > 0 h h r= r > 0 Q > 0 Q Hence for oblate Newtonian polytropes.

41 Conclusions. Numerically for arbitrary strength of gravitational field for reasonable equations of state Q > Q 0. Post-Minkowskian incompressible and polytropic cases compared with Cabezas et. al. and Martín et. al. Q Also showed that > 0 if oblate shape. Q 3. There are asymptotically flat perturbations of Whittaker (not Wahlquist). 4. Only cases found that gets close to Kerr (when getting infinite in size) are those with linear equation of state ρ = d p+ and for d d < 3. p 0c = ( d )