Non-axisymmetric configurations in the Post-Newtonian ... · Non-axisymmetric configurations in the Post-Newtonian Approximation to General Relativity Norman Gürlebeck1 David Petroff2

Newtonian ellipsoidal configuration of equilibriumThe PN equations for Dedekind ellipsoids

Chandrasekhar & Elbert’s ansatz

Non-axisymmetric configurations in the

Post-Newtonian Approximation to General Relativity

Norman Gürlebeck 1 David Petroff 2

1Institut of Theoretical Physics, Charles-University, Prague

2Institut of Theoretical Physics, Friedrich-Schiller University, Jena

MG 12 Paris, 15 July, 2009

N. Gürlebeck, D. Petroff Dedekind ellipsoid to 1PN order



Outline

1 Newtonian ellipsoidal configuration of equilibrium

2 The PN equations for Dedekind ellipsoids

3 Chandrasekhar & Elbert’s ansatz




Possible ellipsoidal configurations

Definition

Solution of the Poisson equation and the Euler equation for aconstant mass density ρ

∆U = 4πρ, (v · ∇)v = −1

ρ∇p0 −∇U

1 McLaurin ellipsoid: axially symmetric, stationary, rigidlyrotating, [Bardeen 1971, Petroff 2003]

2 Jacobi ellipsoids: non-axially symmetric, the ellipsoid is rigidlyrotating [Chandrasekhar 1970]

3 Dedekind ellipsoids: non-axially symmetric, stationary,particles are moving on ellipses, [Chandrasekhar&Elbert 1978]




The Dedekind solution

triaxial ellipsoid with semiaxes a1 > a2 > a3 > 0velocity field ~v = (Ωa1

a2x2,−Ωa2

a1x1, 0)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Ω√ρ

a2

a1

The velocity along the Dedekind sequence




Surface adapted coordinates

In the Newtonian step: confocal ellipsoidal coordinates λ, µ, νwith the coordinate surfaces:

λ = const. confocal ellipsoids

µ = const. confocal one-sheeted hyperboloid

ν = const. confocal two-sheeted hyperboloid

surface of the configuration at λ = a1

obtain the Newtonian solution explicitly

Surface adapted coordinates in PN approximation

Surface of the configuration to any PN order given by:

λ(µ, ν) = a1(1 +∞∑

n=2

S (2n)(µ, ν)ǫ2n)

λ1 = λ (1 +∞∑

n=2

S (2n)(µ, ν)ǫ2n)−1




The projection formalism

(M, gµν) stationary spacetime with the Killing vector ξ = ∂t

F = gtt = ξt

~ξ = (ξa) = (gta) with a, b, ... ∈ 1, 2, 3hab = gab − 1

Fξaξb

twist vector: ~ω = (ωa) = (ǫ[4]aβγδξ

βξδ;γ) withα, β, ... ∈ t, 1, 2, 3stress energy tensor of a perfect fluid:Tαβ = (ρ + pǫ2)uαuβ + gαβp with a constant energy densityρ, the four velocity uα = (ut , ~u) and the pressure p




The exact field equations

F :a,a =

1

2FF,aF

,a − 1

Fωaω

a − 2R[4]tt

2(F−1ξ[a);b] = −(−F )−32 ǫ

[3]abc

ωc

R[3]ab

=1

2FF,a:b − 1

4F 2F,aF,b +

1

2F 2(ωaωb − habω

cωc) + R[4]ab

ω:aa =

3

2FF,aω

a

ǫ[3]abcωb:c = 2√−FhabR

[4]bt

gauge freedom: F−1ξa → F−1ξa + χ,a with χ,aξa = 0

[Geroch 1971] and [Stephani et. al. 2003]




The PN equations for hab and ξ

The expansion

X = X (0) + ε2X (2) + · · · , for X ∈ F , hab, p, ~uX = ε3X (3) + · · · , for X ∈ ~ξ, ~ω

0 = −2h(2)12,12 + a

(2)11,22 + a

(2)22,11

0 = −h(2)12,13 − h

(2)13,12 − h

(2)23,11 + 2h

(2)23,11 + a

(2)11,23 and cyclic

a(2)ii = h

(2)ii + 2U

in Cartesian coordinates

system has the trivial solution ⇒ (h(2)ab

) = −diag(2U, 2U, 2U)

in the gauge ∇ · ~ξ(3) = 0 only Poisson equation:∆~ξ(3) = 16πµ~v




The remaining field equations in brief

Pressure, surface and velocity

∆F (4) = g1(S(2), X (0)), where X (0) is any Newtonian quantity

Bianchi identity: ∇p(2) = g2(F(4), S (2), ~u(2), X (0)) ⇒ two

equations:

integrability condition

vanishing pressure at the surface λ1 = a1

vanishing normal component of the velocity at the surface:u(2)λ1(λ1 = a1, µ, ν) = 0

gi linear in the PN quantities

must be solved simultaneously





The surface:

1 =3

∑

i=1

x2i

a2i

− 2πGρ

c2

[

S1a21

(

x21

a21

− x23

a23

)

+ S2a21

(

x22

a22

− x23

a23

)

+S3

(

x41

3a21

− x21 x2

2

a22

)

+ S4

(

x42

3a22

− x22 x2

3

a23

)

+ S5

(

x43

3a23

− x21 x2

3

a21

)

]

The four velocity:

x1 = v1 +(πGρ)

32

c2((q1 + q)x2x

21 + r1x

32 + t1x2x

23 )

x2 = v2 +(πGρ)

32

c2((q2 − q)x1x

22 + r2x

31 + t2x1x

23 )

x3 =(πGρ)

32

c2q3x1x2x3




The correction of the velocity 1

-15

-10

-5

0

5

10

15

0 0.2 0.4 0.6 0.8 1a2

a1

-15

-10

-5

0

5

10

15

0 0.2 0.4 0.6 0.8 1a2

a1

-15

-10

-5

0

5

10

15

0 0.2 0.4 0.6 0.8 1a2

a1

q1

q in C&E

q

-40

-20

0

0.2 0.4 0.6 0.8 1a2

a1

-40

-20

0

0.2 0.4 0.6 0.8 1a2

a1

r1

t1




The correction of the velocity 2

-12

-8

-4

0

4

8

12

0 0.2 0.4 0.6 0.8 1a2

a1

-12

-8

-4

0

4

8

12

0 0.2 0.4 0.6 0.8 1a2

a1

q2

q3 -4

-3

-2

-1

0

1

0 0.2 0.4 0.6 0.8 1a2

a1

-4

-3

-2

-1

0

1

0 0.2 0.4 0.6 0.8 1a2

a1

r2t2




The surface of the configuration 1

a1 = 1, a2 = 0.7 and ρGc2 = 0.3

1,0

0,5-1,01,0 0,0

x10,5

-0,5

-0,50,0

x30,0

-0,5x2-1,0-1,0

0,5

1,0

1PN Newton

x2K0,6 K0,4 K0,2 0 0,2 0,4 0,6

x3

K0,4

K0,2

0,2

0,4

x1=0




The surface of the configuration 2

1PN Newton

x1K1,0 K0,5 0,5 1,0

x3

K0,4

K0,2

0,2

0,4

x2=0

1PN Newton

x1K1,0 K0,5 0,5 1,0

x2

K0,6

K0,4

K0,2

0,2

0,4

0,6

x3=0




The central pressure

0

3

6

9

12

15

0 0.2 0.4 0.6 0.8 1

p(2)C

ρ3

a2

a1

The central pressure




References

J.M. Bardeen: A Reexamination of the Post-Newtonian MacLaurin

Spheroids, Astrophysical Journal 167, 1971

S. Chandrasekhar: The evolution of the Jacobi ellipsoids by gravitational

radiation, Astrophysical Journal 161, 1970

S. Chandrasekhar, D. Elbert,The Deformed Figures of Dedekind Ellipsoids

in the Post Newtonian Approximation to General Relativity, AstrophysicalJournal 192, 1974 and 220, 1978

R. Geroch: A Method of Generating Solutions of Einstein’s Equation,J.Math.Phys. 12(6), 1971

H. Stephani et. al., Exact Solutions of Einstein’s Field Equations,Cambridge University Press, 2003

D. Petroff: Die MacLaurin Ellipsoide in post-Newtonscher Näherungbeliebig hoher Ordnung, Dissertation, 2003




Thank You!

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Documents

Non-axisymmetric configurations in the Post-Newtonian ... · Non-axisymmetric configurations in the Post-Newtonian Approximation to General Relativity Norman Gürlebeck1 David Petroff2