Relations between Gowdy and Bianchi spacetimes - Alan D. Rendall, Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Am ...

Relations between Gowdy and Bianchi spacetimes

Alan D. Rendall,
Max Planck Institute for Gravitational Physics,
Albert Einstein Institute,
Am Mühlenberg 1,
14476 Golm, Germany.

                                                  1

Introduction

Symmetry reductions of the Einstein equations are often studied

The following is concerned with the vacuum case

Relations between Bianchi and Gowdy solutions will be explored

−dt2 + gij (t)(σ i ⊗ σ j )

σ i are left-invariant one-forms on a 3-dimensional Lie group G

    −1 λ
t     e (−dt2 + dθ2) + t(eP (dx + Qdy)2 + e−P dy 2)
     2 2

P , Q and λ depend on t and θ, periodic in θ. x, y periodic
                                                              2

The functions P and Q satisfy

               Ptt + t−1Pt − Pθθ = e2P (Q2    2
                                         t − Qθ )
               Qtt + t−1Qt − Qθθ = −2(PtQt − Pθ Qθ )

λ can be determined by integration

Q = 0 is called the polarized case

−dt2 + dθ2 + t2dφ2 (auxiliary metric)

Wave map from auxiliary metric to hyperbolic plane (φ-independent)

Target metric dP 2 + e2P dQ2
                                                          3

The definition of wave maps is as follows

Let (M, g) be a pseudo-Riemmanian and (N, h) a Riemannian
manifold

Let Φ be a mapping from M to N

       I ∂ΦJ
L = ∂Φ
    ∂x α   β g αβ h
                   IJ
         ∂x

Euler-Lagrange equations define:

harmonic map (g Riemannian)

wave map (g Lorentzian)
                                                    4

Bianchi I

Consider case of Gowdy where (P, Q)(t, θ) = (P̄ , Q̄)(t)

Equations reduce to ODE

Simplest case of relation to Bianchi

Set σ 1 = dθ, σ 2 = dx, σ 3 = dy

Lie group is T3, abelian, Bianchi type I

On universal cover diagonalization is possible, Q̄ = 0

Diagonalize initial data

Ignoring topology, all θ-independent solutions are polarized
                                                           5

Bianchi type VII0

Circular loop solutions (Chruściel 1991)

These are equivariant (rather than invariant) wave maps

Coordinates (P, Q) poorly adapted to these

Introduce other coordinates on hyperbolic plane as follows:

                               Q2 + e−2P − 1
                    Φ cos Θ = 2
                              Q + (e−P − 1)2
                                   −2Q
                    Φ sin Θ = 2
                             Q + (e−P − 1)2
                                                          6

Field equations are

             −1         1
      Φtt + t Φt − Φθθ = sinh 2Φ(Θ2    2
                                  t − Θθ )
                          2
      sinh2 Φ(Θtt + t−1Θt − Θθθ ) = sinh 2Φ(−ΦtΘt + Φθ Θθ )

Circular loop spacetimes given by

Φ(t, θ) = Φ̄(t), Θ(t, θ) = αθ, α positive real number

Second equation satisfied identically

                                                         7

First equation becomes

d2 Φ̄ + t−1 dΦ̄ = − α2 sinh 2Φ̄
dt2         dt      2

Metric on group orbits takes the form

         1 αθ)dx+(sin 1 αθ)dy]2 +e−Φ̄ [(− sin 1 αθ)dx+(cos 1 αθ)dy]2
eΦ̄[(cos 2            2                       2            2

One-forms in square brackets and dθ are Bianchi type VII0

Conclusion: circular loop spacetimes are spatially homogeneous
of Bianchi type VII0

Any vacuum Bianchi VII0 solution is a circular loop spacetime
                                                              8

Chruściel showed that:

Φ̄ and dΦ̄/dt are O(t−1/2) as t → ∞

tE(t) → E∞ where E is the energy

For a non-trivial solution E∞ > 0

Spacetimes are future geodesically complete

Corresponding statements later obtained for general Gowdy so-
lutions by Ringström (2004)

                                                        9

Late-time dynamics of Bianchi type VII0 vacuum spacetimes
were studied by Ringström (2001)

Follow from earlier Gowdy results

λ = E∞t + O(log t)

λ̇ = E∞ + O(t−1)

         1 −λ
     1
H = 12 t 4 e 4 (λ̇ − 5t−1)

 c = 12E −1 + o(1)           (cf. Wainwright-Hsu)
abH     ∞

 b = 12E −1 + o(1)
acH     ∞
                                                    10

Bianchi II

                       ∂
Equivariance w.r.t. α ∂Q

P (t, θ) = P̄ (t), Q(t, θ) = αθ

Does not fit globally onto torus T 3

Fits onto a T 2 bundle over the circle (for suitable α)

Inhomogeneous generalization

P (t, θ + 2π) = P (t, θ), Q(t, θ + 2π) = Q(t, θ) + 2πα

This has ’twisted Gowdy’ or ’local Gowdy’ symmetry
                                                          11

Bianchi VI0
                                   
                       ∂ −Q ∂
Equivariance w.r.t. α ∂P   ∂Q

P (t, θ) = P̄ (t) + αθ, Q(t, θ) = e−αθ Q̄(t)

Again fits on to T 2 bundle over the circle (a different one)

Inhomogeneous generalization

P (t, θ + 2π) = P (t, θ) + 2πα, Q(t, θ + 2π) = e−2παQ(t, θ)

                                                              12

Global dynamics

Near initial singularity dynamics localizes

Many details of asymptotics are known (Ringström)

Applies without change to the twisted models

Situation is less straightforward for late-time dynamics

  1 R 2π P 2 + P 2 + e2P (Q2 + Q2 )dθ
E=2  0    t     θ          t    θ

Well-defined in twisted case

                                                           13

Global existence on (0, ∞) follows by using domain of dependence

dE = −t−1 R 2π P 2 + e2P Q2 dθ + [P P + e2P Q Q ]2π
dt         0    t         t        t θ       t θ 0

Boundary terms cancel

In Gowdy E = O(t−1)

Does not hold for Bianchi type VI0

E ≥ πα2

Whether it holds for Bianchi type II is unclear

                                                          14

The Gowdy-to-Ernst transformation

Very useful in analysis of initial singularities

Analogue of Kramer-Neugebauer transformation for stationary
and axisymmetic spacetimes

P̃ = − log t − P

Q̃t = te2P Qθ , Q̃θ = te2P Qt

Not always compatible with global topology

Can transform Kasner to Bianchi type II
                                                      15

Can transform Gowdy to twisted Gowdy of type II

Value of 02π te2P Qtdθ restricted to discrete values
         R

Not all twisted solutions obtained

R 2π 2P
 0 e Qθ dθ = 0

Dynamics of large class of solutions determined

Presumably direct analysis of dynamics necessary in general

                                                         16

Remarks on Bianchi Class B

Bianchi Class A models with 2-dim Abelian subgroup discussed

In Bianchi Class B similar formal relation holds

But: area radius is not periodic

Example: Bianchi type V

−dt2 + t2[dθ2 + e2λθ (dx2 + dy 2)]

Milne solution (form of flat space)

                                                       17

Late-time behaviour of Gowdy solutions

Detailed analysis of late-time behaviour in Gowdy based on use
of conserved quantities following from Noether identities
   R 2π
A = 0 2Q(tQt)e2P − 2(tPt)dθ
   R 2π 2P
B = 0 e tQtdθ
   R 2π
C = 0 tQt(1 − e2P Q2) + 2Q(tPt)dθ

Casimir invariant I = A2 + BC classifies behaviour

I < 0 most interesting, I ≥ 0 in Bianchi I

Gowdy-Ernst global implies B = 0, I ≥ 0
                                                         18

Summary

There are several interesting relations between Bianchi models
and (twisted) Gowdy models.

Circular loop spacetimes correspond to Bianchi type VII0

There are twisted generalizations of Bianchi types II and VI0

Global dynamics of large classes of inhomogeneous spacetimes
can be determined

Direct analysis of twisted models remains to be done

                                                           19