2.3 Conformal thin-sandwich decomposition

The two general initial-value decompositions outlined in Section 2.2.1 and Section 2.2.2 require identical freely specified data (&tidle;γij, M&tidle; ij, and K), yet they usually produce different physical initial data. One shortcoming of these approaches is that they provide no direct insight into how to choose the freely specifiable data. All of the data are determined by schemes that involve only a single spacelike hypersurface. The resulting constraint equations are independent of the kinematical variables α and i β that govern how the coordinates move through spacetime, and thus there is no connection to dynamics. York’s conformal thin-sandwich decomposition [115] takes a different approach by considering the evolution of the metric between two neighboring hypersurfaces (the thin sandwich). This decomposition is very similar to an approach originated by Wilson [104Jump To The Next Citation Point46Jump To The Next Citation Point], but is somewhat more general. Perhaps the most attractive feature of this decomposition is the insight it yields into the choice of the freely specifiable data.

The decomposition begins with the standard conformal decomposition of the 3-metric (16View Equation). However, we next make use of the evolution equation for the metric (13View Equation) in order to connect the 3-metrics on the two neighboring hypersurfaces. Label the two slices by t and ′ t, with ′ t = t + δt, then ′ γij = γij + (∂tγij) δt. We would like to specify how the 3-metric evolves, but we do not have full freedom to do this. We know we can freely specify only the conformal 3-metric, and similarly, we are free to specify only the evolution of the conformal 3-metric. We make the following definitions:

u ≡ γ1∕3∂ (γ −1∕3γ ), (41 ) ij t ij
u&tidle;ij ≡ ∂t&tidle;γij, (42 )
and
&tidle;γij&tidle;u ≡ 0. (43 ) ij
The latter definition is made for convenience, so that we can treat ψ, &tidle;γij, and u&tidle;ij as regular scalars and tensors instead of as scalar- and tensor-densities within this thin-sandwich formalism.

The conformal scaling of uij follows directly from (16View Equation), (41View Equation), (42View Equation), (43View Equation), and the identity that, for any small perturbation, δln γ = γijδγij. The result is

uij = ψ4&tidle;uij, (44 )
which relies on the useful intermediate result that6
∂t&tidle;γ = 0. (45 )

Equation (41View Equation) represents the tracefree part of the evolution of the 3-metric, so (13View Equation) becomes

ij ij ij u = − 2αA + (𝕃¯β) . (46 )
Using the conformal scalings (22View Equation), (35View Equation), and (44View Equation), we obtain
ψ6 ( ) &tidle;Aij = --- (𝕃&tidle;β)ij − &tidle;uij . (47 ) 2α

York has pointed out that it is natural to use the following conformal rescaling of the lapse:

α = ψ6α&tidle;. (48 )
This rescaling follows naturally from the “slicing function” that replaces the usual lapse (√ -- α = γ&tidle;α) which has been critical in solving several problems [4]. It also results in the natural conformal scaling (22View Equation) postulated for the tracefree part of the extrinsic curvature. Substituting (48View Equation) into (47View Equation) yields what is taken as the definition of the tracefree part of the conformal extrinsic curvature,
( ) &tidle;Aij ≡ -1- (𝕃&tidle;β)ij − u&tidle;ij . (49 ) 2&tidle;α

Because the tracefree extrinsic curvature satisfies the normal conformal scaling, the Hamiltonian constraint will take on the same form as in (32View Equation). However, the momentum constraint will have a very different form. Combining equations (16View Equation), (19View Equation), (21View Equation), (22View Equation), and (49View Equation) with the momentum constraint (15View Equation), we find that it simplifies toUpdateJump To The Next Update Information

&tidle;Δ βi − (&tidle;𝕃β )ij &tidle;∇ ln &tidle;α = 4&tidle;αψ6 &tidle;∇iK + &tidle;α &tidle;∇ (1&tidle;uij) + 16πG &tidle;α ψ10ji. (50 ) 𝕃 j 3 j &tidle;α

Let us, for convenience, group together all the equations that constitute the conformal thin-sandwich decomposition:

γij = ψ4&tidle;γij, ij − 10 &tidle;ij 1 −4 ij K = ψ A + 3ψ &tidle;γ K, &tidle; ij -1-( &tidle; ij ij) A = 2&tidle;α (𝕃β) − u&tidle; , (51 ) &tidle; i &tidle; ij &tidle; 4 6 &tidle;i &tidle; (1--ij) 10 i Δ 𝕃β − (𝕃 β) ∇j ln &tidle;α − 3α&tidle;ψ ∇ K = &tidle;α∇j &tidle;α&tidle;u + 16 πG &tidle;αψ j, &tidle;∇2 ψ − 1ψ R&tidle;− 1-ψ5K2 + 1ψ −7 &tidle;Aij &tidle;Aij = − 2πG ψ5ρ. 8 12 8
In this decomposition (51View Equation)Update, we are free to specify a symmetric tensor &tidle;γij as the conformal 3-metric, a symmetric tracefree tensor &tidle;uij, a scalar function K, and the scalar function &tidle;α. Solving this set of equations with appropriate boundary conditions yields initial data γij and Kij on a single hypersurface. However, we also know the following: If we chose to use the shift vector obtained from solving (50View Equation) and the lapse from (48View Equation) via our choice of &tidle;α and our solution to the Hamiltonian constraint, then the rate of change of the physical 3-metric is given by
( ) ∂tγij = uij + 2γij ¯∇kβk − αK 3 [ ( )] (52 ) = ψ4 &tidle;uij + 23 &tidle;γij &tidle;∇k βk + 6βk∇&tidle;k ln ψ − ψ6 &tidle;αK .
This direct information about the consequences of our choices for the freely specifiable data is something not present in the previous decompositions. As we will see later, this framework has been used to construct initial data that are in quasiequilibrium.
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