• No results found

Compactified hyperboloidal vacuum initial data

The procedure to obtain vacuum initial data for the spherically symmetric equations (2.76) (or (2.78) or (2.82)), which will of course satisfy (2.77) (or (2.79) or (2.81)), is described in detail in this section. It follows very similar steps as in [171].

3.2.1 General procedure

Transformations of the initial line element

We will calculate spherically symmetric vacuum initial data on a hyperboloidal slice start-ing with the line element on a Cauchy slice:

d˜s2 =−A(˜r)d˜t2+ 1

A(˜r)d˜r2+ ˜r22, where dσ2 ≡dθ2+ sin2θdφ2. (3.10) Both the line element and the coordinates include a tilde to indicate that they measure distances on the physical domain (and not on the conformal one). The form of the initial metric is general enough to consider flat spacetime, the Schwarzschild spacetime and the Reissner-Nordstr¨om (RN) spacetime, among others.

The time coordinate ˜t is transformed into a new time coordinate t using a height functionh(˜r):

t= ˜t−h(˜r), (3.11)

and the line element now reads (substitutingd˜t =dt+h0(˜r)d˜r) d˜s2 =−A(˜r)dt2−2A(˜r)h0(˜r)dt d˜r+


A(˜r) d˜r2+ ˜r22. (3.12) The hypersurfaces of constant time t are now hyperboloidal slices that reach I+. How-ever,I+is still infinitely far away, so that the next step is to compactify the hyperboloidal slices. This is done by rescaling the radial component ˜r by a compactifying factor ¯Ω. Do not confuse the compactifying factor ¯Ω with the conformal factor Ω that rescales the metric in (1.3), as they are not necessarily the same. The new radial coordinateris given by

˜ r= r

Ω(r)¯ , (3.13)

where ¯Ω is such that it vanishes at the value ofrwhere the infinity of ˜r (I+ in this slice) is mapped to, that is ˜r → ∞ ⇔ r → rI so that ¯Ω(rI) = 0. After the radial coordinate transformationd˜r = Ω−r¯ ¯2¯0dr the initial line element reads

d˜s2 =−Adt2−2A h0Ω¯ −rΩ¯0

Ω¯2 dt dr+

1−(A h0)2 A

( ¯Ω−rΩ¯0)2

Ω¯4 dr2 + r2

Ω¯22. (3.14) BothA andh0 are functions of r¯ and ¯Ω depends on r, but for reasons of space and clarity this dependence is not explicitly written. Finally the complete line element is conformally rescaled by Ω2, in an equivalent way as done with the metric in (1.3), as d¯s2 = Ω2d˜s2:

d¯s2 =−AΩ2dt2+Ω2 Ω¯2

−2A h0( ¯Ω−rΩ¯0)dt dr+ h

1−(A h0)2 i A

( ¯Ω−rΩ¯0)2

Ω¯2 dr2+r22

. (3.15)

3.2. Compactified hyperboloidal vacuum initial data 43 Here the overbar indicates that this line element measures distances in the conformally rescaled spacetime.

Initial data for the metric components

Comparing with the line element written in terms of the component variables (2.75), shown here again for convenience,

d¯s2 =− α2−χ−1γrrβr2


rrβrdt dr+γrrdr2θθr22

. (3.16) the initial values of each of the metric components can be directly read off. A convenient choice is

γθθ0 = 1, (3.17a)

χ0 = Ω¯2

2, (3.17b)

γrr0 =

1−(A h0)2 A

( ¯Ω−rΩ¯0)2

Ω¯2 , (3.17c)

βr0 = − A2Ω¯2h0 1−(A h0)2

( ¯Ω−rΩ¯0), (3.17d) α0 = Ω

s A

1−(A h0)2, (3.17e)

where the subscript 0 indicates that these are the expressions for the initial values.

Initial data for the derived quantities

The solution of the Z4 equations only coincides with a solution of the Einstein equations when the constraint fields Θ and Zr are zero. For this reason, their stationary value is expected to vanish and their initial values will also be zero.

The initial values of the componentArrof the trace-free part of the extrinsic curvature and its trace ¯K are expressed in terms of the metric components as

Arr0 = βr0γrr00

0 +2γrr0βr00

0 − βr0γrr0γθθ00

0γθθ0 −2βr0γrr0

0r , (3.18a) K¯0 = βr00

α0 −3βr0χ0

0χ +βr0γθθ00

α0γθθ0 + βr0γrr00

0γrr0 +2βr0

α0r . (3.18b)

They were calculated from the decomposition of (2.21) supposing that the initial values are time-independent. Substituting the initial values for the metric components will provide the explicit expressions for the initial values of the extrinsic curvature. They are not shown here because they are lengthy expressions, but they will be presented in subsection 3.2.3 after performing some simplifications. Note that the initial value of K is the same as ¯K, as Θ0 = 0.

If the background metric is set to the initial values of the evolved metric, ˆ

γrrrr0 =

1−(A h0)2 A

( ¯Ω−rΩ¯0)2

Ω¯2 and γˆθθθθ0 = 1, (3.19) then by definition (2.44) ∆Γr0 = 0, which together withZr0 = 0 sets Λr0 = 0.

3.2.2 Height function approach

The derivation of a height function that provides CMC slices will follow [80, 111], also consider [104]. First we compare (3.12) to the line element

d˜s2 = ˜gµνdxµdxν =−



dt2+ 2 ˜γ¯rrβ˜rdt d˜r+ ˜¯γrrd˜r2+ ˜γ¯θθ22, (3.20) and thus see that (3.12) corresponds to the metric


with determinant ˜g =−˜r4sin2θ. The normal vector in adapted coordinates can be written as ˜nµ= α1˜


. Its expression according to (3.12) is


Contracting the right equation in (2.38), an expression for the trace of the physical ex-trinsic curvature ˜K¯ is obtained:

˜ Substituting the determinant of (3.21) and the expression of ˜nµ (3.22), the previous rela-tion now reads

This expression can be integrated by setting the value of the trace of the extrinsic curva-ture to a constant value ˜K¯ =KCM C: where the parameter KCM C and the integration constant CCM C are set in such a way that KCM C < 0 (according to the convention chosen for the extrinsic curvature) and CCM C > 0. Solving for h0(˜r) and choosing the convenient sign will give us its value to calculate the initial data:

This expression with KCM C = 0 set (maximal slicing) has been widely used in relation with trumpet [89] initial and stationary data. More details are given in subsection 3.3.2.

3.2. Compactified hyperboloidal vacuum initial data 45 The components of the rescaled spatial conformal metric obtained before, (3.17), turn into the following after setting the previous expression for h0(˜r) expressed in terms of r¯:

γθθ0 = 1, (3.27a)

The compactification of the hyperboloidal slices has been performed by rescaling the radial coordinate as in (3.13). In principle, the only conditions that the function ¯Ω(r) has to satisfy is being smooth, positive, going to zero as r goes to the radial location assigned toI+, which is equivalent to ˜r → ∞, and with ¯Ω0 6= 0 to ensure that the transformation is invertible.

A very convenient choice is choosing ¯Ω such that the initial spatial metric is confor-mally flat, equivalent to imposing initial isotropic coordinates. This translates to setting

γrr0 = ( ¯Ω−rΩ¯0)2

This condition can indeed be satisfied and the corresponding ¯Ω can be calculated analyti-cally for flat spacetime and numerianalyti-cally for the Schwarzschild and RN cases (more details in the next section). The derivative ¯Ω0can be isolated from condition (3.28) and expressed in terms ofrand ¯Ω. Substituting it into the initial values for the metric components (3.27) yields

Under the assumption of this condition the initial values for the extrinsic curvature

(3.18) simplify considerably and are written as Arr0 = −2CCM CΩ¯3

r3Ω , (3.30a)

0 = KCM C

Ω +


3 + CCM Cr2 ¯3


A(r¯) +


3 ¯ +CCM Cr2 ¯2



2 . (3.30b)

This choice for the compactification factor ¯Ω suggests setting the flat spatial metric in spherical coordinates for the background metric ˆγij =diag(1, r2, r2sin2θ), as was done in (2.72). The Z4 quantities Θ andZr have to vanish initially and the conformal flatness of the initial data also implies that Λr0 = 0. The initial value of the trace of the physical extrinsic curvature is ˜K¯0 =KCM C (as was set to integrate (3.25)), for any choice ofA(r¯).