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˜s^{2} =−A(˜r)d˜t^{2}+ 1

A(˜r)d˜r^{2}+ ˜r^{2}dσ^{2}, where dσ^{2} ≡dθ^{2}+ sin^{2}θdφ^{2}. (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+h^{0}(˜r)d˜r)
d˜s^{2} =−A(˜r)dt^{2}−2A(˜r)h^{0}(˜r)dt d˜r+

1−(A(˜r)h^{0}(˜r))^{2}

A(˜r) d˜r^{2}+ ˜r^{2}dσ^{2}. (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 → r_{I} so that ¯Ω(r_{I}) = 0. After the radial coordinate
transformationd˜r = ^{Ω−r}^{¯} _{Ω}_{¯}2^{Ω}^{¯}^{0}dr the initial line element reads

d˜s^{2} =−Adt^{2}−2A h^{0}Ω¯ −rΩ¯^{0}

Ω¯^{2} dt dr+

1−(A h^{0})^{2}
A

( ¯Ω−rΩ¯^{0})^{2}

Ω¯^{4} dr^{2} + r^{2}

Ω¯^{2}dσ^{2}. (3.14)
BothA andh^{0} are functions of _{Ω}^{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 Ω^{2}, in an equivalent way as done with the metric in (1.3), as d¯s^{2} = Ω^{2}d˜s^{2}:

d¯s^{2} =−AΩ^{2}dt^{2}+Ω^{2}
Ω¯^{2}

−2A h^{0}( ¯Ω−rΩ¯^{0})dt dr+
h

1−(A h^{0})^{2}
i
A

( ¯Ω−rΩ¯^{0})^{2}

Ω¯^{2} dr^{2}+r^{2}dσ^{2}

. (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¯s^{2} =− α^{2}−χ^{−1}γrrβ^{r2}

dt^{2}+χ^{−1}

2γrrβ^{r}dt dr+γrrdr^{2}+γθθr^{2}dσ^{2}

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

γ_{θθ0} = 1, (3.17a)

χ_{0} = Ω¯^{2}

Ω^{2}, (3.17b)

γ_{rr}_{0} =

1−(A h^{0})^{2}
A

( ¯Ω−rΩ¯^{0})^{2}

Ω¯^{2} , (3.17c)

β^{r}_{0} = − A^{2}Ω¯^{2}h^{0}
1−(A h^{0})^{2}

( ¯Ω−rΩ¯^{0}), (3.17d)
α0 = Ω

s A

1−(A h^{0})^{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 Z_{r} 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 componentA_{rr}of the trace-free part of the extrinsic curvature
and its trace ¯K are expressed in terms of the metric components as

Arr0 = β^{r}_{0}γ_{rr}^{0}_{0}

3α_{0} +2γ_{rr}_{0}β^{r}^{0}_{0}

3α_{0} − β^{r}_{0}γ_{rr}_{0}γ_{θθ}^{0}_{0}

3α_{0}γ_{θθ0} −2β^{r}_{0}γ_{rr}_{0}

3α_{0}r , (3.18a)
K¯0 = β^{r}^{0}_{0}

α_{0} −3β^{r}_{0}χ^{0}

2α_{0}χ +β^{r}_{0}γ_{θθ}^{0}_{0}

α_{0}γ_{θθ0} + β^{r}_{0}γ_{rr}^{0}_{0}

2α_{0}γ_{rr0} +2β^{r}_{0}

α_{0}r . (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

γ_{rr} =γ_{rr}_{0} =

1−(A h^{0})^{2}
A

( ¯Ω−rΩ¯^{0})^{2}

Ω¯^{2} and γˆ_{θθ} =γ_{θθ}_{0} = 1, (3.19)
then by definition (2.44) ∆Γ^{r}_{0} = 0, which together withZ_{r0} = 0 sets Λ^{r}_{0} = 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˜s^{2} = ˜g_{µν}dx^{µ}dx^{ν} =−

˜

α^{2}−˜¯γ_{rr}β˜^{r}^{2}

dt^{2}+ 2 ˜γ¯_{rr}β˜^{r}dt d˜r+ ˜¯γ_{rr}d˜r^{2}+ ˜γ¯_{θθ}r˜^{2}dσ^{2}, (3.20)
and thus see that (3.12) corresponds to the metric

˜

with determinant ˜g =−˜r^{4}sin^{2}θ. The normal vector in adapted coordinates can be written
as ˜n^{µ}= _{α}^{1}_{˜}

1,−β˜^{r},0,0T

. 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 ˜n^{µ} (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 K_{CM C} and the integration constant C_{CM C} are set in such a way
that K_{CM C} < 0 (according to the convention chosen for the extrinsic curvature) and
CCM C > 0. Solving for h^{0}(˜r) and choosing the convenient sign will give us its value to
calculate the initial data:

This expression with K_{CM 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 h^{0}(˜r) expressed in terms of _{Ω}^{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 ¯Ω^{0} 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

γ_{rr0} = ( ¯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 ¯Ω^{0}can 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
A_{rr0} = −2C_{CM C}Ω¯^{3}

r^{3}Ω , (3.30a)

K¯_{0} = K_{CM C}

Ω +

KCM Cr

3 + ^{C}^{CM C}_{r}2 ^{Ω}^{¯}^{3}

r

A(_{Ω}^{r}_{¯}) +

KCM Cr

3 ¯Ω +^{C}^{CM C}_{r}2 ^{Ω}^{¯}^{2}

2

3Ω^{0}

Ω^{2} . (3.30b)

This choice for the compactification factor ¯Ω suggests setting the flat spatial metric
in spherical coordinates for the background metric ˆγ_{ij} =diag(1, r^{2}, r^{2}sin^{2}θ), as was done
in (2.72). The Z4 quantities Θ andZ_{r} have to vanish initially and the conformal flatness
of the initial data also implies that Λ^{r}_{0} = 0. The initial value of the trace of the physical
extrinsic curvature is ˜K¯_{0} =K_{CM C} (as was set to integrate (3.25)), for any choice ofA(_{Ω}^{r}_{¯}).