definition for the scalar field auxiliary variable:

Π =˘ γ_{θθ}√
γ_{rr}
αχ^{3/2}

Φ˙˜ −β^{r}Φ˜^{0}

(3.45) Then it is useful to perform the following variable transformations for the conformal factor (following common prescriptions as in (3.1)) and the trace-free part of the extrinsic curvature

χ→χ_{0}ψ^{−4} and A_{rr} →(A_{rr0}+ψ_{A})ψ^{−6}. (3.46)
As initial value of ˘Π we choose

Π˘_{0} =signγ_{θθ0}√
γ_{rr0}

α_{0}χ^{3/2}_{0} β^{r}_{0}Φ˜^{0}_{0}, (3.47)
where signis -1 gives a mainly ingoing pulse and +1 a mainly outgoing one.

Setting the corresponding initial values for all the remaining variables (and
substitut-ing the value of ¯Ω^{0} determined from (3.28) and ¯Ω^{00} obtained by deriving ¯Ω^{0}), the resulting
Hamiltonian and momentum equations become respectively

ψ^{00} = −π
The momentum constraint is independent of ψ thanks to the specific choice of variables.

This simplifies the procedure a lot, because it allows to first determine ψ_{A} from (3.48b)
and then set the obtained function into the Hamiltonian constraint (3.48a) to calculate
ψ. The initial value of ˜Π (Φ = ˜˙˜ Π) has to be calculated undoing the transformation (3.45):

Π˜_{0} =β^{r}_{0}Φ˜^{0}_{0}(1 + sign

ψ^{6} ). (3.49)

A completely outgoing or ingoing pulse will not be obtained with the choicessign=±1, becauseχis involved in the change of variables (3.45), but the value of signcan be tuned experimentally to vary the ratio between the ingoing and outgoing amplitudes and so the undesired pulses can be minimized.

### 3.5 Initial data for the simulations

### 3.5.1 Initial data perturbations

Gauge wave initial data

Choosing the trace of the physical extrinsic curvature ˜K as variable makes the vacuum constraint equations (2.79) independent of the gauge quantities. This allows us to in-troduce a perturbation in one of the gauge variables, e.g. the lapse α, without having

to solve the constraints for the initial data. The resulting evolution will of course only correspond to gauge dynamics - no actual physical processes take place -, but even these suppose a strong test for our equations.

The initial data for the variables is set as in (3.29) and (3.30), or by the simpler (3.35) in the flat spacetime case. To the initial value of the lapse we add a Gaussian-like compact-support perturbation of the form

δ_{α}_{0} =A_{α}e^{−}^{(r}

2−c2)2

4σ4 . (3.50)

The reason for choosing this particular form for the initial perturbation is its even parity with respect to the origin.

Scalar field initial data

Including non-trivial scalar field initial data requires solving the constraints. For this,
first a Gaussian in r^{2}, the same function as for the lapse perturbation, is set as initial
value for the scalar field,

Φ_{0} =A_{Φ}e^{−}^{(r}

2−c2)2

4σ4 , (3.51)

and then the constraint equations are solved as described in section 3.4.

### 3.5.2 Initial data plots

Examples of the initial data used for the simulations that gave the results presented in
this work are displayed in the following figures. The initial data corresponding to some
of the variables are not shown in the plots due to their simplicity: γ_{rr} and γ_{θθ} (if evolved
by some reason) are always initially 1, which implies that Λ^{r} vanishes. The Z4 quantity
Θ (or used in its “physical” form ˜Θ) is also initially zero.

In the flat spacetime case, the generic choice for the CMC parameter is K_{CM C} =−3,
while Schwarzschild trumpet initial data use K_{CM C} =−1, M = 1 and the critical value
of C_{CM C}. The reason for this smaller |K_{CM C}| in the BH case is the numerical precision
limitation in the calculation of ¯Ω that will be explained in subsection 6.6.1. The initial
perturbations of the scalar field (if present) in the following plots useA_{Φ}= 0.035,σ = 0.1
andc= 0.25, although these values may differ from those used in the actual simulations.

The “mostly ingoing” data usesign=−1 and the “mostly outgoing” ones,sign= +1.

Flat and regular spacetime

Figure 3.15 shows the stationary data corresponding to flat spacetime on the hyperboloidal
slice. The value of lapse and shift at I^{+} depends on the choice of parameter K_{CM C}
(compare to figure 3.18).

The perturbation of flat spacetime by an initially symmetric scalar field is plotted in the initial data in figure 3.16. The scalar field’s presence affects the conformal factor χ, so that it differs from unity in the region close to the origin. The horizontal line at 1 is plotted to mark this difference.

If time asymmetric initial data are chosen for the scalar field, like an ingoing or an outgoing pulse, then the time derivatives of the metric (embodied in the extrinsic curva-ture) are also affected and the momentum constraint has to be solved for the perturbation

3.5. Initial data for the simulations 63

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

r
Β^{r}

Α
Χ
*A*_{rr}

Figure 3.15: Stationary flat initial data (with KCM C =−3).

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

r P

F Χ 1

*A*_{rr}

Figure 3.16: Regular initial data perturbed by a massless scalar field. The effect of the perturbation appears inχ.

onA_{rr}, as described in section 3.4. An example of initial data corresponding to a mostly
ingoing scalar field is plotted in figure 3.17. If the initial scalar field was an outgoing
pulse, the value ofA_{rr} would positive and χ’s profile would also change.

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

r P

F Χ 1

*A*_{rr}

Figure 3.17: Regular initial data perturbed by a mostly ingoing massless scalar field. The
effect of the perturbation appears inχ and A_{rr}.

Schwarzschild spacetime

The trumpet values of a BH spacetime with M = 1 are displayed in figure 3.18. The
conformal factor χ, the lapse α and the shift β^{r} all become zero at the origin, which is
where the trumpet is mapped to. The shift is positive inside of some radius around the
origin (including the BH horizon, which in this case is located atr_{Schw} ≈0.13) and then
becomes negative. The initial value of the variable A_{rr} is plotted in figure 3.20. Figure
3.19 shows the influence of a time symmetric or a mostly ingoing scalar field perturbation
on the conformal factor χ. The Schwarzschild and perturbed trumpet values of A_{rr} are
presented in figure 3.20.

Variable choice for the trace of the extrinsic curvature

The initial values ofK, with expression (3.30b), and ˜K, which is simply ˜K_{0} =K_{CM C}, are
shown in figure 3.21. The latter has a much simpler stationary value, more appropriate
for numerical and visualization purposes. The quantity ∆ ˜K introduced as the variation
of ˜K in (2.80) has a vanishing initial value.

3.5. Initial data for the simulations 65

0.0 0.2 0.4 0.6 0.8 1.0

-0.5 0.0 0.5 1.0

r
Β^{r}

Α Χ 1 0

Figure 3.18: Schwarzschild trumpet BH initial data (with K_{CM C} =−1).

0.0 0.2 0.4 0.6 0.8 1.0

-0.5 0.0 0.5 1.0

r

P Mostly ingoing P Time symmetric F

Χ Mostly ingoing Χ Time symmetric

Figure 3.19: Schwarzschild trumpet BH initial data perturbed by a massless scalar field.

0.0 0.2 0.4 0.6 0.8 1.0 -5

-4 -3 -2 -1 0

r
*A*rr

Mostly outgoing Mostly ingoing Time symmetric

Figure 3.20: Schwarzschild trumpet BH initial data for the variableA_{rr}.

0.0 0.2 0.4 0.6 0.8 1.0

-10 -8 -6 -4 -2 0

r

K

Conformal K -Schwarzschild Conformal K -flat

Physical K

Figure 3.21: Initial data for the trace of the physical extrinsic curvature ( ˜K) and of the
conformal one (K), both with K_{CM C} = −1. The first does not change in presence of a
BH, but the latter does.

## Chapter 4

## Gauge conditions

The gauge variables determine how the spacetime is sliced for the decomposition of the
equations. The lapse α measures the separation of the slices along the proper time of an
observer and the shift β^{a} controls the change in spatial coordinates from one slice to the
next. How these gauge quantities behave during the evolution is not prescribed by the
Einstein equations. For a practical implementation this gauge freedom has to be fixed
by imposing some conditions on the lapse and shift. These conditions will depend on the
physical problem that has to be solved and are usually motivated by either a substantial
simplification in the equations and problem setup or by a good numerical behaviour, or
both at the same time. In any case, the effect of the gauge conditions on the evolution is
huge and the choice has to be made carefully.

In the rest of this chapter I will describe which specific conditions are convenient for the hyperboloidal initial value problem and how some commonly used choices (see [8] for more details) can be adapted to it.

### 4.1 Scri-fixing condition

In the conformal picture future null infinity is an ingoing null surface. From the point of
view of a set of hyperboloidal slices it means that the position of I^{+} on the slice moves
inwards with the speed of light as time goes by. In a numerical implementation this effect
translates to a loss of resolution on the integration domain. A way of avoiding this and
setting the location of I^{+} to a fixed position of the spatial domain is by means of the

“scri-fixing” condition [70].

The basic idea of the scri-fixing condition is to choose the coordinate time vector in
such a way that it flows alongI^{+}. The time vector is given by t^{a} =αn^{a}+β^{a} (2.9), and
it has to be chosen in such a way that it is an ingoing null vector at I^{+}. For simplicity
and to avoid the confusion with complex conjugates we will drop the overbars on the
vectors and covectors in this section, but note that all calculations are performed in the
conformal spacetime.

Newman-Penrose null tetrad

Applying the Newman-Penrose formalism [119] at I^{+} we introduce two real null vectors
k^{a} and l^{a}. The vector l^{a} is defined as the ingoing null vector and k^{a} as the outgoing one,
as displayed in figure 4.1. The tetrad is completed by two complex tangential null vectors,

67

m^{a} and its complex conjugate ¯m^{a}. The only nonzero inner products of the tetrad vectors
are l^{a}k_{a} =−1 and = m^{a}m¯_{a}= +1 (for the metric convention used, (−,+,+,+)).

The tetrad vectors can be expressed in terms of the timelike unit vector n^{a} and the
spacelike unit vectorss^{a}, e^{a}_{θ} and e^{a}_{φ}, for which the only non-vanishing inner products give
n^{a}na=−1, s^{a}sa = +1, e^{a}_{θ}eθ a = +1, e^{a}_{φ}eφ a = +1. (4.1)
The expressions of the tetrad null vectors are

k^{a}= 1

√2(n^{a}+s^{a}), l^{a} = 1

√2(n^{a}−s^{a}), m^{a}= 1

√2(e^{a}_{θ} +ie^{a}_{φ}), (4.2)
Ω = 0

I^{+}

n^{a}
s^{a}

l^{a} k^{a}

t^{a}

αn^{a}
β^{a}

Figure 4.1: Timelike, spacelike and null vectors near I^{+}.
Scri-fixing conditions

Null infinity is given by Ω = 0 and ¯∇_{a}Ω 6= 0. At I^{+} we have that l^{a}|_{I}+ = A∇¯^{a}Ω
I^{+}

(where A is a constant positive proportionality factor) and the condition that the time
vector flows alongI^{+} is given by t^{a}|_{I}+ =B l^{a}|_{I}+ (with B another constant factor). It
can be written as

t^{a}t_{a}|_{I}+ = (∇^{a}Ω) (∇_{a}Ω)|_{I}+ = 0. (4.3)
Using (2.9) it translates to

t^{a}t_{a}|_{I}+ = (αn^{a}+β^{a}) (αn_{a}+β_{a})|_{I}+ = −α^{2}+β^{a}β_{a}

I^{+} = 0. (4.4)
In our spherically symmetric ansatz (see section 2.4.1) it is expressed as

−α^{2}+χ^{−1}γ_{rr}β^{r2}

I^{+} = 0. (4.5)

The scri-fixing condition in our approach can be divided into two parts:

• The behaviour in time of Ω: if the value of Ω changes with time, it may affect
the coordinate position of Ω = 0. In the approach taken here following Zengino˘glu
[168, 170], the conformal factor Ω is a function of the radial coordinate r fixed in
time, so the coordinate location of I^{+} does not change during the evolution.

• Condition (4.5): the evolution quantities χ, γ_{rr}, α and β^{r} have to satisfy at all
times the condition that the time vector is null at the position to which I^{+} has
been fixed.

In a more general setup other than spherical symmetry, corotation is also possible, in
which case the scri-fixing condition is not as strict as t^{a} being a null geodesic generator.