the component Einstein equations also have to be expressed in terms of ¯Φ and ¯Π.

Another possible definition for the auxiliary variable of the rescaled scalar field, ¯Π = ˙¯Φ,
is ¯Π_{adv} = _{α}^{1}

Φ˙¯ −β^{r}Φ¯^{0}

. The evolution equations for the scalar field in terms of this auxiliary variable are given by

Φ˙¯ = αΠ¯adv +β^{r}Φ¯^{0}, (2.92a)

### 2.6 Properties of the spacetime

Schwarzschild or areal radial coordinate

The physical Schwarzschild radial coordinate expressed in our rescaled quantities is given by

RSchw = r Ω

rγ_{θθ}

χ . (2.93)

The previous expression is obtained by comparing the angular part of the physical Schwarz-schild line element,

d˜s^{2} =R^{2}_{Schw}dσ^{2}, (2.94)

with the same quantity expressed in the spherically symmetric metric components of our ansatz (2.75)

In general, the concept of mass cannot be defined quasi-locally due to the equivalence
principle. However, in cases with high symmetry, where all degrees of freedom are fixed
-like in spherical symmetry, where no gravitational radiation exists -, it is possible to define
a quasi-local concept of mass. The Misner-Sharp mass function [114], a special case of
the Hawking quasi-local mass [91], is constant on each round sphere and represents the
gravitational mass contained by the sphere of areal radius R_{Schw}. Its definition is

˜

g^{ab}( ˜∇_{a}R_{Schw})( ˜∇_{b}R_{Schw}) = 1− 2M_{M S}

R_{Schw} , (2.96)

so that its actual expression is given by The Misner-Sharp mass coincides with the ADM mass at spacelike infinity and with the Bondi mass at null infinity.

The explicit expression of (2.97) in our evolution quantities is MM S = r

In simulations involving a BH or a scalar field strong enough to collapse into a BH it is very useful to have a quantity that detects the creation of a horizon and locates it in the domain. The actual event horizon can only be determined a posteriori, because it is a global quantity of the spacetime, but the apparent horizon, located inside of the event horizon if weak cosmic censorship holds, can be calculated along with the simulation.

The apparent horizon is observer dependent and is defined as the outermost marginally trapped surface, a smooth closed 2-surface whose outgoing null geodesics have zero expan-sion. In spherical symmetry it can be calculated explicitly in terms of the Misner-Sharp mass and the areal radius. The horizon will be located at the outermost point along the radial coordinate where the following expression equals zero:

1− M_{M S}

RSchw

2

= Ω^{2}g¯^{ab}( ¯∇_{a}R_{Schw})( ¯∇_{b}R_{Schw}). (2.99)
The final explicit expression of this quantity is straightforwardly calculated from (2.98).

## Chapter 3 Initial data

### 3.1 Solving the constraints

In a Cauchy formulation, the Einstein equations are decomposed into evolution equations and constraint equations, as derived in section 2.2. The constraint equations are such that if satisfied by the initial data, they will be satisfied at all times at the continuum level due to the Bianchi identities (see e.g. [8]). They are commonly expressed as elliptic equations, so that they have to be solved globally and require boundary conditions.

The constraint equations are given by the scalar Hamiltonian constraint and the vector momentum constraint, that is, four equations in total. The metric and extrinsic curvature have a total of twelve components to be determined, which means that there are eight degrees of freedom to be fixed and the solution of the constraint equations will provide the four remaining quantities.

In order to separate the freely specifiable data and provide a convenient set of elliptic equations to solve, variable transformations are performed on some of the quantities.

Among the widely used constraint decompositions are the York-Lichnerowicz conformal
decompositions. Their first ingredient is the conformal decomposition of the Hamiltonian
constraint proposed by Lichnerowicz [109]. This is performed via a decomposition of the
spatial metric ¯γ_{ab}into a conformal factorψand an auxiliary conformal background spatial
metric γ_{ab} [109, 159, 160]:

¯

γ_{ab} =ψ^{4}γ_{ab}. (3.1)

The Hamiltonian constraint is expressed in terms of the new variables as

∇^{a}∇_{a}ψ− ^{1}_{8}ψR[γ] +^{1}_{8}ψ^{5} K¯_{ab}K¯^{ab}−K¯^{2}

−2πψ^{5}ρ= 0. (3.2)
The second ingredient is the decomposition of the extrinsic curvature suggested by York
[161, 162], which takes the form

K¯_{ab} = ¯A_{ab}+^{1}_{3}γ¯_{ab}K.¯ (3.3)
The trace-free part A^{ab} is split into a transverse-traceless tensor and a longitudinal part.

The longitudinal part can be defined in terms of the original metric ¯γ_{ab} or the conformal
one γ_{ab}, thus giving the physical [121, 122] and conformal [164, 166] transverse-traceless
variants, respectively. Making the same initial choices for the freely specifiable data in
the two formulations will in principle provide results with different physical properties.

39

The conformal thin-sandwich decomposition by York [165] takes into account the change in the metric between two neighboring hypersurfaces in the form of the quan-tity

u_{ab} =∂_{t}γ_{ab}, (3.4)

which is used in the decomposition of the trace-free part of the extrinsic curvature. This formulation provides a more direct relation between the choices for the freely specifiable data and their effects on the initial physics of the system.

For more details about these formulation see e.g. [60] or [8].

Bowen-York initial data

The Bowen-York initial data solution [44] is obtained under the assumptions of conformal and asymptotic flatness and maximal slicing. This conditions allow to find an analytical solution to the momentum constraint for a BH with given spin and linear momentum.

Moreover, as the momentum constraint is linear under these assumptions, it allows to obtain initial data for more than one BH by superposition.

Another important ingredient is the puncture approach by Brandt and Br¨ugmann [46], a generalization of the Brill-Lindquist data [47]. It assumes a certain form for the conformal factor and solves the Hamiltonian equation for a smoother quantity.

The assumptions used to obtain the Bowen-York puncture initial data are too restric-tive for some initial data configurations. For instance, these methods are not suited to obtain initial data for a very rapidly spinning BH, because the Kerr-Newman solution accepts no conformally flat slices [151]. The Bowen-York solution of a spinning BH is a superposition a Kerr BH and some gravitational radiation, commonly called “junk” radi-ation. A consequence is that there is a maximum limit in the BH’s spin achievable with this initial data calculation, namely 0.93 out of 1 [61, 64, 88]. Going beyond this limit requires dropping the conformal flatness assumption and solving the constraints with a more general approach [110, 137].

Black hole excision and moving punctures

The singularities present in the BHs require a special numerical treatment. The most common choices are excision and puncture evolution.

BH excision was first used in spherical symmetry in [140]. It consists basically of two ingredients: first the central singularity is excised by setting a boundary inside of the BH’s (apparent) horizon and then the shift vector is given non-zero values such that it fixes the BH’s horizon to a given coordinate location. The excision boundary is spacelike (it is located inside of the BH) so no boundary conditions are required, provided that no gauge modes with superluminal speeds have outgoing characteristics at the excision surface.

The excision approach has been successful in spherical symmetry [17, 112, 136], but its implementation into 3+1 three-dimensional codes becomes more complicated [16] due to the difficulty in expressing the spheroidal excision surface in the Cartesian coordinates used in the code. The excision method is used by the Spectral Einstein Code [3, 123].

Unlike the static puncture method, where the singular behaviour of the metric at the BH’s singularity is factored out and a regular part is evolved separately [86, 142], the moving puncture approach puts the singular part into a dynamical conformal factor. The

3.1. Solving the constraints 41 use of the puncture method to evolve Bowen-York initial data with a conformally flat metric was proposed by [51]. In the BSSN formulation, the singularity can be evolved in the conformal factor χ (2.40) as done in [57, 56] or using the variable ϕ as in [22, 23].

It is important that the puncture is staggered, so that it never coincides with an actual gridpoint. Singularity avoiding slicings, as those given by maximal slicing and the 1+log gauge condition (see subsection 4.3.1), are needed in order to prevent the slices from reaching the singularity and ruining the simulation. Also a non-vanishing shift, like e.g.

the Gamma-driver condition (see subsection 4.4.2), is required to allow the punctures to move across the numerical grid. In unconstrained evolutions, the 1+log slicing condition and the Gamma-driver shift are common choices, as in [52].

Hyperboloidal initial data

To illustrate, following [72, 100], the kind of problems that may arise when solving the
constraint equations to obtain initial data from the conformal equations, let us consider
the subclass of hyperboloidal slices where the initial value of the extrinsic curvature is
pure trace, ˜Kab = ^{1}_{3}γ˜abK˜. The vacuum momentum constraint is

∇˜^{b}K˜ab−∇˜aK˜ = 0, (3.5)
and it requires that the trace of the extrinsic curvature ˜K is a non-vanishing constant.

The Hamiltonian constraint

R[˜γ]−K˜abK˜^{ab}+ ˜K^{2} = 0, (3.6)
is reduced to a single second order elliptic equation by the modified Lichnerowicz ansatz

˜

γ_{ab} = Ω^{−2}¯γ_{ab} =ω^{−2}φ^{4}γ¯_{ab}. (3.7)
Of the two conformal factors introduced, an appropriate value that vanishes atI^{+}will be
set for ω and the Hamiltonian constraint, now transformed into the form of the Yamabe
equation [158], will be solved for φ:

4ω^{2}4φ¯ −4ω( ¯∇^{a}ω)( ¯∇_{a}φ)−
1

2R[¯γ]ω^{2} + 2ω4ω¯ −3( ¯∇^{a}ω)( ¯∇_{a}ω)

φ= 1 3

K˜^{2}φ^{5} (3.8)
The Yamabe equation is degenerate at I^{+}, as ω|_{I} = 0 holds. It allows to determine the
boundary value as

φ^{4} = 9

K˜^{2}( ¯∇^{a}ω)( ¯∇_{a}ω). (3.9)
A positive and unique solution for the Yamabe equation (3.8) has been proven to exist
in [15]. In spite of the equation’s degeneracy, its numerical resolution does not pose any
problems and the Yamabe equation can be solved using standard procedures.

Among the efforts to obtain initial data on hyperboloidal slices are the first implemen-tations by Frauendiener [74, 72] and H¨ubner [94, 96]; the work in Schneemann’s Diploma thesis [139], where solutions in spherical and in axial symmetry are presented; the gen-eralization of Bowen-York initial data to hyperboloidal slices for binaries of boosted and spinning BHs in [53]; and perturbed Kerr initial data in [138].