Outlook

9.2 Outlook

Gauge conditions

The treatment of the gauge conditions at I⁺ is much better understood now, but other problems arise due to the presence of strong field initial data. The next step is the tuning of the gauge conditions in the interior part of the integration domain according to common prescriptions and the improvement of some of the derived gauge conditions, especially those that satisfy the preferred conformal gauge. A possibility would be to test a matching of the 1+log slicing condition and the (integrated) Gamma-driver shift condition in the interior part with harmonic-like gauge conditions in the neighborhood of I⁺. Another problem to solve is to understand what stable stationary solutions for the Schwarzschild spacetime on a hyperboloidal slice exist and how they relate to the gauge source functions.

Regularity conditions

A brief description of the regularity conditions atI⁺ required by the equations has been included in this work. They are especially important in a non-staggered implementation, because they have to be explicitly satisfied at I⁺ so that the equations attain regular limits there. However, a more general study of of the regularity conditions in a [2+1]+1 decomposition will allow to understand much better the treatment required by I⁺ and prepare the path for more complicated setups other than spherical symmetry. Work in this direction has already started.

Implementation into a three-dimensional code

A three-dimensional implementation of the setup used in this work using the Einstein Toolkit framework [1] is one of the next steps planned. This will take advantage of all the knowledge about the problem obtained so far and will require a spherical boundary in the numerical setup. This can be implemented using the multipatch framework in theLlama Code [2]. Another possible option is to implement it using three-dimensional spherical polar coordinates [25], which allow for a clear separation of the radial direction that will hopefully simplify the regularizations at I⁺. The coordinate singularities (not present in a Cartesian grid) can however pose difficulties. The development of more complicated hyperboloidal-based codes will require suitable initial data, which means that previous work [53] on the hyperboloidal elliptic equations will be continued.

Simulations in Anti-deSitter spacetimes

In the case of a negative cosmological constant Λ, I⁺ is not a null surface but a timelike one and appropriate boundary conditions at I⁺ have to be prescribed (this is one of the reasons for using a non-staggered grid). The required boundary conditions have to be reflecting-like and current numerical results in four-dimensional AdS of a constrained evolution of the Einstein equations coupled to a scalar field in spherical symmetry [34]

indicate that AdS is non-linearly unstable. Evolving AdS with the procedures presented in this thesis and appropriate boundary conditions in spherical symmetry could complement the stability results of AdS obtained so far and maybe even provide new results.

Appendix A

Construction of Penrose diagrams

Here I will show the explicit expressions used to create the Carter-Penrose diagrams presented in chapters 1 and 3.

A.1 Kruskal-Szekeres-like coordinates

Omitting angular dimensions, we consider a line element of the form d˜s² =−A(˜r)d˜t²+ 1

A(˜r)d˜r². (A.1)

First we eliminate the coordinate singularity at the horizon introducing a tortoise coordinate, in terms of which the line element takes the form

d˜s² =A(˜r) −d˜t² +d˜r_∗²

, (A.2)

and thus is related to the original radial coordinate ˜r as d˜r∗ = d˜r

A(˜r). (A.3)

The integrated expression of ˜r∗ depends explicitly on the form of A(˜r) and includes an integration constant that will be set to convenience for each case.

A transformation to the null coordinates ˜uand ˜v is performed

˜

u= ˜t−r˜∗, v˜= ˜t+ ˜r∗, (A.4) so that the line element reads

d˜s² =−A(˜r)d˜u d˜v. (A.5)

The quantityA(˜r) is to be expressed in terms of ˜uand ˜v. Its exact expression will depend on the specific form of ˜r_∗(˜r) and will be explicitly given in the following sections.

It is convenient to perform a coordinate transformation on the null coordinates ˜uand

˜

v that leaves the null cone structure invariant. This transformation will determine the precise form of the metric and can thus serve to simplify it. The transformations that will be considered here are of the form

U˜ =−e^−Bu˜, V˜ =e^Bv˜, (A.6) 157

where B is some given function of the parameters specific to each case. The choice B = 4M in the Schwarzschild spacetime gives the Kruskal-Szekeres coordinates.

The following change will allow us to express the line element in a form similar to Minkowski spacetime:

In terms of the original ˜t and ˜r coordinates, their expressions reduce to T˜=e^˜r∗B sinh

The final expressions of ˜T and ˜R are compactified in the same way as done with the Minkowski spacetime in subsection 1.2.1, i.e. using (1.5) (with ˜T and ˜R instead of ˜t and

˜

r), (1.7) and (1.11). To obtain the Carter-Penrose diagrams,Ris plotted in the horizontal axis andT in the vertical one.

For some values of the radial coordinate - inside of the horizon ˜r < 2M for the Schwarzschild case and among horizons ˜r− < r <˜ ˜r+ for the non-extreme RN one - the radius ˜rbecomes the timelike coordinate and the time ˜tturns into the spatial one, because A(˜r)<0 there. In these ranges of ˜r, the sign of ˜uand ˜U in the coordinate transformations (A.4), (A.6) and (A.7) is the opposite one. However, the resulting ˜T and ˜R yield exactly the same expression as in (A.8). As ˜T is spacelike and ˜R is timelike, their expressions in this case have to be interchanged before applying the compactification procedure of subsection 1.2.1.