1. Introduction
2.4 The situation of frontline professionals in the Obstetric-Paediatric
3.2.1 Data Generation the in-depth discursive interviews
In order to measure the extent of the GL instability, we need to characterise the size of the transverse S2 spheres along the ring’s horizon. For the stationary axisymmetric solution, the spheres are obtained simply by fixing the angular coordinate constant. However, for our dynamical horizon, taking a constant- section is no longer a sensible choice, as the axisymmetry is broken. Generally speaking, there is no preferred way to slice a transverse section through an arbitrarily chosen point onH. One may contemplate using spacelike geodesics alongH to define a preferredS2, however the section obtained by slicing along such geodesics will generically not form a smoothS2. Our approach here is to instead extremise the sphere’s area over all the possible smoothS2 that can be sliced onH. Specifically, consider an apparent horizonH with topology S2S1, which is defined as the level setDF .; /as previously described. The line element onH is given by
dsH2 Dhd2Ch d 2C2 h dd CZ2hwwd2 (8.6) ZWD R sin
cosC1=tanh; (8.7)
wherehdenote the components of the metric on†pulled back ontoH. Recall also that we allow RDR. /in order to accommodate stretched horizons.
We parametrise anS2section onH via D‰./. The line element on theS2is given by dsS22 D h ‰0./2C2 h ‰0./Ch
d2CZ2hwwd2; (8.8)
and thus its area is given by AS2 D4
Z
d q
h ‰0./2C2 h ‰0./Ch
Z2hww: (8.9)
To extremise the area, we varyAS2 with respect to‰./to obtain the Euler-Lagrange equation, which schematically takes the form
2 Z
hh h2
‰00./CA ‰0./3CB ‰0./2CC ‰0./CDD0; (8.10)
whereA,B,C, andDare some complicated expressions involving only the metric components,Z,
Figure 8.1: The red curve shows the transverse S2 with minimum and maximum area of a particularly deformed ring horizon, as found by the variational method.
and their derivatives (i.e., independent of‰./). In practice, the exact expression can be produced using computer algebra. The boundary conditions are obtained by requiring that the solutions are smooth spheres, in particular that there are no singularities at the poles. Expanding (8.8) near D0, we can see that the smoothness condition is given by
h ‰0.0/2C2 h ‰0./Ch D R2hww
1C1=tanh; (8.11) which can be solved to obtain a Neumann boundary condition
‰0.0/D 1 h
0
@ h C s
h2 h
h R2hww
1C1=tanh
1
A: (8.12)
Note that the choice of positive square root is dictated by the direction of the black ring’s rotation.
Likewise, by considering (8.8) near D, we get
‰0./D 1 h
0
@ h C s
h2 h
h R2hww
1C 1=tanh
1
A: (8.13)
The resulting boundary value problem can be discretised using finite differences and solved via Newton’s method. In general, we expect there to be multipleS2solutions corresponding to local extrema and saddle points. For example, with themD2perturbation, the symmetry of the system is such that we should expect the minimal and maximalS2 to have another copy on the opposite side of the ring, giving a total of four solutions during the first generation of GL instability. Since solutions obtained by Newton’s method depend on the initial guess used, we proceed by running the Newton solver on multiple initial guesses around the ring in order to converge to all possible
Figure 8.2: To measure elastic deformation of the ring horizon, we use distances along spacelike geodesics from the centre of the ring to the point where they intersect the innerS1of the horizon.
These are shown by the red curves.
solutions. A good choice for this is to use constant- curves as initial guesses. Even though these curves do not satisfy our boundary conditions, we found that the Newton solver can quickly correct for this. We can then pick out the solutions with minimal and maximal areas, as displayed in Figure 8.1, to characterise the GL instability.
Over the course of our simulation of thin black rings, we identified a new type of non-axisymmetric instability which deforms the shape of the ring but does not cause its thickness to vary like in GL.
This elastic mode instability is somewhat more mathematically straightforward to characterise compared to the GL instability. Let.r; /be the polar coordinates on thez D0plane. For each 0 0 < 2, let.r./; .//be the affinely-parametrised spacelike geodesic emanating from the origin in the direction 0, i.e. r.0/D0, .0/ D 0and 0.0/ D0. The geodesic intersects H at some parameter Dwith ./ D , and we denote length from D0toD by`. 0/. By treating as a bijective function of 0, we can use`. 1. //to characterise the distance from the centre of the ring to various points along its innerS1. Any variation in this function is indicative of the deformation in the shape of the ring, which is precisely what we are after. This procedure is depicted in Figure 8.2.
t =p
M D0 t =p
M D5:53 t =p
M D17:3 t =p
M D17:4
Figure 8.3: Evolution of a fat ring with D 0:7, subject to an m D 2 perturbation mode with amplitudeAD0:02. The ring initially grows in coordinate size as the evolution goes through the gauge adjustment phase. This ceases att =p
M 5. We can then clearly see the radial instability kicking in, causing the ring to become thicker over time. Eventually, the hole in the middle closes att =p
M 17 and the geometry settles down to that of a Myers–Perry black hole. Here, we show snapshots corresponding to the coarsest time steps immediately before and after a spherical horizon is formed. Note that there is no substantial deformation in the shape of the ring’sS1 or non-uniformity in theS2size in this case.