Initial data for an asymptotically flat black string
by
Shannon Potter
A Thesis
Presented to
The University of Guelph
In partial fulfilment of requirements
for the degree of
Master of Science
in
Physics
Guelph, Ontario, Canada
c Shannon Potter, September, 2014 ABSTRACT
Initial data for an asymptotically flat black string
Shannon Potter Co-advisors: University of Guelph, 2014 Dr. Luis Lehner and Dr. Eric Poisson
It has been argued that a black string in a compact fifth dimension is unstable and that this instability can lead to a bifurcation of the event horizon. Such an event would expose a naked singularity and – if observable by an asymptotic observer – would constitute a violation of cosmic censorship. While the previously studied black string appears to bifurcate, strictly there are no asymptotic observers when one dimension is compact. With this motivation, we consider an asymp- totically flat black string in a non-compact fifth dimension, as an instability in this black string may lead to a true violation of cosmic censorship — an event of theoretical interest. We limit ourselves to the construction of initial data — a metric representing the black string at a moment in time.
While we do not consider the time evolution, and hence do not observe the instability, we construct initial conditions which we expect to be unstable. To assess this, we find the apparent horizon and evaluate curvature invariants along the horizon. To my family, my friends and to Vincent:
for forgiving me when I wasn’t there.
To Linda Allen:
who gave advice, and made exceptions, indiscriminately.
And to everyone else in the library, in our office, and in the coffee shops:
you have been the borders of my life.
iii Acknowledgements
Thank you to my advisor Luis Lehner for his unwaverable patience and support. And to the rest of my advisory committee: Martin Williams — who gave always needed encouragement, and Eric
Poisson for reviewing my thesis. And thank you to Reggi Vallillee, for her help with so many things over the last few years.
This work was supported in part by a Queen Elizabeth II Graduate Scholarship in Science and
Technology, in part by an NSERC Discovery Grant to Luis Lehner, and through departmental bursaries.
iv Contents
Acknowledgements iv
1 Introduction 1
1.1 Cosmic censorship ...... 2
1.2 5D Schwarzschild solutions ...... 3
1.3 Gregory-Laflamme Instability ...... 6
1.4 Asymptotically flat black string ...... 8
2 Notation and Definitions 10
2.1 Discretisation ...... 11
2.2 Matrix Operations ...... 13
2.3 Derivatives ...... 14
2.4 Newton’s Method ...... 17
2.5 Sylvester’s Equation ...... 19
2.6 Derivatives of matrix products ...... 22
2.7 Separating a Hadamard product ...... 25
2.7.1 Hadamard product as Kronecker product ...... 25
2.7.2 Hadamard product as matrix product ...... 27
2.7.3 Hadamard product as matrix product for a separable matrix ...... 27
2.8 Separating matrix triple product ...... 30
3 Initial data for an asymptotically flat black string 34
3.1 Cauchy decomposition and constraint equations ...... 34
3.2 Obtaining consistent data from the constraint equations ...... 38
v 3.2.1 Analytic Method ...... 40
3.2.2 Numerical Method ...... 44
3.2.3 Solutions common to both ...... 52
4 Physical properties 55
4.1 Mass ...... 55
4.2 Apparent Horizon ...... 58
4.2.1 Method to solve for apparent horizon ...... 65
4.3 Curvature Invariant ...... 70
5 Tests and Applications 71
5.1 Conformal factor ...... 71
5.1.1 Analytic solutions ...... 71
5.1.2 Numerical Solutions ...... 77
5.2 Physical properties ...... 79
5.3 Concluding Remarks ...... 87
Bibliography 88
Appendices 89
A Numerical Methods 90
A.1 Gauss-Seidel Newton ...... 90
A.2 Linear Least Squares Regression ...... 92
A.3 Interpolation ...... 93
vi List of Tables
2.1 Derivatives and J(X˜ )H˜ for differnt forms of F(X)...... 24
vii List of Figures
1.1 Gregory-Laflamme Instability ...... 8
5.1 Analytic solutions for conformal factor and density ...... 73
5.2 Convergence of analytic solutions for conformal factor ...... 74
5.3 Examples of analytic solutions for an AFBS of varrying magnitudes ...... 75
5.4 Examples of analytic solutions for a deformed AFBS ...... 76
5.5 Numerical conformal factor and unphysical density ...... 79
5.6 Estimate of ADM mass ...... 81
5.7 Analytical conformal factor, unphysical density and Kretchmann scalar ...... 82
5.8 Apparent horizon and residue of conformal factor for analytical solution ...... 83
5.9 Kretchmann scalar evaluated on the horizon for numerical solutions ...... 84
5.10 Numerical conformal factor and unphysical density ...... 85
5.11 Apparent horizon and residue of conformal factor for numerical solution ...... 86
viii Chapter 1
Introduction
In this thesis, we are going to construct a model of a black string, for simplicity, at a single moment in time. A black string is a type of black hole which extends in a hypothetical extra fifth dimension; each cross-section of the string is a four dimensional black hole. We do this because, independent of the plausibility of this black string or a fifth dimension, the black string is known to be possibly unstable and the instability is of theoretical interest. It has been argued that an unstable black string, initially described by a single, continuous horizon, can separate into multiple black holes — an event which exposes a naked singularity.
We work within the framework of General Relativity (GR). The defining equations of GR are the Einstein Field Equations (EFE) which relate the energy of a spacetime — mostly contained in the stress-energy tensor Tαβ with its curvature. The curvature is encoded in the Riemann tensor
α Rβγδ, and in quantities derived from it Rαβ and R. By a spacetime we mean a collection of points which (except within a black hole) form a smooth convex manifold, along with a metric tensor — a mathematical object which specifies how far apart neighbouring points are in the manifold. The
EFE are, 1 R − Rg = 8πT . (1.1) αβ 2 αβ αβ
For a given stress-energy tensor, if there is a unique solution, it is given by a manifold and a metric tensor, which converts coordinate displacements (dxα) to physical distance (ds) by,
2 α β ds = gαβdx dx .
1 α The repeated indices above (α and β) imply summation over the coordinates x . The value of gαβ depends on the coordinate system chosen and the curvature of the spacetime; the curvature tensor
α Rβγδ can be obtained from gαβ. By equation (1.1), if a spacetime is curved, it must contain energy, and likewise the spacetime around an object with energy must be curved. According to GR, massive objects do not exert any force on one another, but rather they curve the spacetime in their vicinity, and it is this curvature which influences the paths of bodies, such as the orbit of the moon or the planets.
A distinct feature arising in GR is the existence of black holes. Owing to the highly curved spacetime where there is a black hole, and the limited speed of light, it is impossible for anything inside to escape the event horizon (even light — which, while massless, is still constrained to follow paths in spacetime).
The interface between the outer region — where the path of an outgoing observer, particle, or ray of light can continue outward from the black hole, and an interior region where it cannot, defines the event horizon. Outside the event horizon, the physics of the black hole may be similar to other massive objects – like a star, other bodies may orbit it etc. Inside the horizon, all paths are converging and focused at the interior. It is not realistic for multiple things to occupy the exact same location, and so at this point — the singularity — GR breaks down. While the theory can predict physical phenomena outside the horizon well, the future of anything which crosses the event horizon to the interior of the black hole is uncertain, as the path inevitably leads to a point which
GR cannot say anything about.
1.1 Cosmic censorship
In four dimensions (4D) — the three familiar spatial dimensions, plus time — the concern that
GR is unable to predict the fate of things entering the black hole is mitigated by the existence of an event horizon. Despite their being a point where GR loses predictability, nothing from the singularity can affect anything outside the horizon — as, by definition, all of the paths are converging to, and terminating at, the singularity — and so anything happening outside the horizon (orbits and paths of observers, particles or light) can be correctly predicted. It is thought that for any
2 spacetime singularity a horizon exists surrounding it and this is formalised by the Cosmic Censorship
Conjecture (CCC). It is in this sense that GR can be safely used: despite having a point (or region) of unpredictability, this region is censored by the event horizon and the external spacetime can still be described by GR.
In 4D, the CCC appears to hold under generic conditions. Stationary black objects are very simple and as a result, only a few different types exist. A non-rotating black hole is a Schwarzshild black hole, and by Birkoff’s theorem this only has a single parameter, mass. A black hole can also rotate, and possess an electric charge – giving rise to Kerr and Reisner-Nordstrom black holes. The fact that there is a unique solution for a given set of parameters is consistent with these black holes being stable. If any of these black objects were found to be unstable — so that a small perturbation to the horizon, which could be caused by in-falling matter, grows in time — then it is plausible that the black hole may undergo changes which expose the singularity. One approach to test the stability of a black object is take the known solution which describes it, perturb it by a small quantity, and determine if the perturbations will grow – indicating it is unstable — or radiate away, leaving a stable black hole (need not radiate). If it were found that a small variation of the initial conditions resulted in a wildly different solution, the black hole would be seen to be unstable. A viable way to search for realistic scenarios of naked singularities, is to appeal to perturbations of the Schwarzschild/Kerr solutions. If a singular solution were found this way, then it would suggest that cosmic censorship is an artifact of spherical symmetry. However no solutions of perturbed Schwarzschild or Kerr spacetimes have been found which have naked singularities. And it has been shown that, consistent with the CCC, that static black holes are stable so that small perturbations to the black hole are dampened, the event horizon remains intact, and the singularity censored. While some violations have been found, they require delicate conditions and in this sense are not realistic. [18].
1.2 5D Schwarzschild solutions
Although classical GR is able to correctly predict physical phenomena on a macroscopic scale, several modifications to the theory have been proposed, with the aim of having a theory consistent with quantum mechanics (a theory of quantum gravity). Some of these such as string theory include more than three spatial dimensions, and these additional spatial dimensions are often taken to be
3 compact — small, toroidal (doughnut-shaped) dimensions which, while not easily observed, are desirable for the theory. The inclusion of higher dimensions give rise to black objects with distinct physical features, not present in 4D. This is a consequence of the different rotational dynamics afforded by new dimensions, and the freedom to extend 4D horizons in higher dimensions. Several higher dimensional black objects are possible mathematically and for this larger class of solutions, it is not clear that stability and cosmic censorship, present in 4D, hold. Next, we give the simplest higher dimensional black hole solution — the extension of the Schwarzschild black hole – and subsequently review how it has been shown to be unstable.
In 4D outside of a spherically symmetric distribution of energy of total mass M, the metric is given by the Schwarzschild solution,
r ds2 = −V (r)dt2 + V (r)−1dr2 + r2dΩ2,V = 1 − Sch , (1.2) Sch Sch 2 Sch r
2 2 2 2 where rSch = 2M4 is the location of the event horizon, dΩ2 = dθ + sin (θ)dφ , and units are such that c = G4 = 1. For a given mass, M4, the solution is unique.
The static solution to the vacuum EFE which maintains the 4D symmetry gives one of the simplest higher dimensional black holes, and it is an extension of the Schwarzschild black hole to one additional spatial dimension, 5D. It is symmetric in the 4D radial coordinate but, unlike the
Schwarzschild solution, it is not unique. The spacetime can be either spherically symmetric in the 5th dimension — a hyperspherical black hole — or it can be taken as independent of the 5th dimension
— a black string.
In 5D we can either use spherical or cylindrical coordinates. For the black string we simply add the dz2 to the line element in equation (1.2) and have an extension of the Schwarzschild metric which is translationally invariant in the fifth dimension z. In this case cylindrical coordinates (t, r,
θ, φ and z) are more convenient.
r ds2 = −V (r)dt2 + V (r)−1dr2 + r2dΩ2 + dz2,V = 1 − Sch . (1.3) Sch Sch 2 Sch r
4 The event horizon is the same as the Schwarzschild black hole, rSch = 2M4, where M4 is the mass in a z = constant slice of the spacetime. As the metric is independent of z this trivially satisfies the vacuum field equations because they are satisfied on each 4D (z = constant) slice. Because the hyperspherical black hole is symmetric in the 5th dimension, it is better expressed in spherical coordinates: an angular coordinate χ and a radial coordinate R, which are related to the cylindrical coordinates by r = Rsin(χ), z = Rcos(χ) and R2 = r2 + z2. The metric of the hyperspherical black hole is, R2 ds2 = −V (R)dt2 + V (R)−1dR2 + R2dΩ2,V = 1 − BH , (1.4) BH BH 3 BH R2
2 2 2 2 2 2 where dΩ3 = dθ + sin (θ)dφ sin (χ) + dχ .
This non-uniqueness leads to the question of stability: if two static configurations are possible, and a black object in one state is perturbed, what is there to prevent it from transitioning to the other state? As we will discuss next, the stability of the 5D Schwarzschild black hole (string) depends on the ratio of the length of the black object to its mass, L/M. This has been studied in a compact 5th dimension, where it is supposed that the black string extends the entire 5th dimension, forming a closed loop. In this context a long, thin black string may become unstable (large L/M) as may a heavy, hyperspherical black hole in a small compact dimension (small L/M)). The former is analogous to a thinning stream of tap water; the evidence of the instability is the transition of the continuous stream to beads. The later is as a large sphere constrained to a small torus, the two poles are forced closer to one another, inevitably pulling together, with this deformation illustrating that the spherical configuration is unstable.
For later use, we note a few aspects about the properties of the metric in equation (1.4). It is a special case of a Schwarzschild-Tangherlini metric, a generalisation of the Schwarzschild metric into
D dimensions. The relationship between the mass of the black hole, M5, and the horizon radius is
[15], 2 3πRBH M5 = . (1.5) 8G5
The metric in equation (1.4) can also be written in a conformally flat form as,
R2 ds2 = −V (R)dt2 + Ψ2 dR2 + R2dΩ2 ,V = 1 − BH . (1.6) BH 3 BH R2
5 Where Ψ is the conformal factor, which by symmetry of the spacetime, depends only on R. To find the conformal factor we follow [12] and seek a Ψ(R) so that,
−1 2 2 2 2 2 2 2 VBH (r) dr + r dΩ3 = Ψ(R) dR + R dΩ3 .
−1 2 2 2 2 2 2 r For this to hold it must be that VBH (r) dr = Ψ(R) dR and r = Ψ(R) R , or Ψ(R) = R where
dr r p r and R are related by, dR = R VBH (r). Integrating and choosing the integration constant so that Ψ → 1 as R → ∞, the solution is, R2 Ψ = 1 + BH . (1.7) BH 4R2
For the black string metric in equation (1.3) we cannot find a conformal factor ΨBH in the same way.
1.3 Gregory-Laflamme Instability
As shown by the metrics given in equations (1.3) and (1.4), for a black object having the same mass (and no charge or angular momentum) there are two distinct solutions possible. However it could be that one solution is omitted on physical grounds: given a black hole and a black string of the same mass M, is it expected that the one that would actually result from gravitational collapse would be the one with greater entropy. The comparison of the entropy for a black hole and a black string was done by Gregory and Laflamme[7][6], and they found a critical value of L/M, above which a hyperspherical black hole is entropically preferred, and below which a black string is. Because black strings above the critical length could attain higher entropy as their black hole counterpart, it was conjectured and shown perturbatively that black strings above this length are unstable. The entropy of a black hole is proportional to the surface area of the event horizon, which can be found from integrating the invariant volume element, dV = p|g|dxn, where g is the determinant of the metric. For a surface of constant radius (the horizon radius: rBS or RBH ) at constant time t,
4 2 6 2 4 |gBS| = r sin (θ) and |gBH | = R sin (θ)sin (χ) correspond to the areas,
Z π Z 2π Z L 2 2 ABS = rBS sin(θ)dθdφdz = 4πrBSL, 0 0 0 Z π Z π Z 2π 3 2 2 3 ABH = RBH sin(θ)sin (χ)dθdχdφ = 2π RBH . 0 0 0
6 The location of the horizon of the black string is the same as in 4D (except with G = G5): rBS = 2G5M4, as each z = constant slice is just a 4D Schwarzschild black hole. M4 is the mass contained in the 4D slice, so for a black string of length L the mass would be M = M4L. For the black hole, the mass is M = M5 with M5 that in equation (1.5). Thus the areas are:
M 2 A = 16π G2, BS L 5 r 8 A = 16πM 2G3/2 . BH 5 27πM
The black hole is entropically preferred (ABH > ABS) if,
L2 27πG > 5 . M 8
In the compact case, G5 = L and the inequality above is that in [6]. This suggests that a black string is unstable (since a black hole is entropically preferred) if the length of the string is large with respect to the mass or equivalently—as the mass is proportional to the radius —with respect to the radius so that long, thin black strings are unstable. This was shown in 1991 by Ruth Gregory and
Raymond Laflamme and is known as the Gregory-Laflamme instability. They showed the instability explicitly in by perturbing the metric in equation (1.3) by adding γαβdλ, where dλ is small[7, 8].
Requiring that gαβ also satisfy the vacuum Einstein Equations and solving for the perturbation,
Ωt+iµixi they found that γαβdλ ≈ e . If Ω > 0 the perturbation grows in time (t), and they found that Ω > 0 corresponds to long wavelength perturbations in the 5th dimension (see Figure 1.1). This is consistent with the entropy argument where strings above a critical length are unstable, as long wavelength perturbations can only propagate on correspondingly long strings.
While Gregory and Laflamme showed that an instability exists they only speculated on the end state of the unstable black string, and they conjectured it would bifurcate. In conjunction with their argument based on the entropy of the black string, their analysis was convincing and it was supposed that such a scenario would arise, and that the details of the bifurcation—and any implications for cosmic censorship—were waiting on a full theory of quantum gravity.
7 Interest in the classical problem was revived in 2001 when Horowitz and Maeda showed that bifurcation cannot occur in finite proper time along the horizon[10]. They argued that the more likely end state is a non-uniform string, but did not rule out that the horizon pinches off in infinite proper time along the horizon. Infinite proper time on the horizon may correspond to finite proper time for an asymptotic observer. If the horizon pinches off there will be a momentarily naked singularity, and if this is observed asymptotically, violates cosmic censorship: the event horizon ensures that the predictability of the asymptotic spactime is not compromised by the singularity, and if this is lost — even momentarily — GR fails to model the spacetime.
Since then the full non-linear evolution of the Einstein Field Equations showed that an unstable black string evolves into a chain of hyperspherical black holes attached by thing string-like necks[14].
The thin necks are subject to the Gregory-Laflamme instability as well, and a fractal structure develops with successive generations of unstable necks. By looking at the change in string thickness from generation to generation, it was argued that the thickness does reach zero, and the horizon bifurcates. [14]
Figure 1.1: Plot of Ω versus wavenumber scaled by the horizon radius (r+). This shows that there are unstable modes (Ω > 0) for large wavelength (small m). Figure reproduced from [6].
1.4 Asymptotically flat black string
The evolution in [14] was done in a compact fifth dimension. Strictly, the CCC applies to asymp- totic observers — those infinitely far away. Because the dimension is compact, it is not possible to have an asymptotic observer along the z-direction. In this sense, a black string bifurcating in a compact dimension could be argued to not constitute a violation of cosmic censorship. Here we consider the non-compact scenario: an elongated, but asymptotically flat, black string extended in
8 the 5th dimension. The expectation is that the fractal structure found in [14] is not an artifact of the compactness but a generic outcome of the instability. If this is the case, it should arise in the non-compact dimension as well.
Our objective in the next chapters will be to construct consistent data on a given hypersurface of the spacetime which represents an AFBS. In our case, the spacetime will have four spatial dimensions and one time, and we construct data on a slice of the spacetime which is a hypersurface of constant time. If such data is sufficiently similar to that for a black string in a compact dimension, our results will suggest that the end state which has been conjectured for the compact case is likely to hold for the AFBS. Unlike the compact black string, the AFBS is (topologically) a S(3) black hole. We aim to find data for the AFBS which, far from the ends, will resemble a black string.
However, the AFBS is intuitively not a static solution to the EFE: because it is topologically a highly distorted hyperspherical black hole, it is expected to contract to a hyperspherical black hole.
However if the AFBS is sufficiently long, we expect that the black string will be unstable and tend more rapidly towards a bifurcation of the horizon rather than a contraction to a hyperspherical black hole. That this is the case can be seen by nothing that the time for the AFBS to contract would be proportional to L (linear), but the time for the instability is exponential since the perturbation goes like eΩt. Therefore, while we do not know the exact length at which the instability precedes the contraction it is natural to expect that, in principle, a long enough string can be constructed.
In Chapter 2, we discuss some notation and definitions which underly some of the methods used in Chapters 3 and 4. In Chapter 3, the Cauchy decomposition (Section 3.1) is used to establish constraint equations that the gravitational field must satisfy on the hypersurface, and from these we obtain initial data for the metric of the AFBS. Simplifying assumptions are made along the way
(which are described in Section 3.2) so that the constraints reduce to a single equation which, when supplied with an energy density, determines the metric of the AFBS. While analytic solutions can be constructed for a few specific density distributions, ultimately we rely on numerical methods to solve for generic cases. The methods we use are described in Sections 3.2.1 and 3.2.2. Finally, in order to assess whether the data found resembles a black string, we find the shape of the apparent horizon (Section 4.2) and determine the characteristics of curvature invariants evaluated on the horizon (Section 4.3).
9 Chapter 2
Notation and Definitions
In Chapter 1 we motivated our interest in an asymptotically flat black string — a black object in a noncompact 5th dimension which is analogous to the previously studied black string in a compact
5th dimension. Our objective is to study asymptotically flat black strings (AFBS) by constructing consistent data on a spacelike hypersurface in the spacetime. A novel extension of standard methods are introduced to solve numerically for the constraint on such hypersurface, and for the apparent horizon. In this chapter we describe the notation, general methods, and some useful relationships which are used in Chapters 3 and 4. These chapters describe the problems we address in this work. In particular, we will show in Section 3.2 that, under certain conditions, the constraint on the hypersurface reduces to a second order nonlinear equation for a function of two variables – a radial coordinate, r, and a coordinate along the 5th dimension, z. Similarly, when looking for an apparent horizon (Section 4.2) we solve another second order equation for a different function of r and z.
Here we define some notation and operations which we will use in both methods, and we do so in terms of a general function which we denote by χ(r, z). We introduce a discrete approximation to χ(r, z) (Section 2.1) which allows us to write the PDEs as algebraic equations. General matrix operations which we use are defined in Section 2.2, and differentiation is done using discrete derivative operators (Section 2.3). By writing the PDEs algebraically we are able to solve them using Newton’s
Method (Section 2.4). We find that, under some conditions, the equations used in implementing
Newton’s method are of the form of the Sylvester equation (Section 2.5). In order to take advantage of existing methods developed for solving the Sylvester equation, additional relationships are worked
10 out which help us to see how, and under what conditions, the equations can be written this way
(Sections 2.6–2.8).
2.1 Discretisation
We formulate an algebraic problem by considering the unknown function χ(r, z) over a discretised domain. We are interested in finding χ over a real domain (rmin, rmax) × (zmin, zmax) and we solve for the discrete values of χ represented by an nr × nz grid. Explicitly, the discrete approximation to χ(r, z) is the nr × nz matrix which we denote by X,
χ(r , z ) χ(r , z + ∆z ) ··· χ(r , z ) min min min min j min max χ(rmin + ∆ri, zmin) χ(rmin + ∆ri, zmin + ∆zj) χ(rmin + ∆ri, zmax) X = . . . . .. . . . . χ(rmax, zmin) χ(rmax, zmin + ∆zj) ··· χ(rmax, zmax)
We define the coordinates so that the origin (r = 0, z = 0) lies at the centre of the distribution of density. As a result, both the functions that χ(r, z) will represent have most variation in the vicinity of the origin. We do this because otherwise the points near the central boundary of X are subject to larger numerical error than the bulk points. This is a result of having to use less accurate approximations to derivatives — forward and backward difference equations instead of central, as shown below— at the boundary. To avoid having the part of the domain where χ(r, z) varies most correspond to where the derivatives of X are determined with less accuracy, we extend r to negative values and let rmin = −rmax. One additional, but important, added benefit is that this automatically preserves regularity of the solution at r = 0. We also adopt zmin = −zmax, and so the origin corresponds to the maximum of the density. This increases our domain, and the number of points representing X that we need to solve for, by a factor of 4. While it would be quicker to solve on the smaller domain, we use the larger one because we can be more confident that numerical issues associated with the boundary are not influencing the solution at the interior. In the case of the
Hamiltonian constraint, far from the source the solution χ(r, z) ≈ r−p for some positive power p, and so by placing boundaries far from the source we ensure that the forward and backwards derivatives are not introducing significant error. For the function used in finding the horizon we are not able to
11 use any boundary conditions, and so it is particularly important that the issues associated with the boundary are far removed from the region of interest.
For the same reason — anticipating the region of interest to be near the origin — we adopt a grid spacing which is non-uniform and gives higher resolution (smaller spacing) near the origin. The spacing ∆ri and ∆zj is such that within some region around the origin a fine spacing is used, and outside a coarse spacing is used:
fine ∗ ∆r |r| ≤ r ∆ri = coarse ∗ ∆r |r| > r
fine ∗ ∆z |z| ≤ z ∆zj = coarse ∗ ∆z |z| > z
This allows us to obtain high resolution where it is most needed, while otherwise keeping nr and nz as small as possible (as compared to the alternative of using ∆rfine and ∆zfine everywhere because we need it in the interior). The derivative matrices, defined below, are modified accordingly around these cutoffs to account for the abrupt change in ∆r or ∆z. With this in mind, we employ a coarse spacing which is an integer multiple of the fine spacing as this ensures that points on the fine grid can be treated as an extension of the coarse grid as necessary.
In formulating the algebraic problem we need to arrange the unknowns — the nr × nz elements of X — in vector form. We define the vector form of X as the nr · nz × 1 vector,
T X˜ = . X1,1, X2,1, ··· , Xnr ,1, X1,2, X2,2, ··· , Xnr ,2, ··· , X1,nz , X2,nz , ··· , Xnr ,nz
In what follows X˜ denotes the vector form of a matrix X defined as above — effectively by “stacking columns” instead of rows.
12 2.2 Matrix Operations
The operation A◦B represents the Hadamard (element-wise) product of matrices A and B (pro- vided they are of the same size), and A B Hadamard (element-wise) division. For an m×n matrix
A, A ⊗ B is the Kronecker product,
a B ··· a B 11 1n . .. . A ⊗ B = . . . . am1B ··· amnB.
Basic properties of the Kronecker product are available in [19], and a more extensive discussion, with some proofs, is given in [2]. For later reference we list a few properties here. The inverse of a
Kronecker product is the Kronecker product of the inverses:
(A ⊗ B)−1 = A−1 ⊗ B−1. (2.1)
The transpose of a Kronecker product is the Kronecker product of the transposes:
(A ⊗ B)T = AT ⊗ BT . (2.2)
A function applied to a Kronecker product (of this form) may be applied to the argument instead
[2],
f (A ⊗ 1) = f (A) ⊗ 1. (2.3)
Similarly, f (1 ⊗ B) = 1 ⊗ f (B), but this does not hold generally for f (A ⊗ B). For matrices of compatible sizes,
(A ⊗ B)(C ⊗ D) = (AC ⊗ BD) . (2.4)
When formulating the PDEs as algebraic problems, the Kronecker product arises naturally due to the block structure of the matrix needed to differentiate the vector X˜ (shown in the next section).
A very useful property of the Kronecker product, which we use frequently, is the following:
Y = AXB ↔ Y˜ = BT ⊗ A X˜ . (2.5)
13 Note that the first equation contains all matrices, whereas in the second X and Y are in their vector forms, X˜ and Y˜ . Equation (2.5) is used to switch between an algebraic problem with respect to the vector X˜ , and the equivalent one with respect to the matrix X. That equation (2.5) holds is easily understood in the context of differentiation, as shown below, but a general proof is given in [2]. The
Kronecker sum is an mn × mn matrix defined in terms of an n × n matrix P and m × m matrix Q as,
P ⊕ Q = 1n ⊗ Q + P ⊗ 1m. (2.6)
The Kronecker sum satisfies1,
exp(P ⊕ Q) = exp(P) ⊗ exp(Q). (2.7)
It has been shown that the eigenvalues of P ⊕ Q can be expressed in terms of those for P and Q.
From theorem T2.14 in [2], “βk ⊗ αi is an eigenvector of P ⊗ Q with eigenvalue αiβk and is also
2 an eigenvector of [P ⊕ Q] with eigenvalue αi + βk.” where βk is an eigenvalue of P and αk is an eigenvalue of Q.
2.3 Derivatives
To differentiate, derivative operators composed of the appropriate finite difference coefficients are defined [20]. Recall that we have defined the matrix representing χ(r, z) as having r along the
first dimension and z along the second. While this choice is arbitrary, it is the case that the first dimension of the matrix represents a particular spatial dimension and the choice must be kept in mind when taking derivatives.
As an example of what we mean by a derivative operator or matrix, the fourth-order accurate,
first derivative operator for a general n × q matrix — with the variable being differentiated changing
1by exponentiating (2.6) and using (2.3) and (2.4) 2There is a typo in [2], where the bracketed term was P ⊗ Q but should read, as written above, P ⊕ Q[3].
14 1 along the first dimension — is the n × n matrix dn,
− 25 4 −3 4 − 1 0 ··· 0 12 3 4 . 1 1 . − 0 0 ··· . 2 2 1 2 2 1 12 − 3 0 3 − 12 0 0 1 − 2 0 2 − 1 0 ··· 0 12 3 3 12 1 −1 . .. . dn = diag(∆) × . . . . ··· 0 1 − 2 0 2 − 1 0 12 3 3 12 1 2 2 1 0 − 0 − 12 3 3 12 . . 1 1 . 0 0 0 − 2 0 2 1 4 25 0 ··· 0 4 − 3 3 −4 12 where diag of a vector denotes a diagonal matrix which has the vector along the diagonal and × is regular matrix multiplication. In this case, diag(∆)−1 is the matrix with the inverse grid spacing
1 , 1 , ..., 1 on the diagonal. To see that this operator will work, note that when applied to a ∆1 ∆2 ∆n vector X˜ , − 25 X + 4X − 3X + 4 X − 1 X /∆ 12 1 2 3 3 4 4 5 1 1 − X1 + X3 /∆ 2 2 1 2 2 1 12 X1 − 3 X2 + 3 X4 − 12 X5 /∆ 1 . d X˜ = . . n 1 2 2 1 Xn−4 − Xn−3 + Xn−1 − Xn /∆ 12 3 3 12 − 1 X + 1 X /∆ 2 n−2 2 n−1 1 4 25 4 Xn−4 − 3 Xn−3 + 3Xn−2 − 4Xn−1 + 12 Xn /∆
The result is a vector containing all of the derivatives at each point, as represented by the finite difference approximations with forward (backward) coefficients being used for the first (last) element, and lower order accuracy used for the second and second-to-end points (second order accurate). We have left the spacing (∆) general in the above expressions, but, as mentioned, we will use only four distinct values: ∆rfine, ∆rcoarse, ∆zfine and ∆zcoarse. Notice that, as defined, the derivative matrices require the same spacing for each row – for example, in the first row above ∆ must be the spacing between X1 and X2 as well as the spacing between X2 and X3 and so on — and when the
15 th ∆coarse coarse and find grids meet we will use the coarse derivatives and evaluate every ∆fine point where necessary to ensure the same spacing is used for each row. We are careful to ensure that where the grids overlap corresponds to a region that we expect to be relatively flat, as to minimise any error introduced by the abrupt transition of grid spacing. When solving for the conformal factor this expectation is based on the magnitude of the density distribution, and when solving for the apparent horizon it is based on the magnitude of the gradient of the conformal factor.
2 2 −1 A second derivative operator dn is defined similarly, using appropriate coefficients and diag(∆ ) in place of diag(∆)−1, as well as operators for second order accurate derivatives. Because X is de-
fined with r along the first dimension, left multiplying a derivative matrix with X approximates the
first (d1 X) or second (d2 X) derivatives with respect to r. The first and second derivatives with nr nr respect to z are (d1 XT )T = Xd1 T and (d2 XT )T = Xd2 T, as the derivative operators must be nz nz nz nz applied to the transpose of X, XT, and then the result transposed to be have consistent coordinates with X.
Derivative operators used for the vector X˜ are defined differently, as X˜ has elements along both r and z in the first dimension. The second derivative along r is,
2 d 0n ······ 0n nr r r . 0 d2 . nr nr 2 2 . .. . D = 1nz ⊗ d = . . . , nr nr . . 2 . dn 0nr r 0 ······ 0 d2 nr nr nr where 0 is an n × n matrix of zeros, and the first derivative is D1 = 1 ⊗ d1 . nr r r nr nz nr
To differentiate along z, X˜ first must be permuted (just as, in the matrix case, X was transposed).
The permutation matrix which achieves this is the sparse matrix P with non zero elements as
16 indicated schematically below.
1 2 ··· nr +1 nr +2 ··· (nz −1)nr +1 (nz −1)nr +2 ··· nr ∗nz 1 1 0 ········· 0 2 0 0 1 . . . . . . nz 0 1 nz +1 0 1 P = . nz +2 0 1 . . . . . . 2n 0 1 z . . .. . . . . . nr nz 0 ······ 0 ··· 1
The first and second derivatives with respect to z are PTD1 P X˜ and PTD2 P X˜ . While the nz nz same result could be achieved by transformations between the desired vector and the matrix repre- sentation of the z-derivatives of X, the advantage of explicitly defining everything with respect to X˜ is that to implement Newton’s method we will need to identify a Jacobian. This is straightforward if we can distribute all derivative operators to the left of X˜ .
2.4 Newton’s Method
Newton’s method is an iterative method to solve for the roots of an equation or a system of equations[9]. It also enables a nonlinear system of equations to be replaced by a sequence of linear equations. For any function F : Rn → Rm of an n × 1 vector x, the system of m equations,
F(x) = 0, is amenable to Newton’s method. It is not generally possible to solve for x in one step (unless it is linear), and this method requires multiple iterations yielding an approximate or trial solution xk at
∗ ∗ each step k, for k = 1, 2, ..., kmax. Letting x be the true solution (in the sense that F (x ) = 0), the
∗ ∗ solution converges to x if the trial solutions xk approach x as the iteration number k increases.
17 The method requires an initial guess x0, and whether or not the solution converges can depend on
∗ how close x0 is to x .
Given an initial guess x0, Newton’s method provides the incremental change, ∆x, that should be added to the guessed value to obtain a better approximation of x∗. It is based on the Taylor expansion of F around x,
F (x + ∆x) ≈ F (x) + J(x)∆x, where J(x) is the Jacobian of F (x),
∂F1 ∂F1 ... ∂F1 ∂x1 ∂x2 ∂xn ∂F ∂F ∂F 2 2 ... 2 ∂x1 ∂x2 ∂xn J(x) = . . . . . . ... . ∂Fm ∂Fm ... ∂Fm ∂x1 ∂x2 ∂xn
With the goal of finding the solution x∗, for any trial solution xk, Newton’s method prescribes that a better estimate will be x = xk + ∆x for which F (xk + ∆x) = 0, or, from above,
x = xk + ∆x, ∆x = −J(x)−1F (x). (2.8)
If the function F is linear, then the Jacobian above does not depend on x and it will be the case that x∗ = x0 + ∆x and the equation is solved in one iteration. Otherwise, for a nonlinear equation, this is repeated until an x is found which is close enough to the unknown x∗, in the sense that F (x) is sufficiently close to zero. We assess this convergence by monitoring the norm of F at each iteration k, 1 m 2 X 2 L2 F (xk) = Fi(xk) , i=1
∗ and consider an xk to be the solution x when LF (xk) < for a set error tolerance .
Notice however that to solve for the step ∆x by inverting the Jacobian in equation (2.8) would be computationally costly and another method for solving the linear system,
J(x)∆x = F (x),
18 is often used. These include iterative methods (such as Gauss-Seidel) in which the Jacobian is decomposed in a convenient way so as to only require solving triangular matrices, requiring a much lower computational cost. Here, we use Newton’s method but solve J(x)∆x = F (x) by implementing a method used to solve Sylvester’s equation[21] which entails an even lower computational cost, particularly for large matrices.
2.5 Sylvester’s Equation
Sylvester’s equation (also known as a Lyapunov equation in the case B = AT) is a linear equation of the form,
AX + XB = −F, (2.9) for an m × m matrix A, n × n matrix B, and X being m × n. Rewriting equation (2.9) in vector form and using equation (2.5),
AX˜ + XB˜ = −F˜,
˜ ˜ ˜ AX1nz + 1nr XB = −F,
T ˜ ˜ 1nz ⊗ A + B ⊗ 1nr X = −F.
Recognising that the last line is the Kronecker sum, equation (2.9) is equivalent to,
BT ⊕ A X˜ = −F˜. (2.10)
To check that equation (2.9) has a unique solution X, we can check that the equation (2.10) has a unique solution X˜ . Inspecting equation (2.10), it is evident that the condition for a unique solution is that the matrix BT ⊕ A is invertible. Following [2] as in Section 2.2 the system in equation (2.10)
T has a unique solution if the eigenvalues of B ⊕ A are all non-zero. Letting βk denote the eigenvalue
T of B and αi denote those of A, we conclude that the solution exists as long as:
αi + βk 6= 0, ∀ i = 1..nr, k = 1..nz; (2.11)
T where nr and nz are the number of (possibly repeated) eigenvalues of A and B .
19 While equation (2.10) could be solved by inverting BT ⊕A, it is much quicker to solve in the form of equation (2.9) using the Bartels-Stewart algorithm [21, 1]. Decompose A and B using a Shur decomposition[22],
A = QUQ−1,
B = RVR−1, where U, V are upper triangular matrices and R, Q are orthogonal matrices. Substituting these expressions into equation (2.9) yields,
QUQ−1X + XRVR−1 = −F.
We can now multiply from the left by Q−1 and from the right by R,
UQ−1XR + Q−1XRV = −Q−1FR.
Defining Y = Q−1XR, this can be rewritten as,
UY + YV = −Q−1FR.
The left-hand side is equivalent to the Kronecker sum and so with respect to Y˜ the above is,
VT ⊕ U Y˜ = −Q−1˜FR. (2.12)
Equation (2.12) is similar to equation (2.10) but, since U and V are triangular, equation (2.12) is much faster to solve than equation (2.10). This algorithm is used by Matlab [4], which in turn employs SLICOT [11] and LAPACK [13] routines.
Notice that the same approach can not be taken for a general equation of the form,
AX + XB + CXD = −F, (2.13)
20 because there are three terms on the left hand side, except in the special cases below. However, under some conditions, the system can be made triangular. This is the advantage of the Bartels-Stewart algorithm, and so (in principle) may be solved as quickly as the system in equation (2.12).
In the case where C and D share orthogonal matrices with A and B (C = QWQ−1, D = QZQ−1 where W and Z are upper triangular), equation (2.13) becomes,
UY + YV + WYZ = −Q−1FR, Y = Q−1XR.
Or, in vector form, VT ⊕ U + ZT ⊗ W Y˜ = −Q−1˜FR. (2.14)
Since Z and W are upper triangular, ZT ⊗ W is triangular (or very nearly), and so a similar algorithm to the Bartels-Stewart would apply to equation (2.14). However, the Matlab algorithm which solves the Sylvester equation is substantially quicker than what we wrote ourselves, and so it would follow that an extension to solve (2.14) would be slow as well. This only works in the case that
C and D have the same orthogonal matrices in their Schur decompositions as A and B respectively.
While the Schur decomposition is not unique and so it is possible this could be achieved through the choice of Z and W, it is certainly not guaranteed for any C and D.
Another case of C and D, which is discussed more later, is that they are diagonal. In this case DT ⊗ C is guaranteed to be diagonal and the matrix VT ⊕ U + DT ⊗ C is triangular (it is redundant to decompose C and D in this case). Still, implementing the Bartels-Stewart algorithm ourselves turned out much more computationally expensive than the Matlab algorithm.
Lastly we note that the Bartels-Stewart algorithm, while quick, does not take advantage of the repeated structure of A ⊗ B. Rather it solves an mn × mn system by putting it into a triangular form. Equation (2.1) shows how easily a Kronecker product can be inverted but the same does not hold for a Kronecker sum. Note that if we could rewrite the Kronecker sum as a Kronecker product, we would simply have an m × m and n × n matrix to invert instead of the full nm × nm system to
21 solve. This would require solving for C and D where:
AX + XB = CXD,
1 ⊗ BT + A ⊗ 1 = CT ⊗ D.
However this has m2 + n2 unknowns and m2n2 equations and so it not generally solvable (although a least squares solution could be used). The opposite problem — solving for A and B given C and
D — is solvable (with some freedom in the choice of A and B) and this related problem is discussed at the end of Section 2.8.
Finally, while an explicit solution does not appear to exist in term of A, B and their inverses,
A ⊕ B has an inverse in terms of the eigenvectors of AB and their transposes:
1 2 1 2T −1 X X a ⊗ a b ⊗ b (A ⊕ B) = k i k i . αk + βi i k
1 2 1 2 T T Here ak, ak, bi , bi are the eigenvectors of A, A , B, and B respectively, and αk and βi are the corresponding eigenvalues. Since the eigenvalues of A ⊕ B are all combinations of the sum of the eigenvalues of A and B (Section 2.2), and all eigenvalues of A ⊕ B must be non-zero for the matrix to be invertible, it follows that (A ⊕ B) is invertible if A and B do not have any pairs of eigenvalues which add to zero.
We conclude that there seems to be no quick method available for solving equation (2.13). The relationships we work out in Sections 2.7 and 2.8 are used to rewrite (where possible) equations of the form (2.13) in the (easily solved) form of equation (2.10). Section 2.6 summarises some useful relationships which are used in Sections 2.7 and 2.8.
2.6 Derivatives of matrix products
In finding the Jacobian to use Newton’s method, we differentiate F(X) or F(X˜ ) with respect to
X˜ to get J(X˜ ), and multiplying this with H˜ . The result can be put back into a matrix form. We tabulate these results for the forms we use in Table 2.1.
22 The first column shows terms containing the matrix X and the second column is the vector form of the first (using equation (2.5)). Terms which, in the second column, are disassociated to the left of X˜ have straightforward Jacobians (third column). The last column shows the result of applying the Jacobian to the vector representation H and returning to matrix form (through equation (2.5)).
With the goal of using the Bartels-Stewart algorithm, we seek a matrix form of the Jacobian applied to H˜ which is “linear with respect to H” — but specifically in the sense that H can be left- or right-disassociated from the remaining part of the term. While CHDT is linear, this does not hold. The simplest examples where it does hold are the first two terms, AX → AH and
XB → HB. The matrix operators which represent derivatives are (except for the mixed derivatives) of the form AX or XB. Note that the only way this is preserved within a Hadamard product is if the underlying function represented by the matrix is a function of the variable corresponding to the type of derivative. This is shown in rows 6-8 of Table 2.1, where P represents an underlying function depending only on r and Q, one on z.
As before X is nr × nz, as is H, which as vectors are X˜ and H˜ . Here x and y are introduced which denote vectors along r and z respectively, and h is the change for these vectors (nr × 1
T or nz × 1). Other derivatives are defined for X = xδ , as would be the case if X represents a function which is separable and the variable being differentiated with respect to changes along r.
As well as the case that X = γxT (where is it separable and the variable being differentiated is
along z). For the vector form of the nr × 1 vector x, let ˜x = lnz ⊗ x and ˜y = y ⊗ lnr (where ln is an n × 1 vector of ones) so that these have dimensions consistent with how X is defined. C is nr × nr and D is nz × nz (not necessarily diagonal). In the last two lines we have used equation
˜ T (2.4) so that diag(δ) ⊗ 1nr h = diag(δ) ⊗ 1nr (lnz ⊗ h) = (δ ⊗ h) = δh and, similarly for ˜ 1nz ⊗ diag(γ) h. The matrices K and L (defined in the next section) act to select the elements of the Kronecker products which correspond to the Hadamard product.
23 F(X) F(X˜ ) J(X˜ ) J(X˜ )H˜ (matrix form) Regular matrix multiplication ˜ AX (1nz ⊗ A) X (1nz ⊗ A) AH T ˜ T XB B ⊗ 1nr X B ⊗ 1nr HB CXDT (D ⊗ C) X˜ (D ⊗ C) CHDT
Hadamard products (M)◦X = L (M ⊗ X) K KT ⊗ L (M ˜⊗ X) diag (M) M◦H
24 ¯ ˜ ¯ ¯ (AX)◦P 1nz ⊗ diag(P(r))A X 1nz ⊗ diag(P(r))A diag(P(r))AH ¯ ¯ T ˜ ¯ T ¯ (XBdiag(Q(z)))◦Q diag(Q(z))B ⊗ 1nr X diag(Q(z))B ⊗ 1nr HBdiag(Q(z)) (CXDT)◦Q◦P Ddiag(Q¯(z)) ⊗ diag(P¯(r))C X˜ Ddiag(Q¯(z)) ⊗ diag(P¯(r))C diag(P¯(r))CHDTdiag(Q¯(z))
Outer products T T xδ diag(δ) ⊗ 1nr ˜x diag(δ) ⊗ 1nr hδ T T γy 1nz ⊗ diag(γ) ˜y 1nz ⊗ diag(γ) γh ˜ ˜ Notes: A and C are nr × nr, B and D are nz × nz, X and H are nr × nz, X and H are nrnz × 1. For the outer products: x is nr × 1, y is nz × 1, ˜x = lnz ⊗ x ¯ ¯ and ˜y = y ⊗ lnr . The underlying function for P depends on r, that for Q depends on z. P(r)(P(r)) is a column (row) from P (Q).
Table 2.1: Derivatives and J(X˜ )H˜ for differnt forms of F(X) 2.7 Separating a Hadamard product
We now show that it is not possible to rewrite a Hadamard product as a regular matrix product.
This is desirable because in solving both equations numerically we will need to determine a Jacobian and when differentiating terms with Hadamard products it would be useful to have done so. To see this, suppose that we could write,
H◦M = AH.
That is, we could find a matrix A so that H◦M = AH where A depends only on M. The vector form would be,
H◦~M = AH˜ = (1 ⊗ A) H˜ , and the Jacobian, ∂H◦~M = (1 ⊗ A), ∂H~T or as a applied to a matrix X, A (using equation (2.5) or Table 2.1).
Below, we consider the case of rewriting the Hadamard product as a regular matrix product. We show that this is not possible and instead find an approximation for C and D so that,
H◦M = CHD,
H◦~M = CHD˜ , = DT ⊗ C H˜ ,
∂H◦~M = DT ⊗ C . ∂H~T
First we note that without making any approximations, it is possible to rewrite a Hadamard product using a Kronecker product.
2.7.1 Hadamard product as Kronecker product
That the Hadamard product can be expressed as a Kronecker product was pointed out in [17], and originally in [5]. It is done by writing the Kronecker product A ⊗ B, which contains the products of all elements of A with all those of B, and left and right multiplying this with a matrix which
25 removes the unwanted combinations. Consider A and B to be m × n matrices (they have to be the same size to be able to take the Hadamard product).
m Let ei be a vector of length m with the only non-zero entry being the i − th which is equal to 1. It is evident that the Hadmard product H◦M can be obtained from,
emT 0mT ··· 0mT h M h M ··· h M en 0n ··· 0n 1 11 12 1n 1 mT mT mT n n n 0 e ··· 0 h21M h22M ··· h2nM 0 e ··· 0 2 2 H◦M = . . . . . . , . .. . . .. . . .. . . . . . . . . . . mT mT mT n n n 0 0 ··· em hm1M hm2M ··· hmnM 0 0 ··· en where the first matrix on the left hand side selects the proper row of M and the matrix on the right hand side selects the column. Equivalently, since M and H have the same dimensions and
H◦M = M◦H, M and H could switch places on the right hand side. Denoting the left multiplying matrix by J and the right by K,
H◦M = J (H ⊗ M) K = J (M ⊗ H) K.
Proceeding as above to obtain the Jacobian,
∂H◦~M ∂J (M ⊗~ H) K ∂ KT ⊗ J (M ~⊗ H) = = . ∂H~T ∂H~T ∂H~T
Since H◦~M = diag(M˜ )H˜ = diag(H˜ )M˜ , this must be equivalent to,
∂H◦~M = diag(M˜ ). ∂H~T
26 2.7.2 Hadamard product as matrix product
For a general M, separating H◦M = AH can not be done3 without the elements of A depending
Pnr on those of H. This would be equivalent to solving mijhij = k aikhkj for all of the aik,
mijhij = ai1h1j + ai2h2j + ai3h3j + ... + ai4h4j, n n n n n Xz Xz Xz Xz Xz mijhij = ai1 h1j + ai2 h2j + ai3 h3j + ... + ai4 h4j. j=1 j=1 j=1 j=1 j=1
This is the linear system,
n n Xz Xz Ax = b, bi = mijhij, xi = hij, j=1 j=1 which has a least squares solution,
A = bxT(xxT)−1.
This shows that, in general, the elements of A will depend on those of H, through x and b.
2.7.3 Hadamard product as matrix product for a separable matrix
However, if the function represented by the matrix M is separable — if M is representing f(r, z), and f(r, z) = g(r)d(z) — then we can write,
M = γδT, where γ is the vector representation of g(r), and δ is that for d(z). In this case the Hadamard product can be written,
H◦M = CHDT where C = diag(γ) and D = diag(δ). To see this note that left (right) multiplication of a matrix
H with a diagonal matrix corresponds to a row (column) scaling of the elements of H. As will be shown later in this chapter, when writing either of the PDEs as an algebraic equation, we will need to separate the Hadamard products which arise. As was shown above, they can only be expressed as matrix products in the case that the underlying function is separable. As this will not always be
3In the same way, it can be shown that this can not be done for H◦M = HA either.
27 the case, we will sometimes use an approximation. Independent of whether the result is exact or approximate, we solve the following problem. We would like to find vectors γ and δ which satisfy,
γ δ γ δ ··· γ δ 1 1 1 2 1 nz γ2δ1 γ2δ2 ··· γ2δnz M = . . . (2.15) . .. . . . . γnr δ1 γnr δ2 ··· γnr δnz
Because the system in equation (2.15) is overdetermined (nr × nz equations and only nr + nz unknowns), if the function underlying M is not exactly separable, there will be an error introduced by imposing the constraint in equation (2.15). The implications of this error depend on the application
— what the M is that we are constraining, and what we are using it for— and are discussed when equation (2.15) is used.
We solve equation (2.15) as follows. We want to choose vectors γ and δ in order to best equate the elements of equation (2.15). To do this we minimise the squared 2-norm of the element-wise difference of the right and left hand side of equation (2.15). To obtain the optimal vectors, which we denote γ∗,δ∗, we solve the minimisation problem:
nr nz ∗ ∗ X X 2 [γ , δ ] = argmin (L) ,L = (γiδj − Mij) . (2.16) γ,δ i=1 j=1
Setting the first derivatives of L to zero yields the following system of nr + nz equations,
Pnz k=1 (γiδk − Mik) δk i ≤ nr, fi(x) = Pnr k=1 (γkδi − Mki) γk i > nr,
T TT where fi(x) = 0 and we have defined x = γ , δ . While in the context of equation (2.15) we are interested in the elements of γ and δ, the optimisation is better expressed with respect to a single vector (we will choose the x which minimises the error, L).
28 Next we verify that the Hessian is positive definite, assuring that we have found a minimum of L. ~ The Hessian of L or, equivalently, the Jacobian of fi(~γ, δ) is block diagonal, with elements:
P δ2 i = j ≤ n , k k r P 2 k γk i = j > nr, ∂f (~γ, ~δ) i = 2γjδi−n − Mj,i−n i > nr, j ≤ nr, ∂xj r r 2γ δ − M i ≤ n , j > n , i j−nr i,j−nr r r 0 otherwise.
2 P 2 2 2 P 2 2 Letting kδk = k δk and ∆ = kδk 1nr and kγk = k γk and Γ = kγk 1nz , the Hessian is of the form, ∆ Q H(L) = , (2.17) QT Γ
T T T where Q is the nr × nz matrix with elements Qi,j = 2γiδj − Mi,j. Letting z = [z1 z2 ] , we evaluate zT Hz, ∆ Q z1 zT Hz = T T = zT ∆z + zT Qz + zT QT z + zT Γz . z1 z2 1 1 1 2 2 1 2 2 T Q Γ z2
2 2 T Equivalently, because ∆ = kδk 1nr and Γ = kγk 1nz , and at the optimum, Q = γδ ,
T 2 2 2 2 T T T T z Hz = kδk kz1k + kγk kz2k + z1 γδ z2 + z2 δγ z1,
2 2 2 2 T T = kδk kz1k + kγk kz2k + 2(z1 γ)(δ z1).
By the Cauchy-Schwartz inequality,
2 2 2 2 T 2 kδk kz1k + kγk kz2k ≤ z Hz ≤ δkkz1k + kγkkz2k , (2.18) and so H(L) is positive definite (all elements of γ and δ are non-zero and so the left hand side is strictly positive).
29 T We use Newton’s method (see Section 2) to solve iteratively for x∗ = γ∗T, δ∗T . Given an
T −1 T initial guess, we update xi+1 = xi + h where h = − J(x) J(x) J(x) f(x). Because this system is only nr × nz, Matlab’s linsolve function provides a quick way to solve for h.
2.8 Separating matrix triple product
As mentioned, unlike equation (2.9), there is not a quick method to solve equation (2.13) in one step. For this reason we decompose,
CHDT = MH + HN, (2.19) which allows an equation of the form (2.13) to be written in the form (2.9). To find the conditions under which this is possible, we rewrite the above as vectors and use equation (2.6). This lets us remove H, as we want expressions for M and N in terms of C and D only.
CHD˜ T = MH˜ + HN˜ ,
˜ ˜ T ˜ (D ⊗ C) h = (1nz ⊗ M) h + N ⊗ 1nr h,
T D ⊗ C = 1nz ⊗ M + N ⊗ 1nr .
Explicitly the last line is,
T T D11C ··· D1n C M ··· 0n N 1n ··· N 1n z r 11 r 1nz r . .. . . .. . . .. . . . . = . . . + . . . (2.20) D C ··· D C 0 ··· M N T 1 ··· N T 1 nz 1 nz nz nr nz 1 nr nz nz nr
It is evident (after some inspection) that there is no exact solution, but we can still obtain an approximate one. Solving for M and N can be done by minimising the sum of squared errors, where the error E˜ is the vector version of CHDT − MH − HN, with respect to M and N,
T ˜T ˜ T T T E E = H D ⊗ C − 1nz ⊗ M − N ⊗ 1nr D ⊗ C − 1nz ⊗ M − N ⊗ 1nr H. (2.21)
30 This is done using the expression derived in [2] for differentiating Kronecker products,
∂A ⊗ C ∂A ∂C = ⊗ C + (1 ⊗ P) ⊗ A (1 ⊗ P). ∂B ∂B ∂B
The result of minimising (2.21) is equivalent to “solving” for M and N in a more straightforward, but less justifiable, way as follows. Note that from equation (2.20) it must be that N and D are diagonal (ignoring the irrelevant case where C is the identity), and so the diagonal elements of N and D are related by,
T DiiC = M + Nii 1nr ,
T M = D11C − N111nr ,
T = D22C − N221nr ,
= ...
= D C − N T 1 . nz nz nz nz nr
Or, denoting by ln an n × 1 vector of ones, by,
n M = lT Dl C − lT Nl 1 . z nz nz nz nz nr
Therefore, assuming diagonal N and D, we can find M as,
1 1 M = lT Dl C − lT Nl 1 , nz nz nz nz nr nz nz ¯ ¯ M = DC − N1nr ; where D¯ and N¯ are the average values of the elements of D and N.
A similar expression arises for N. This is a result of the least squares minimisation of equation
(2.21) , where the minimisation yields an equation for M and N, but can also be seen by noting that if (2.19) holds than so should its transpose. We have that,
T ¯ ¯ N = CD − M1nz ,
31 and combining this with the expression above for M,
M = M¯ + D¯C − C¯,
N = N¯ + C¯D − D¯ .
To fix N¯ and M¯ we impose that the average of the element-wise error is zero,
M¯ + N¯ + CD¯ jj + DC¯ ii − CiiDjj Hij = 0.
While there is still a choice of M¯ or N¯, we choose them to be equal and,
1 M = D¯C − C¯D,¯ (2.22) 2 1 N = C¯D − C¯D.¯ (2.23) 2
Thus we can replace (approximately),
1 1 CHDT ≈ D¯C − C¯D¯ H + H C¯D − C¯D¯ , 2 2 where C = diag(γ∗), D = diag(δ∗), and γ∗ and δ∗ are solutions to equation (2.16). Note that if the underlying function is actually independent of z then D = 1 (the identity), and M and N are,
1 M = C − C,¯ 2 1 1 N = C¯ − C¯ = C¯; 2 2 and, 1 1 CHDT = C − C¯ H + H C¯ = CH. 2 2
Similarly if the function is independent of r, MH + HM = HDT.
In summary, if we have a Hadamard product H◦M and we know (or can approximate) the underlying function of M as separable, then we can write H◦M = CHDT (using equation (2.16) and C = diag(γ∗) and D = diag(δ∗)). If we would further like to express CD = MH + HN, we
32 use M and N in equations (2.22). If we know that the underlying function really only depends on r or z, then the expression CHDT = MH + HN is equivalent to either CH or HDT, and these can be simply used instead.
33 Chapter 3
Initial data for an asymptotically
flat black string
Here we discuss how to exploit the methods described in Chapter 2 to obtain consistent data in
GR. We make some assumptions about the physical problem which (mathematically) simplify the constraint equations. We discuss how a single equation needs solving which, under some further conditions, is amenable to the methods of separation of variables (Section 3.2.1). More generally, we must use the numerical methods of Chapter 2. The application of these methods to the constraint equation is shown in Section 3.2.2.
3.1 Cauchy decomposition and constraint equations
In order to pose an initial value problem for the metric, one can make use of coordinate trans- formations to single out a spacelike solution. We adopt the Cauchy decomposition, in which the full spacetime manifold is foliated into a series of non-intersecting spacelike hypersurfaces (denoted by Σ) which, along with the metric gαβ, define the spacetime. Any hypersurface can be regarded as defining an initial configuration and its future development obtained via evolutions equations obtained from Einstein’s equations.
A hypersurface in a spacetime with coordinates xα is defined by placing a restriction on the coordinates of the form Φ(xα) = 0. Since Φ, by definition, only changes off of the hypersurface the
34 timelike normal vector to the hypersurface, nα, is proportional to the derivative of Φ,
Φ q n = − ,α , k∇Φk = gαβΦ Φ . (3.1) α k∇Φk ,α ,β
α The factor of proportionality ensures that nα is normalised (nαn = −1) and that the normal vector
α points in the direction of increasing Φ (n Φ,α > 0). The hypersurface, Σ0, that will be used as the initial condition is a hypersurface of constant time, x0 = x¯0:
Φ(xα) = x0 − x¯0.
The metric is separated into components defined on and off the hypersurface by first transforming to convenient coordinates. On each hypersurface, coordinates ya are defined. Points of a fixed ya on each hypersurface are connected by a curve parametrised by a scalar field t(xα). Tangent vectors
α ∂xα α α ∂t(x ) α α on the hypersurface are ∂ya |t = ea , and tangent vectors to t(x ) are ∂t |ya = t . The tangent t is thus defined off of the hypersurface, but it is not necessarily orthogonal to it: tα can have both a
α α component that is tangent (along ea ), and a component normal (along n ) to the hypersurface,
α α a α t = Nn + N ea . (3.2)
The vector N a is called the shift vector and it measures how much a point of fixed ya shifts with each subsequent hypersurface. The scalar N, the lapse, measures the proper time between hypersurfaces
(the parameter t(xα)). With these definitions of coordinates, ya on Σ and tα off of Σ, the coordinates of the full spacetime are parametrised as xα = xα(t, ya). Then a displacement dxα can be written as, ∂xα ∂xα dxα = dt + dya = tαdt + eαdya. ∂t ∂ya a
Or, upon substituting the expression for tα from (3.2),
α α a a α dx = Nn dt + (N dt + dy )ea , where coordinates in the full spacetime are distinguished by Greek letter and in the 4D spacetime,
2 α β Latin. The line element ds = gαβdx dx is written in these coordinates. Using the expression
35 α α for dx , that the tangent and normal vectors to Σ are orthogonal (ea nα = 0), that the normal is
α α β timelike (n nα = −1), and by identifying hab = gαβea eb as the induced metric on Σ, we have that,
2 2 2 a a b b ds = −N dt + hab(N dt + dy )(N dt + dy ). (3.3)
Just as hab is obtained by projecting the metric gαβ onto Σ by taking the inner product with
α β ea eb , any tensor in the full spacetime can be projected onto the hypersurface by contraction with the tangent vectors. Objects and operations analogous to those in the full spacetime exist which
a a pertain only to the hypersurface, including Christoffel symbols Γbc, covariant differentiation A|b and
c a Riemann tensor Rdab,
1 Γa = ham h + h − h , (3.4) bc 2 mb,c mc,b bc,m
a a a m A|b = A,b + ΓmbA ,
c c c c m c m Rdab = Γdb,a − Γda,b + ΓmaΓbd − ΓmbΓda,
α The Einstein Feild Equations are written with respect to the full Riemann tensor Rβγδ. As we are only considering a single slice (constant time), we require a form of the EFE which pertain to
a α a hypersurface. The relationship between Rbcd on Σ and the full Reimann tensor Rβγδ leads to the Gauss-Codazzi equations which in turn lead to the constraints imposed by the EFE on Σ that we are seeking. Projecting Rαβγδ onto Σ, we have the Gauss equation,
α β γ δ ad Rαβγδea eb ec ed = Rabcd + KacKbd − K Kbc. (3.5)
Equation 3.5 illustrates that the projection of the full Riemann tensor onto the hypersurface is given by the intrinsic curvature Rabcd—the Riemann tensor defined in terms of the induced metric hab— and the extrinsic curvature Kab. The extrinsic curvature is the projection of the covariant derivative
µ β of the normal onto the hypersurface Kab = nµ;βea eb = na|b. Referring to equation (3.4), this can
a be non-zero due to curved spacetime (Γbc 6= 0), or due to the surface Σ being embedded in such a
b way that the partial derivative of the normal has a component tangent to the surface (na,bv 6= 0).
36 Projecting along three tangents and one normal vector, we have the Codazzi equation,
µ β γ δ Rµαβγ n eb ec ed = Kab|c − Kac|b. (3.6)
The Gauss-Codazzi equations (3.5 and 3.6) are used to connect the Einstein field equations to Σ.
The field equations, written in terms of the full spacetime metric gαβ, are equivalent to the constraint
Mα = 0, where, 1 M = R − Rg − 8πT nβ. (3.7) α αβ 2 αβ αβ
The normal and tangent components give the Hamiltonian and momentum constraints, respectively:
1 M nα = R − Rg − 8πT nβnα, (3.8) α αβ 2 αβ αβ
1 M eα = R − Rg − 8πT nβeα. (3.9) α a αβ 2 αβ αβ a
µ αβ By using Rµαβγ , as given by the Gauss-Codazzi equations, to calculate Rαβ = Rαµβ and R = g Rαβ and substituting the expressions into the field equations (3.5 into 3.8 and 3.5 into 3.9) the constraints become,
α β 4 2 ab 16πTαβn n = R + K − K Kab,
α β b 8πTαβea n = Ka|b − K,a.
α β α β ρ˜ = Tαβn n is the energy density, T00, and ja = Tαβea n is a current. With these, the above become,
b 8πja = Ka|b − K,a, (3.10)
4 2 ab 16πρ˜ = R + K − K Kab.
These constraints —the Hamiltonian and momentum constraints — must all be satisfied on the hypersurface. The functions to be specified are the current ja and densityρ ˜ as well as the induced metric hab and extrinsic curvature Kab. How this is done, and the simplifications we make, is discussed in Section 3.2
37 3.2 Obtaining consistent data from the constraint equations
To implement equations (3.10) we make some assumptions on the hypersurface: the extrinsic cur-
a vature Kab, and current j , vanish and the density (and therefore conformal factor) are independent of θ and φ, depending only on the radius (r) and extra dimension (z). This simplifies the problem at no real cost; as discussed in Chapter 1, there are no instabilities associated with perturbations of the horizon in the angular coordinates. Also, the physical density ρ(r,˜ z) is suitably rescaled by the conformal factor in order to bring the constraint equation from (3.10) into a mathematically well-posed form. We motivate and discuss these assumptions in turn below.
1 The Einstein field equations (EFE), Rαβ − 2 R = 8πTαβ when provided with source functions Tαβ, are 25 equations for the unknown components of the metric gαβ. As the metric tensor is symmetric, only 15 of these are unique. When the Cauchy decomposition is adopted, as described in Section
3.1, the EFE are embodied in the Hamiltonian and momentum constraints (3.10) and another set of evolution equations which contain second order time derivatives. In this representation the unknown functions are the 10 independent components of the induced metric hab subject to second order equations1, the lapse N and 4 components of the shift N a. There are 15 unknown functions for the initial value problem, but the Hamiltonian and momentum equations (3.10) only provide five equations which constrain them. The first assumption we make is equivalent to a restriction on the form of the metric. To reduce the number of unknown functions, we assume that the induced metric is conformally related to the flat metric in 4 spatial coordinates,
2 2 2 2 2 2 ds = ψ dr + r dΩ2 + dz . (3.11)
Through equation (3.11), the 10 unknown components of the induced metric are reduced to a single unknown function, the conformal factor ψ. Notice that if ψ → 1 and the flat metric is recovered.
As neither of the initial constraints contains a lapse or a shift, but only hab and Kab, the remaining assumptions are made on the extrinsic curvature. The following convenient and typically adopted
(but not necessarily realistic) scenario is assumed: It is supposed that some form of energy is at a turning point in its evolution (which could be, for example, a radially oscillating star immediately
1 Alternately the unknowns could be taken to be hab and Kab (10+10 unknown functions) supplemented with the specification of the five components of the lapse and shift.
38 before gravitational collapse), such that the time derivative of the metric is momentarily zero. We make this assumption because it greatly simplifies the problem: since the extrinsic curvature is proportional to the Lie derivative of the metric with respect to the time coordinate, at the moment of time symmetry Kab = 0. With Kab = 0, and by assuming a conformally flat induced metric
(3.11), the initial value problem simplifies. Because the extrinsic curvature Kab vanishes, so does
a the current j . The only constraint on the initial metric hab is,
16πρ(r,˜ z) = 4R; (3.12)
4 4 4 ab where R is the Ricci scalar defined on a hypersurface Σ0 by hab. Finding R from R = R hab
4 with Rabcd defined as in equation (3.4), the initial value problem results,
8 ∇2ψ(r, z) = − πψ(r, z)3ρ(r,˜ z). (3.13) 3
We seek a density distribution ρ(r,˜ z) and a conformal factor ψ(r, z) which satisfy equation (3.13).
In particular, we are interested in solutions which are “similar” to the black string in a compact dimension. Similarity is judged by the apparent horizon and curvature invariants calculated from the metric. We specify a density distribution which, for compact enough mass, we expect to yield an apparent horizon. By tuning the density distribution we expect to change the shape of the apparent horizon. Since the event horizon is outside of the apparent horizon, the shape of the apparent horizon is expected to roughly correspond to shape of the initial event horizon, allowing favourable initial conditions to be controlled by the choice of density distribution. The location of the event horizon would have to be checked by studying the time evolution, but this is beyond the scope of the present work. For densityρ ˜(r, z) and conformal factor ψ(r, z), equation (3.13) becomes:
ψ ρ˜ ψ + ψ + 2 ,r = − ψ3 (3.14) ,rr ,zz r 6
ρψ˜ 4 Equation (3.14) is ill-posed[23]. Rescaling the density as ρ = 6 guarantees a well-posed problem,
ψ ρ ψ + ψ + 2 ,r = − , ψ(r → ±∞, z) = 1, ψ(r, z → ±∞) = 1. (3.15) ,rr ,zz r ψ
39 As discussed in Section 2.1 we used coordinates r and z which run negative to positive, and so the boundary conditions in equation (3.15) are imposed at these four boundaries. By using a symmetric density, we automatically have φr = 0 at r = 0, which is required for the physical solution to be locally flat. Because we are interested in constructing a function ψ which is suitable for an initial condition of a long asymptotically flat black string, and not in the particular density distribution that sources it, a post hoc specification of the physical density is sufficient.
We now discuss two approaches to solve for the conformal factor.
3.2.1 Analytic Method
With the assumptions made in Section 3.2, the problem of finding the conformal factor is reduced to solving equation (3.15). The difficulty in solving this lies in its nonlinearity, which enters through the density term, ρ/ψ. If ρ = 0, we have a linear equation which is straightforward to solve, but only admits the solution ψ = 1. We require a suitable non-zero ρ as this will determine the existence and shape of an apparent horizon — which, in turn, indicates that we have found initial data corresponding to a black object. In the next section we show how this term influences the numerical method. Here we consider a convenient choice of non-zero ρ which simplifies equation (3.15) so that we can use the method of separation of variables.
Consider equation (3.15) and assume that a separable solution ψ(r, z) = R(r)Z(z) exists. Differ- entiating this ψ(r, z) and substituting into equation (3.15), we have,
2 R0(r) R00(r) Z00(z) ρ + + = − (3.16) r R(r) R(r) Z(z) R2(r)Z2(z)
If the right hand side were a function of only r or z, this would be a separable equation. While, as written in equation (3.16), it is not, this can be imposed by constraining the density to depend on some arbitrary functions G(z) and f(r) in the following way,
ρ = f(r)R(r)Z2(z) + G(z)R2(r)Z(z).
40 Substituting this expression for ρ into equation (3.16) and rearranging we have,
2 R0(r) R00(r) f(r) Z00(z) G(z) + + = − − . r R(r) R(r) R(r) Z(z) Z(z)
Notice that the left side depends only on r and the right side depends only on z, and this can only be the case if both are equal to a constant — the separation constant, λ. By setting both sides equal to λ, equation (3.16) is reduced to two ordinary differential equations, one depending on r and one depending on z,
2 R00(r) + R0(r) + f(r) = λR(r), r Z00(z) + G(z) = −λZ(z).
The solutions, where c1 − c4 are constants are given by,
c √ c √ R(r) = 1 sinh( λr) + 2 cosh( λr) + I , (3.17) r r R √ √ Z(z) = c3 sin( λz) + c4 cos( λz) + IZ .
The particular solutions IR and IZ are the integrals,
1 √ Z √ √ Z √ IR = √ cosh( λr) r sinh( λr)f(r)dr − sinh( λr) r cosh( λr)f(r)dr , λr 1 √ Z √ √ Z √ IZ = √ cos( λz) sin( λz) − sin( λz) cos( λz)G(z) . λ
As we discuss next, the boundary conditions for these solutions are difficult to impose. For this reason, we find it useful to first simplify the above integrals in order to elucidate the dependence of the solutions on the choice of functions F (r) and G(z). Letting F (r) = rf(r), after repeated application of integration by parts, we find that,
00 ! ∞ 1 F (r) F (r) F (4)(r) F (6)(r) 1 X F (2i−2)(r) I = + + + ... = , R r λ λ2 λ3 λ4 r (λ)i i=1 00 ∞ G(z) G (z) G(4)(z) G(6)(z) X G(2i−2)(z) I = − + − + − ... = . Z λ λ2 λ3 λ4 (−λ)i i=1
41 Substituting these expressions into the equations (3.17) yields,
∞ 1 √ √ X F (r)(2i−2) R(r) = c sinh( λr) + c cosh( λr) + , (3.18) r 1 2 (λ)i i=1 ∞ √ √ X G(z)(2i−2) Z(z) = c sin( λz) + c cos( λz) + . 3 4 (−λ)i i=1
The boundary conditions on ψ(r, z) greatly constrain these solutions. As we discussed in Chapter
2, we set the boundaries to be rmin = −rmax and zmin = −zmax to prevent the boundary from coinciding with the region of high density at the origin. As a result, all boundaries are representing
(nearly) flat space and so the conformal factor is fixed to 1 here. To have ψ(r, z) = 1 on the boundary we require that R(r) = 1 at rmin and rmax and that Z(z) = 1 at zmin and zmax. While this is possible to impose through the the choice of c1 −c4, F (r), G(z) and the separation constant λ, we lose considerable freedom as well as the ability to have solutions which decay smoothly at the boundary2.
Note that a seemingly easy way to impose the boundary conditions on R(r) and Z(z) would be to take any solution which approaches zero and add 1 — ie. R(r) = 1+A(r), Z(z) = 1+B(z) — which
2 would make desirable functions (such as f(x) ≈ e−ax ) solutions. However, this would contradict the assumption that the solution is separable.
With this in mind, we define a new function ψ˜(r, z) = R˜(r)Z˜(z) = ψ(r, z)−1 and rewrite equation
(3.15) as, 2 ˜ ˜ ˜ ρ ψ,r + ψ,rr + ψ,zz = − . (3.19) r ψ˜ + 1
This separates into two equations for R˜(r) and Z˜(z), identical to those for R(r) and Z(z), and a modified expression for ρ(r, z),
2 R˜00(r) + R˜0(r) + f(r) = λR˜(r), r Z˜00(z) + G(z) = −λZ˜(z),
ρ(r, z) = f(r)Z˜(z)(ψ˜ + 1) + G(z)R˜(r)(ψ˜ + 1).
2 For example, we could take G(z) = 0 and choose λ and c3, c4 such that Z = 1 at the boundary, but this would mean that Z(z) changes rapidly just inside the boundary. While this would be valid solution with the proper boundary conditions, qualitatively it does not have the characteristics we want for an asymptotically flat solution.
42 This is the same as the previous system, since ψ˜ + 1 = ψ, but we have shown that we can solve for ψ˜(r, z) = R˜(r)Z(˜z) with R˜(r) = 0 and Z˜(z) = 0 as boundary conditions, and add 1 to get the solution ψ(r, z) with the proper boundary conditions.
Because the trigonometric and hyper-trigonometric functions in equations (3.18) are not particu- larly desirable, and we still have a lot of freedom since the general forms of F and G have yet to be specified, we simplify the expressions in equation (3.18) by setting c1 = c2 = c3 = c4 = 0. We now have the following solution for the conformal factor and density,
∞ ∞ 1 X F (2i−2)(r) X G(2i−2)(z) ψ(r, z) = 1 + , (3.20) r (λ)i (−λ)i i=1 i=1 ∞ ∞ F (r) X G(z)(2i−2) 1 X F (2i−2)(r) ρ(r, z) = ψ(r, z) + G(z) . r (−λ)i r (λ)i i=1 i=1
We will choose values of λ that are convenient, and typically this will mean very large. This is because for a given accuracy (truncation error) of the solution, a large λ reduces the number of terms required to calculate in the sums, if infinitely or highly differentiable functions are used. In choosing F (r) we note that functions which have a factor of r in all even order derivatives and F (r) itself are convenient3 to cancel the r−1 factor and avoid divergences. The sinusoidal functions from equation (3.18) could have also been included in the z dependence, if we were interested in this kind of dependence but, as mentioned, this comes at the cost of asymptotically flat and regular solutions.4
In Chapter 5 we show some asymptotically flat solutions which do contain sinusoidal z-dependence through the choice of G(z).
Equations (3.20) give a large set of solutions. The specific choices of F (r) and G(z) which give rise to an elongated and asymptotically flat conformal factor are shown in Chapter 5. Using this method we are constrained to solutions which are separable, and for this reason we also use a numerical method which can yield more general conformal factors though, as we will discuss next, some assumptions are made here as well. In Section 3.2.2 we describe the numerical method and in
Section 3.2.3 we show a set of solutions of the form in equation (3.18) which are consistent with the
3This is convenient but not necessary; Any function which satisfies f(0) = 0, f 0(0) 6= 0,f 00(0) = 0, f 000(0) 6= 0, f (4)(0) = 0, f (5)(0) 6= 0, etc. will also work as we can use L’Hopital’s Rule to remove the r−1 factor. 4If we had imposed periodic boundary conditions, as in the case of the compact fifth dimension, this would work well, particularly to get perturbed initial conditions for the black string.
43 assumptions in the numerical method.
3.2.2 Numerical Method
There are a few methods that we could use to solve equation (3.15) numerically. One standard approach is, instead of solving (3.15) directly, to solve for the steady state of the related parabolic problem, ∂ψ ψ ρ = ψ + ψ + 2 ,r + . ∂τ ,rr ,zz r ψ
By introducing an artificial time τ, equation (3.15) is recast as a time-dependent problem and the
∂ψ steady state solution — when ∂τ = 0 — is found by evolving ψ(τ, r, z) until it is stationary (or nearly stationary). While this is a simple and reliable method, the introduction of the additional parameter τ, and the time required to evolve ψ(τ, r, z) to a sufficiently stationary state, makes it a rather slow method5.
A direct method of solving equation (3.15) is Newton’s method (see Section 2.4). To see that this
ρ is necessary for equation (3.15), note that if the last term were simply ρ (and not ψ ) then, using the notation defined in Section 2, we could rewrite equation (3.15) as the linear system,
T d2 Ψ + Ψd2 + d1 Ψ r = ρ. nr nz nr
We would solve for Ψ by using equation (2.5) and following Chapter 2, we would have the linear system, D2 + 2D1 R + PTD2 P Ψ˜ = ρ. nr nr nz
Here R is a matrix which is the coordinate vector r repeated in each column,
R = lT ⊗ r, nr
where lnr is a is a nr × 1 vector of ones. Note that, as was discussed in Section 2.6, the only reason we are able to interchange the derivative operator with the Hadamard division by R, is because R
5In order to obey the CFL condition, it must be that δτ ≤ ∆2 where ∆ is the spatial step (δr or δz). As a result, to obtain a solution at a given value of τ, and a high spatial resolution (small ∆), the number of τ steps required is large.
44 changes only along the first dimension6 of ψ.
Equation (3.15) can be written as F (Ψ˜ ) = 0 where,
F (Ψ˜ ) = D2 + 2D1 R + PTD2 P Ψ˜ + ρ˜ Ψ˜ . nr nr nz
Notice that, because of the last term on the right hand side, this is not linear and we can not solve it in a single step. We use Newton’s method as described in Section 2.4. The Jacobian of F (Ψ˜ ) is,
J(Ψ˜ ) = D2 + 2D1 R + PTD2 P − diag(ρ˜ Ψ˜ ◦Ψ˜ ). (3.21) nr nr nz
The solution can be found iteratively,
Ψ˜ k+1 = Ψ˜ k + H˜ ,J(Ψ˜ k)H˜ = −F (Ψ˜ k).
Before proceeding, the boundary conditions have to be considered. Instead of solving for the nrnz components of the vector Ψ˜ , notice that 2nr + 2nz − 4 of these are fixed. These are the boundaries of the matrix representation of ψ. When these components of Ψ˜ are kept fixed, the Jacobian loses
2nr + 2nz − 4 columns (but not rows – as the number of equations is unchanged) and the system
J(Ψ˜ k)H˜ = F (Ψ˜ k) is overdetermined. The way to obtain an exactly determined system is to multiply
T by J(Ψ˜ k) ,
T T J(Ψ˜ k) J(Ψ˜ k)H˜ = J(Ψ˜ k) F (Ψ˜ k), (3.22)
T which has the added benefit that the matrix J(Ψ˜ k) J(Ψ˜ k) is positive semidefinite.
A viable approach is to simply solve the linear system in equation (3.22) using equation (3.21).
T This can be done by explicit matrix inversion of J(Ψ˜ k) J(Ψ˜ k) — which is known to be compu- tationally costly for large matrices — or by using an iterative method, such as Gauss-Seidel. The algorithm for Gauss-Seidel (described in the appendix) provides a way to solve the above and avoid inverting J(Ψ˜ )T J(Ψ˜ ). Still, because this matrix is very large, this is computationally costly. In
6As a counterexample, if this were instead division by a matrix representing r2 + z2, we would not be able to simply apply the Hadamard division to the derivative matrix.
45 practice we found that solving equation (3.22) by Gausds-Seidel, while quick for small matrices, lim- ited the size of system we could solve in a reasonable time frame. Our ability to solve a large system is important because, for a highly curved spacetime, we anticipate a sharply peaked conformal factor and so we require a large matrix in order to resolve it.
We proceed in another way: notice that the Jacobian in equation (3.21) is sparse, and contains many repeated matrices. This suggests it may be possible to reduce the size of the system, or to simplify J(Ψ˜ ) in another way. Algorithms to solve linear systems require a number of steps (and therefore computation time) which is proportional to some power of the size of the system. Having a method with a lower power than explicit matrix inversion is the motivation for methods like Gauss-
Seidel, and here we seek a way to rewrite the equation so that it is solvable by a method with a lower power than Gauss Seidel. We modify the Jacobian and obtain a form similar to the Sylvester equation (Section 2.5). This is done at the cost of some freedom, and we will show that the added constraint is that the factor ρ/ψ2 be separable. This constraint is distinct from that required for the analytic solutions (that ρ/ψ is additively separable), and so we can find solutions of a different form than in the previous section.
To simplify J(Ψ˜ ) we write the nrnz × nrnz derivative matrices in terms of the nr × nz ones,
D1 = 1 ⊗ d1† , (3.23) nr nz nr
D2 = 1 ⊗ d2 , (3.24) nr nz nr
D2 = PT1 ⊗ d2 P, (3.25) nz nr nz where we have introduced the abbreviation d1† = 2d1 r. Since ⊗ is bilinear and associative nr nr equation (3.21) can be written,
J(Ψ˜ )H˜ = 1 ⊗ d2 + d1† H˜ + PT1 ⊗ d2 PH˜ − diag(ρ˜ Ψ˜ ⊗ Ψ˜ )H˜ nz nr nr nr nz
Next we use equation (2.5) to rewrite J(Ψ˜ )H˜ = −F (Ψ˜ ). First, we simplify things by removing the permutation matrices. Since,
PH˜ = H˜T,
46 and
PTM˜ = M˜T, we can rewrite,
˜ 1 ⊗ d2 + d1† H˜ = d2 + d1† H1 , nz nr nr nr nr nz
PT1 ⊗ d2 PH˜ = PT d2 H˜T1 = 1 Hd˜ 2 . nr nz nz nr nr nz
Substituting these expressions into J(Ψ˜ )H˜ above, J(Ψ˜ )H˜ = −F (Ψ˜ ) becomes,
d2 + d1† H1 + 1 H(d2 )T − H◦(ρ Ψ ⊗ Ψ) = −F. nr nr nz nr nz
Abbreviating A = d2 + d1† , B = (d2 )T, and since H1 = H and 1 H = H, we have, nr nr nz nz nr
AH + HB − H◦(ρ Ψ◦Ψ) = −F. (3.26)
Equation (3.18) is similar to Sylvester’s equation introduced in Section 2.5. The difference is that we have the additional term H◦(ρ Ψ◦Ψ) or, defining M = ρ Ψ◦Ψ, H◦M. If this term could be re-written as regular matrix multiplication, equation (3.26) would be amenable to methods used to solve Sylvester’s equation. In the next section we show how this is can be done using the relationships found in Sections 2.7 and 2.8. First we note that it is possible to solve equation (3.26) iteratively by using a type of “predictor-corrector” method and diverting the Hadmard product term to the right hand side. At each iteration, the Hadamard product would be taken with the value of H from the last iteration. While in principle this would work, it is time consuming, intuitively because the Hadamard product term (which contains H) is not considered when solving for H. If we were to solve (3.26) this way, the first iteration effectively treats H◦ρ Ψ◦Ψ as zero and only after iterating does this term become included, and even then, with a lag. In practice we find that as compared to this strict predictor-corrector method, we solve the problem in less overall steps by using the modification described next.
47 Modifying the Hadamard product term
Since H is what we need to solve for in equation (3.26), we need to express H◦M as matrix multiplication with H where the other matrix (or matrices) are independent of H and depend only on the elements of M. As shown in Section 2.7, this is possible only when the underlying function of
M is separable. For the density distributions we are interested in, M = ρ Ψ◦Ψ can be assumed seperable. As discussed in Section 2.7, if it is not exactly separable there is an error introduced by this constraint. However, since we are defining the physical density post hoc already, that it might change from the density we specify prior to solving for the conformal factor, is not important. We simply recalculate the unphysical density as ρ = M◦Ψ◦Ψ before we calculate the physical density.
Assuming ρ Ψ◦Ψ is separable, or can be approximated as separable, we decompose,
H◦M = (D(CH)T )T = CHDT, where C = diag(γ∗) and D = diag(δ∗) and the vectors γ∗ and δ∗ are the solutions to equation
(2.16) obtained, as described in Section 2.7, by Newton’s method.
With this, we now have an equation in the form of a generalised Sylvester equation,
AH + HB − CHDT = −F. (3.27)
The drawback is that first we must solve for the matrices C and D, which is a nonlinear optimisa- tion problem. However it is a small problem (only nr +nz unknowns) and for the size of matrices we use, we find it quicker to solve several of these small problems than the alternative — not treating this H as a variable and solving more large systems by a predictor-corrector method.
Returning to the system AH + HB − CHDT = −F with C and D known (according to equation
(2.16)), we solve for H. There are no methods developed for solving this as a single equation — other than the large linear system we started with — and this can be understood by considering the methods used for Sylvester’s equation. As was shown in Chapter 2, A and B are decomposed using a Shur decomposition and when there are only two terms, multiplying through by the orthogonal matrices from the right and left gives a coupled system of equations for a new matrix. In general,
48 the same strategy is not possible when there are three terms. The exception, as discussed in Chapter
2, is when A and C as well as B and DT share eigenvectors which of course will not hold under most conditions. In our case, with C and D diagonal, we can solve equation (3.27) directly using a modification of the Bartel-Stewart algorithm as noted in Chapter 2. As we can not implement the algorithm nearly as efficient as the existing Matlab/LAPACK routine, it turns out to be quicker to use the existing routine and modify equation (3.27) to suit it.
We recall under what conditions it is possible to re-write the last term on the left of equation
(3.27) to make the system exactly in the form of Sylvester’s equation. We already know this is not possible, as this has been discussed when we wanted to separate the Hadamard product previously and settled for an approximation (CHDT in equation (3.27)) (Section 2.7). However keeping in mind that H is a small quantity and does not necessarily need to be correct7, we may be able to make due with an approximation. As shown in Section 2.8, we can decompose:
CHDT = MH + HN, 1 M = D¯C − C¯D,¯ 2 1 N = C¯D − C¯D¯; 2 where C¯ and D¯ are the average of the elements of C and D. Recall that this is only possible because
C and D are diagonal. While this is not an exact solution, it can speed up the algorithm by letting us start with a better initial guess than solving AH0 + H0B − CH0DT = −F by using either of
AH0 + H0B = −F or CH0DT = −F alone. Instead we have simply Sylvester’s equation,
(A − M) H0 + H0 (B − N) = −F. (3.28)
Before describing the algorithm to solve iteratively for H, starting with this H0, we consider if our equations satisfy existence and uniqueness, and for this we can use the result for Sylvester’s equation.
First we need to recall what each matrix above is: We have expressed J(Ψ˜ )H˜ = −F (Ψ˜ ), where H
7An incorrect step H can be viewed as a starting point farther from the solution and, so long as this does not cause the trial solutions to diverge from the true solution, it is innocuous. It is reasonable to assume that using a simplified equation to obtain a guess of H is safe in this sense.
49 is the matrix version of H˜ , as the following,
AH + HB − CHDT = −F, (3.29)
A0H + HB0 = −F, where
A = d2 + 2d1 r, nr nr
B = d2 , nz 1 A0 = A − δ¯∗diag(γ∗) + δ¯∗γ¯∗, 2 1 B0 = B − γ¯∗diag(δ∗) + δ¯∗γ¯∗, 2 and γ∗ and δ∗ are the solutions to (2.16) with M = ρ Ψ◦Ψ, bars denote averages and as usual
C = diag(γ∗), D = diag(δ∗). From Section 2 we know that the second equation in (3.29) above will have a solution if the matrix B0T ⊕ A0 is invertible or, equivalently, if no combination of an eigenvalue of B0T and A0 sums to zero. Because of the unknowns γ∗ and δ∗ in A0 and B0 this can not be guaranteed — the values of the matrices depend on the specific density. For the equation
AHk + HkB = −F + CHk−1DT, where we treat the third term as a source term, this condition can be checked ahead of time since,
A = d2 + 2d1 r, nr nr
B = d2 . nz
However, the eigenvalues still depend on r, and so while the condition is not obviously violated, we would need to verify for each r that it holds. For A0 and B0 we would need to check before each iteration of H, as the density may be modified at each step (to maintain the constraint ρ Ψ◦Ψ =
δ∗T γ∗). In practice we note that the algorithm does not generally fail, and, since there is no reason to suppose it would, we omit these checks as to not add unnecessarily to the computational time.
(In applying a similar method to the apparent horizon, we do observe failure under generic but predictable conditions, as discussed in Chapter 4).
50 Algorithm
Because we know that while the constraint ρ Ψ◦Ψ = δ∗T γ∗ can be satisfied exactly, the de- composition CHDT = MH + HN will not generally be, we implement a type of predictor-corrector method. We obtain the initial guess of H0 from equation (3.28), and we can correct this by iterating either,
• AHk + HkB = −F + ρ Ψ◦Ψ, or,
k −1 k−1 k−1 T−1 −1 k−1 −1 −1 −1 • H = C˜ AH + H B + F D˜ + αcC˜ H + αDD˜ − αC αDC˜ D˜ .
where we have defined C˜ = C + αC 1 and D˜ = D + αD1, for small αC and αD because they may otherwise not be invertible due to near-zero elements arising from very small densities. Which equation we should iterate in the second step above depends on which term is deemed more important to solve for “properly” and which can be treated as a modification to the source term, F. While there is no particular reason to use one or the other8, we take the first which treats ρ Ψ◦Ψ as a modification to the source. In practice, this turns out to be much quicker and so we choose to solve
Sylvester’s equation iteratively, by diverting the CHDT term to the source after the first iteration.
At each step in Newton’s method, before solving for Hk, we determine C and D. While we start with a density distribution which should be separable, the matrix M changes on each iteration within
Newton’s method because the solution is changing. Hence, we must repeatedly solve for C and D.
After each iteration, when Ψ has been updated, we then use C, D and Ψ to determine M and ρ.
In this way we do not need to assume that an initially separable density remains so throughout the iterations, as we will always end up with a density which is consistent with the conformal factor and the constraint (equation (2.16)) through the matrices C and D.
As described, we have two layers of iterations: we find the solution Ψ˜ iteratively by adding an H˜i, and we also find each H˜i iteratively. The numer of iterations is determined by setting a stopping condition — instead of fixing the maximum number of iterations in both layers, we simply continue updating (Ψ˜i or H˜k) until the norm of the change in the variable between iterations is less than a tolerance, which depends on the order of accuracy of the finite difference approximations (D) and
8Without knowing the solution it is not possible to know whether the principal term or the density term dominates.
51 the relative grid size as: 0 = . 2(N−1)D
Here N is an integer numbering the solution for a set of array sizes. That is for N = 1 we have spacing ∆, and for N = 2 we have ∆/2 and for N = 3, ∆/4 etc. This stopping condition ensures that our solutions are comparable at different grid sizes in the following way. We know that the error in the solution depends on the grid spacing ∆r and that for a given order of accuracy of the derivatives D, we expect the error to go as ∆D, so in the above the reduction of the tolerance at each level is equal to the reduction in the error of the solution. Intuitively a larger system requires more iterations to “solve”, and this condition makes this explicit as we do not apriori know how many more iterations we would need to use. If we think of the allowable tolerance as being set relative to the error in the solution — for example, if we wanted to set the tolerance to twice the error we expect to get (going below becomes meaningless) — it is obvious that as the error we expect shrinks, so must the tolerance. Furthermore, as the error is absolute (proportional to the grid spacing, and not relative to the solution), we set the condition on the absolute change in the solution (or the step H).
3.2.3 Solutions common to both
We would like to be able to compare solutions obtained from the numerical method with the analytic solutions, where possible. To solve the equation numerically we have imposed the following constraint, where P and Q are any functions,
ρ(r, z) = P (r)Q(z). ψ(r, z)2
This is quite different from the relationship between ρ and ψ assumed for the analytic solutions, but there are a cases which satisfy both constraints. These common solutions will be used as a test of the numerical method.
The assumption we made in Section 3.2.1 when solving for a separable conformal factor was that
ρ(r,z) f(r) G(z) ψ(r,z)2 is equivalent to the sum of a function of r, R(r) , and a function of z, Z(z) . Equating these
52 definitions of ρ(r, z),
P (r)Q(z)R2(r)Z2(z) = f(r)R(r)Z2(z) + G(z)R2(r)Z(z), f(r) G(z) P (r)Q(z) = + . R(r) Z(z)
There are two cases for which this can hold,
ρ(r, z) G(z) f(r) = αR(r), = α + , R2(r)Z2(z) Z(z) ρ(r, z) f(r) G(z) = βZ(z), = β + . R2(r)Z2(z) R(r)
Except for the cases where one of α or β is zero, there appear to be no solutions which are consistent with the boundary conditions. However, a set of non-physical solutions can still be used as a check that both the analytic method and the numerical method yield consistent solutions.
F (r) Consider the first equation above: f(r) = αR(r) or, since F (r) = rf(r), R(r) = αr . This gives the differential equation for R(r),
∞ F (r) 1 X F (2i−2)(r) = . αr r (λ)i i=1
Choosing F (r) = Ae−βr and solving for α,
" # ∞ !i Ae−βr Aβ2e−βr Aβ4e−βr Aβ6e−βr Ae−βr X β2 Ae−βr 1 Ae−βr = α + + + + ... = α = α . λ λ2 λ3 λ4 λ λ λ β2 i=1 1 − λ
53 It follows that α = λ−β2, and the requirement α > 0 is equivalent to λ > β2. So we have a solution, with λ > 0, α > 0,
∞ A √ X G(2i−2)(z) ψ(r, z) = 1 + e− λ−αr , αr (−λ)i i=1 A √ R˜(r) = e− λ−αr, αr ∞ X G(2i−2)(z) Z˜(z) = , (−λ)i i=1 ! G(z) ρ(r, z) = ψ(r, z)2 α + . Z˜(z)
While this will not satisfy the boundary conditions — ρ(r, z) should be zero at the boundaries, requiring α = 0 — it still provides a set of solutions which are common to both methods.
Alternatively we can set α = β = 0. To keep some dependence on both r and z, we can use the non-homogeneous part of Z˜(z) and set β = 0 (equivalently we could do this with R˜(r)). Considering
2 the form of ρ(r, z) which we will use in the numerical method, functions like e−cx for x = r, z seem to be a good choice for the form of F (r) and G(z). Because of the 1/r in R˜(r), we choose
2 2 F (r) = Are−ar (f(r) = Ae−ar ) as both F (r) and its even derivatives come with a factor of r. The solution is,
2 F (r) = Are−ar , ∞ 1 X F (2i−2)(r) R˜(r) = , r (λ)i i=1 √ √ π 2n + 1 Z˜(z) = c4 cos( λz), λ = , n ∈ Z, 2 zmax
2 ρ(r, z) = Ae−ar R˜(r)Z˜2(z).
These solutions can be used as a test of the consistency of the analytic and numerical methods.
Having found solutions to our problem we next discuss quantities which we compute to verify their physical properties.
54 Chapter 4
Physical properties
To characterise the solutions from Chapter 3 we look at the mass (Section 4.1), the location of the apparent horizon (Section 4.2), and curvature invariants on the horizon (Section 4.3).
4.1 Mass
The mass of the spacetime constructed in Chapter 3 can be calculated via the ADM mass [16].
AB This is computed by a surface integral of the difference of the extrinsic curvature scalar k = σ kAB from its value in flat space (k˜), in the limit that the surface is at infinity,
I 1 ˜ p 2 MADM = − lim k − k det(σ)dσ . (4.1) 4π2 R→∞
2 π 1 The constant 1/4π is that in [16] divided by 2 to account for the additional angular coordinate .
a b In equation (4.1), σAB = habeAeB is the metric on the 3-sphere —parametrised by θ, φ and χ — and the limit R → ∞ takes the surface of the sphere to infinity. Because we are taking a surface of constant R, it is convenient to use spherical coordinates. Recall that in spherical coordinates the non-zero components of the metric are along the diagonal,
2 2 2 2 2 2 2 hab = ψ(R, χ) diag 1,R ,R sin (χ),R sin (χ)sin (θ) . (4.2)
1The factor of 8π in [16] is twice the integral of the area of a 2-sphere, here we have the additional integral over R π 2 π the angle χ which gives 0 sin (χ) = 2
55 The surface is at constant R, and the normal vector to this surface sa is given by equation (3.1). For this surface the only non-zero component of the normal is in the radial direction, and from equation
(4.2) we have that, 1 s = −ψ, s = −ψ , sR = − . (4.3) R R,R ,R ψ
AB ˜ AB In equation (4.1), k = σ kAB and k = η kAB. The extrinsic curvature is [16],
a b kAB = sa|beAeB.
The induced metric is, analogous to equation (3.4),
σAB = hab − sasb.
The covariant derivative of the normal is from equation (??),
u sa|b = sa,b − Γabsu.
Using the above relationships we obtain k as follows,
AB k = σ kAB, (4.4)
AB a b = σ sa|beAeB,
AB a b = sa|bσ eAeB,
ab a b = sa|b h − s s ,
u ab a b = sa,b − Γabsu h − s s ,
ab a b ab u u a b = h sa,b − sa,bs s − h Γabsu + Γabsus s .
56 It follows from equations (4.2) and (4.3) that the first two terms in the last line of equation (4.4)
R cancel out. For the last two terms, the required Christoffel symbols (Γii) are,
Ψ ΓR = ,R , RR Ψ Ψ ΓR = − R + R2 ,R , χχ Ψ Ψ ΓR = − R + R2 ,R sin2(χ), θθ Ψ Ψ ΓR = − R + R2 ,R sin2(χ)sin2(θ). φφ Ψ
Using these expressions in equation (4.4) we have that,
u a b ab k = Γabsu s s − h , 3 1 Ψ = − + ,R . Ψ R Ψ
˜ To find k — which is k for flat space, we take Ψ = 1 and Ψ,R = 0 and so,
3 k˜ = − . R
To ensure that we are comparing κ andκ ˜ at the same value of R, R is scaled by Ψ in the expression
˜ 3 ˜ forκ ˜: k = − RΨ . Substituting k − k into equation (4.1),
Z π Z π Z 2π 3 3 2 MADM = 2 lim Ψ,RΨR sin (χ)sin(θ)dχdθdφ, (4.5) 4π R→∞ 0 0 0 where we have used pdet(σ) = R3Ψ3sin2(χ)sin(θ). Integrating over θ and φ we arrive at the final expression for the ADM mass,
Z π 3 3 2 MADM = lim Ψ,RΨR sin (χ)dχ. π R→∞ 0
As a check of this expression we use the conformal factor for the hyperspherical black hole, as the relationship between mass and horizon radius is known. As in equation (1.7) the conformal factor is, R2 Ψ = 1 + BH . 4R2
57 Substituting this into the expression for MADM , taking the limit, and integrating over χ gives,
3 M = R2 . ADM 4 BH
Recall from Chapter 1 or from [15] that the mass of a hyperspherical black hole is2,
2 3RBH M5 = . 4G5
And so, with G5 = 1, we obtain the correct expression.
In practice we calculate the mass for large a series of large R and observe what it is approaching as R increases. As to not waste time using an excessively large domain — as we would to see the result at very large R — we fit the results and are satisfied with the assumption that we can roughly predict the value as R → ∞. See Appendix for details.
Having discussed how to compute the mass in our spacetime we now turn our attention to the existence and shape of a possible black hole.
4.2 Apparent Horizon
The solutions of the conformal factor found in Sections 3.2.1 -3.2.3 determine the metric through equation (3.11), and the metric describes the spacetime at a slice. The defining feature of a black object is the event horizon, and so to determine if the resulting spacetime contains a black object we look for the event horizon. Because our objective is to construct a metric which corresponds to an
AFBS, we anticipate elongated horizon. As discussed in Chapter 1, depending on the length of the string, the AFBS may be subject to the Gregory-Laflamme instability and, independent of the length of the string, the AFBS will tend to contract to a hyperspherical black hole. The conformal factors for the AFBS therefore do not correspond to static spacetimes. For these non-static spacetimes, we rely on looking for the apparent horizon. If an apparent horizon exists, then we know that there is an event horizon outside of it, which may be roughly of the same shape.
2 2 3πRBH π This is M5 = in [15], corrected with the factor of . 8G5 2
58 The apparent horizon is a hypersurface on which outgoing null rays are marginally trapped. Inside the apparent horizon, outgoing rays will have a have a negative divergence so that while they are initially outgoing, they are redirected inward. A ray sent outward from outside of the apparent horizon will, in the absence of further energy ingoing to the region, continue to be locally outgoing, with a positive divergence. Denoting the tangent to a null ray as kα, a ray is marginally trapped if it has zero divergence or expansion, which we denote by κ,
α κ = k;α = 0.
Because of our assumption that the initial condition is a moment of time symmetry (Section 3.2), this condition simplifies. First we note that the null vector kα can be expressed as a combination of spacelike (sα) and timelike vectors (nα)[16],
kα = sα + nα.
The zero expansion of the null vector in terms of the spacelike and timelike normals is,
α α s;α + n;α = 0.
When the extrinsic curvature is zero, as at a moment of time symmetry, the expansion of the
α timelike normal is also zero, since its expansion n;α is related to the extrinsic curvature Kab through
α α β ab ab α n;α = nα;βea eb h = Kabh . The covariant derivative with respect to the full spacetime, s;α, can be written in terms of quantities on the hypersurface as,
α a b a a s;α = s|a + s s Kab = s|a,
where the last equality follows from Kab = 0. Putting this together we have have that the apparent
a horizon is the surface on which s|a is zero,
a κ = s|a = 0.
59 Because of the spherical symmetry of the spacetime, just as with the conformal factor, the horizon will be independent of the angular coordinates and depend only on r and z. With respect to a specific value of the radial coordinate R (where R2 = r2 +z2), the apparent horizon is described by a function
φ = R¯ − h(r, z) where the apparent horizon coincides with φ = 0. The expansion κ is found by taking the covariant derivative of the normal to the surface φ. The normal is proportional to the
a derivative of the function φ — as φ only changes off the surface — and is normalised (s sa = −1) by the factor k∇φk,
φ q sa = habs , s = − ,b , k∇φk = habφ φ . (4.6) b b k∇φk ,a ,b
From equation (2.12), the covariant derivative of the normal is,
a a a u s |a = s ,a + Γ ubs .
And so we have,
a a u κ = s ,a + Γ ubs . (4.7)
Our objective is to find a function φ(r, z) which satisfies κ = 0 with κ as defined in equation
(4.7). In the next section we look at the cases of the hyperspherical black hole and black string and subsequently we obtain an expression for κ for the generic case of the AFBS. We then show how this can be done using the methods described in Chapter 2.
Apparent horizon of black string and hyperspherical black hole
For a hyperspherical black hole, the conformal factor is symmetric in r and z, and depends only on
R, where R2 = r2 + z2. It follows that the apparent horizon will also depend only on R,
φ = φ(R).
The non-zero components of the metric are along the diagonal,
2 2 2 2 2 2 2 hab = ψ(R) diag 1,R ,R sin (χ),R sin (χ)sin (θ) .
60 The normal vector, from equation (4.7), only has one component,
φ q sR = hRRs , s = − ,R , k∇φk = hRRφ2 , R R k∇φk ,R
2 RR 1 or, since hRR = ψ and h = ψ2 ,
1 sR = − , s = −ψ. ψ R
a a R a R Because s only has an R component, the sum in equation (4.7) is Γ uas = Γ Ras . These components of Γ are,
1 ψ ΓR = hRRh = ,R , RR 2 RR,R ψ 1 ψ 1 Γχ = hχχh = ,R + , Rχ 2 χχ,R ψ R 1 ψ 1 Γθ = hθθh = ,R + , Rθ 2 θθ,R ψ R 1 ψ 1 Γφ = hφφh = ,R + . Rφ 2 φφ,R ψ R
It follows that κ is, from equation (4.7),
R R χ θ φ R κ = s ,R + Γ RR + Γ Rχ + Γ Rθ + Γ Rφ s , ψ ψ 1 1 = ,R − 4 ,R + 3 . ψ2 ψ R ψ
Therefore, for a hyperspherical black hole,
ψ 3 κ (ψ) = −3 ,R − . (4.8) BH ψ2 Rψ
With the conformal factor for the hyperspherical black hole from equation (1.7), we have that
1 κBH (ψ) = 0 at R = 2 RBH . For the black string, we use cylindrical coordinates (r, θ, φ, z) and assume the horizon only depends on r,
φ = φ(r).
61 The non-zero components of the metric are in cylindrical coordinates are,
2 2 2 2 hab = ψ(r) diag 1, 1, r , r sin (θ) .
As before, sa only has a component in the coordinate that φ depends on, — in this case, r. Where
r 1 r 1 r > 0, s = − ψ and sr = −ψ, but where r < 0, s = ψ and sr = ψ. This ensures that the normal vector is pointing outward from the surface, and the sign change is required to compensate for the change in the sign of the coordinates. The components of Γ needed in equation (4.7) are of the form
a Γ ra,
1 ψ Γr = hrrh = ,r , rr 2 rr,r ψ 1 ψ Γz = hzzh = ,r , rz 2 zz,r ψ 1 ψ 1 Γθ = hθθh = ,r + , rθ 2 θθ,r ψ r 1 ψ 1 Γφ = hφφh = ,r + . rφ 2 φφ,r ψ r
r 1 Substituting this into equation (4.7) and writing sr = −sgn(r)ψ, s = −sgn(r) ψ , we have,
ψ 2 κ (ψ) = −sgn(r) 3 r + . (4.9) BS ψ2 rψ
In this case, we cannot write down where κBS(ψ) = 0, because we do not have an expression for the conformal factor for the black string.
Note that in both of these cases, the expressions for κ do not depend on the specific function φ, and only on the argument of φ. In the hyperspherical black hole case, as long as φ is a function of
R, then κ is given by (4.8), and in the black string case, as long as φ is a function of r, then κ is given by equation (4.9). To reiterate, once we specify the coordinate which is constant on a contour level of the surface — in the above, r and R — then we know κ everywhere.
We extend this argument to any function of h(r, z): if the expansion can be expressed as a function of h(r, z) only — the levels of φ correspond to h(r, z) = constant, and the horizon is on a curve h(r, z) = constant — then the expansion is independent of the function φ. This is because, as
62 √ above, if φ is expressed as a function of any one variable — R = r2 + z2, r, h(r, z), or f(R), f(r), f(h(r, z)) etc. — then any explicit φ dependence cancels out. Of course this is as we would expect since the shape function φ has only been introduced as a way to find the location of a single contour h(r, z). We emphasise this because in Section 4.2.1 we will use this conclusion, that κ is a well-defined function independent of φ.
Apparent horizon of asymptotically flat black strings
For the AFBS, we proceed in the same way as above, except with the function φ having arguments either r and z or R and χ, depending on which coordinates we choose for the metric. While either cylindrical (r, z) or spherical (R, χ) would work, we choose to use the cylindrical coordinates with the idea that the solutions we will be most interested in are the elongated AFBS, making cylindrical coordinates a more natural choice. In these coordinates the normal vector has two components,
φ φ sr = −sgn(r) ,r , s = −sgn(r) ,r , ψ2k∇φk r k∇φk φ φ sz = −sgn(z) ,z , s = −sgn(z) ,z . ψ2k∇φk z k∇φk
Again we use sgn(·) to correct the sign for the r components, and here we correct for z as well. The expansion is, from equation (4.7),
r r χ θ φ r z r χ θ φ z κ = s ,r + Γ r + Γ rχ + Γ rθ + Γ rφ s + s ,z + Γ zr + Γ zχ + Γ zθ + Γ zφ s .
Working out the terms for sr we have,
! 1 ψ k∇φk sr = −sgn(r) φ + 2 ,r − ,r φ , ,r k∇φkψ2 ,rr ψ k∇φk ,r ψ 1 Γr + Γχ + Γθ + Γφ = 4 ,r + 2 , r rχ rθ rφ ψ r 1 1 ψ 1 Γr + Γχ + Γθ + Γφ sr = −sgn(r) 4 ,r + 2 φ . r rχ rθ rφ ψ2 k∇φk ψ r ,r
Thus the first two terms of κ are,
! 1 ψ 2 k∇φk sr + Γr + Γχ + Γθ + Γφ sr = −sgn(r) φ + 2 ,r + − ,r φ . ,r rr rχ rθ rφ k∇φkψ2 ,rr ψ r k∇φk ,r
63 An analogous expression is found for the z terms. Putting these together we have that κ is,
! sgn(r) ψ 2 k∇φk κ = − φ + 2 ,r + − ,r φ (4.10) k∇φkψ2 ,rr ψ r k∇φk ,r ! sgn(z) ψ k∇φk − φ + 2 ,z − ,z φ k∇φkψ2 ,zz ψ k∇φk ,z
There is no information about what φ(r, z) should be, however equations (4.8) and (4.9) are useful to get an idea of what κ is like. The requirement for φ(r, z) is that the surface has an isoline which is congruent to h(r, z) on the horizon. Since (so far) we are only interested in one level of φ(h(r, z))
— that which gives κ = 0 — then how the surface is constructed outside of this does not matter.
In Section 4.2.1 we will assume that we can set φ = κ, and in this case the assumption is that the expansion has levels of constant h(r, z). Based on this being the case for the black hole and black string, it seems a reasonable assumption. That is, even asymptotically, the expressions for
κ above only depend on r or R. Our assumption will be that for the asymptotically flat black string — something qualitatively in between these two solutions — the expansion only depends on h(r, z). Because the expansion is determined by the conformal factor — which is localised within some region, it would be strange if far from the origin where the gradient of the conformal factor is very small, the dependence on h(r, z) was altered. To further support this, we note that if the expansion can be expressed as a function of h(r, z) — not just on the horizon — and if we further assume that the conformal factor also depends only on this same function, then we could express,
" # 1 ψ 2 κ = 3 ,h + . AF BS q 2 2 2 ψ rψh,r h,r + h,z
Furthermore, under the assumption that the conformal factor can be expressed as ψ(r, z) =
ψ(h(r, z)), we know what the function h(r, z) is: it is given by any level of ψ(r, z).
We next discuss the method that we use to find the apparent horizon, h(r, z).
64 4.2.1 Method to solve for apparent horizon
Using the methods described in Chapter 2, we write the equation for the discrete vector version of φ, Φ˜, obtained from setting κ in equation (4.10) to zero, in the form,
F(Φ˜) = 0.
From this we need to obtain the Jacobian. Because κ involves products of derivatives of φ and functions which depend on r and z, it is not possible to analytically determine the Jacobian. This follows from Chapter 2 where we noted that a Hadamard product can not be separated, except in the case that one matrix represents a separable function.
As we would like an analytic Jacobian3, we will assume that the terms in κ are separable where needed. As there is no justification for them to be so, we will not be solving the equation exactly.
However, an approximate solution may be sufficient if it yields a solution Φ and a corresponding κ which has suitable closed curve — suitable in the sense that they do not appear to be artifacts of the numerical method. We judge this by monitoring how the closed curves change as the residue of the solution approaches zero.
We find that, most likely as a result of the approximations made, we do not solve the equation exactly. As a result the φ we obtain does not exactly correspond to κ, but they are similar. We verify that when solving for the case of a hyperspherical black hole or black string, we do obtain
φ ≈ κ, differing only near R = 0 or r = 0 (the expansion and conformal factor diverge here, so we would not expect to find good agreement anyway). Furthermore we are using an iterative method and we find that the residual is always shrinking as the iteration number increases — that is, φ → κ.
In practice, the evaluation of κ after only a few iterations yields suitable closed curves, so in general we do not attempt to solve until the residual is zero but rather until a stopping condition is met.
3The Jacobian could be found numerically from first principles, by adding small quantities to each component of F , but this must be done for each component at each step which would likely render the time savings of doing this “direct” solve negligible.
65 Our starting equation is equation (4.10), which we rearrange slightly as,
! ! ψ 2 k∇φk ψ k∇φk κk∇φkψ2 = −sgn(r) φ + 2 ,r + − ,r φ −sgn(z) φ + 2 ,z − ,z φ . ,rr ψ r k∇φk ,r ,zz ψ k∇φk ,z (4.11)
About this we make two assumptions: that the conformal factor, ψ, is separable and that the norm of the gradient of φ, k∇φk, is as well. It is reasonable to assume that the conformal factor is separable because in the numerical method we use densities which are separable, and in the analytic method we have already used this assumption to use the methods of separation of variables. The assumption that k∇φk is separable is not so innocuous, but we use it to get an initial guess of the step H, as shown below. Furthermore, we are treating φ as fixed — really it should be differentiated and used in the Jacobian. This is the most offencive assumption, but can be seen as similar to the predictor-corrector method, though with the prediction and correction occurring at the level of φ and not of H. At subsequent steps, where we are intent on finding an exact solution, we divert the terms with k∇φk to the source (in a predictor-corrector type method, as was done in solving for the conformal factor).
Before describing the method, we note that these assumptions and approximations are not harmful to our results. This is because ultimately we will be evaluating the exact expression for κ and taking its zero-crossings as the apparent horizon. The utility of setting φ = κ is to help us obtain a suitable shape on which to evaluate κ.
As in Chapter 2 we denote matrices by bold capital letters and the vector form with a tilde. The matrix version of equation (4.11) is shown in equation (4.12). Note that the sgn(·) functions are absent, as we will include this implicitly in our definition of Φ.
k∇Φk Φk∇ΦkΨ2 = − d2 Φ + 2d1 Ψ Ψ + 2 r ◦d1 Φ − ,r ◦d1 Φ (4.12) nr nr nr k∇Φk nr
T T T T k∇Φk − Φd2 + Φd1 ◦ 2Ψd1 Ψ − Φd1 ◦ ,z . nz nz nz nz k∇Φk
66 Before making the approximations we note that the exact F(Φ˜), which we will monitor for conver- gence, and use in Newton’s method, is,
k∇Φk Fexact(Φ) = − Φ◦k∇Φk◦Ψ◦Ψ − d2 Φ + 2d1 Ψ Ψ + 2 r ◦d1 Φ − ,r ◦d1 Φ nr nr nr k∇Φk nr (4.13)
T T T T k∇Φk − Φd2 + Φd1 ◦ 2Ψd1 Ψ − Φd1 ◦ ,z . nz nz nz nz k∇Φk
Next we adopt the notation M [·] and N [·] to mean the matrices containing the r and z dependence respectively and obtained as in equation (2.22), using equation (2.16). Using these we have an approximation of F(Φ),
F(Φ) = − Φ◦k∇Φk◦Ψ◦Ψ (4.14) ! h i k∇Φk − d2 + M 2d1 Ψ Ψ + M [2 r] d1 − M ,r d1 Φ nr nr nr k∇Φk nr ! h i k∇Φk − d1 ΦN 2d1 Ψ Ψ − d1 ΦN ,r nr nr nr k∇Φk ! T T h T i T k∇Φk − Φ d2 + d1 N 2Ψd1 Ψ − d1 N ,z nz nz nz nz k∇Φk ! h T i T k∇Φk T − M 2Ψd1 Ψ Φd1 − M ,z Φd1 . nz nz k∇Φk nz
The equation J(H) = −Fexact(Φ) is,
! h i k∇Φk − sgn(r)◦ d2 + M 2d1 Ψ Ψ + M [2 r] d1 − M ,r d1 H nr nr nr k∇Φk nr ! T T h T i T k∇Φk − H d2 + d1 N 2Ψd1 Ψ − d1 N ,z ◦sgn(z) nz nz nz nz k∇Φk ! h T i T k∇Φk T = −Fexact(Φ) + M 2Ψd1 Ψ Hd1 − M ,z Hd1 nz nz k∇Φk nz ! h i k∇Φk + d1 HN 2d1 Ψ Ψ − d1 HN ,r + H◦k∇Φk◦Ψ◦Ψ. (4.15) nr nr nr k∇Φk
Note the reappearance of the sgn(·) functions on the left hand side, as here Φ is absent and we need to ensure that a correct H results. We solve this iteratively (since H appears on the right hand side), using the method for Sylvester’s equation (Section 2.5).
67 As an initial guess of Φ, Φ0, we use the discrete form of,
φ0 = 0.05 αsgn(r)κBS(ψ) + (1 − α)κBH (ψ) , (4.16)
where κBS(ψ) and κBH (ψ) are as in equation (4.8) and (4.9), which are evaluated with the conformal factor of the AFBS. The weighting is such that 0 ≤ α ≤ 1, and is decided arbitrarily (based on whether we think the conformal factor is more “string-like” or “hole-like”) and the factor of 0.05 helps to get a smaller residual quicker — that is, the expansion we find is smaller in magnitude than that for a black hole or a black string.
We solve for Φ iteratively,
Φi = Φi−1 + H.
We solve for H iteratively, starting with an H0 which is the solution to,
! 2 h 1 i 1 k∇Φk,r 1 −sgn(r)◦ d + M 2d Ψ Ψ + M [2 r] d − M d H0−M k∇Φk◦Ψ◦Ψ H0 nr nr nr k∇Φk nr ! 2 T 1 T h 1 T i 1 T k∇Φk,z − H0 d + d N 2Ψd Ψ − d N ◦sgn(z) − H0N k∇Φk◦Ψ◦Ψ nz nz nz nz k∇Φk
= −Fexact(Φ). (4.17)
On subsequent iterations, we move the terms containing k∇Φk terms to the right hand side, with the hope that the initial H0 has captured them to some extent, and the subsequent iterations are done without this approximation. The terms which include derivatives of the conformal factor can be reasonably assumed separable and so we continue with these on the left hand side (and drop h i h i M 2Ψd1 T Ψ and N k∇Φk,z , as under this assumption they would be zero). For k ≥ 1 we nz k∇Φk
68 solve,
! h i − sgn(r)◦ d2 + M 2d1 Ψ Ψ + M [2 r] d1 H nr nr nr k
T T h T i − H d2 + d1 N 2Ψd1 Ψ ◦sgn(z) k nz nz nz
exact k∇Φk,r 1 1 T k∇Φk,z = −F (Φ) − sgn(r) ◦d Hk−1 − sgn(z)Hk−1d ◦ + k∇Φk◦Ψ◦Ψ◦Hk−1. k∇Φk nr nz k∇Φk (4.18)
To encourage the solution we want, φ = κ, each iteration we shift φ by a constant which makes the average of κ and φ outside of a fixed r, equal. We do not attempt to match them everywhere because we observe that κ is divergent in places. We also know that κ is symmetric, and we impose this on φ periodically — although it does not develop any asymmetries. When we make the initial
1 1 condition as in equation (4.16), we use modified functions in place of the R and r terms, as these diverge in our domain. In an attempt to prevent Newton’s method from failing, we restrict the solution H to not exceed a fixed fraction of the average of Φ (in absolute values). This prevents the solution from rapidly diverging. Since our initial guess agrees well on the boundaries, and the residual is not large anywhere, we find that the trial solutions are able to remain close enough to the true solution that it does converge. That is, we avoid the potential issue with Newton’s method
(noted in Chapter 2) of having an initial guess which is too far off to ever converge.
As we are solving the equation, we monitor kappa at each iteration. We find that within the first few iterations there are closed curves where κ = 0. After the “solution” is found (we have iterated until a stopping condition is met), we take the minimum of all of the κ across all iterations. The closed curve where this is zero yields the outermost surface of κ = 0. To be reasonably confident we have found the outermost curve, we look at all of the curves — if they are continuing to expand as the iteration number increases, then we use a more strict stopping condition. In most cases they will shrink, and so subsequent iterations do not contribute anything more to finding the outermost curve.
69 4.3 Curvature Invariant
Once we have found the function h(r, z) which represents the apparent horizon, we evaluate the
Kretchmann scalar — an invariant measure of curvature on this surface. Because there are known values for the Kretchmann scalar in the case of a hyperspherical black hole and in the case of a black string, this evaluation helps us characterise whether or not we have constructed a metric representing one or the other.
The Kretchmann scalar is,
abcd K = RabcdR .
With the metric of the AFBS this evaluates to,
Kψ8r2 3 3 = ψ ψ + ψ 2 + ψ 2 + 2 ψ 2 ψ2r2+ ψ 2 (ψ − 3 ψ ) − 8 ψ ψ ψ + ψ 2 (ψ − 3 ψ ) ψr2 8 rr zz 2 rr 2 zz zr r zz rr r z zr z rr zz 2 2 2 2 2 2 2 2 + 2 ψrψ (ψrr + ψzz ) ψ + ψz + ψr r + 3 ψz + ψr r + 4 ψ ψr . (4.19)
For a hyperspherical black hole the conformal factor is (from equation (1.7)),
R2 ψ = 1 + BH . 4R2
1 With this conformal factor and setting R = 2 RBH as the horizon (from Section 4.2), the Kretchmann scalar evaluates to, 3 1 K = . (4.20) BH 2 R4
For a black string, we use the metric in equation (1.3) and where r = rSch,
1 K = 6 . (4.21) BS r4
70 Chapter 5
Tests and Applications
In this chapter we show some representative results of the methods described in Chapters 3 and
4. In Section 5.1 we obtain solutions for the conformal factor describing an asymptotically flat black string. We show that our solutions for the conformal factor obtained by the analytic and numerical methods behave as expected as we vary the step size, number of iterations etc. In Section 5.2 we obtain further physical insights of the solutions by evaluating the ADM mass, obtaining the location of the apparent horizon and evaluating the curvature invariant.
5.1 Conformal factor
5.1.1 Analytic solutions
Because our analytic solutions in equations (3.20) contain infinite sums that we truncate, we verify the truncated solutions by checking the norm of the residual. As in equation (3.20), we find the conformal factor Ψ(k, N,p) and density ρ(k, N,p) as,
T imax 2 (i−1) imax 2 (i−1) X dN,p F X dN,p G Ψ(k, N, p) = 1 + r, (5.1) (λ)i (−λ)i i=1 i=1 T T imax 2 (i−1) imax 2 (i−1) X dN,p G X dN,p F ρ(k, N, p) = Ψ(k, N, p)◦ F + G r, (5.2) (−λ)i λi i=1 i=1
71 where G and F are the N × 1 vectors representing F (r) and G(z), imax has replaced infinity, and
2 dN,p is the matrix of second order derivatives defined in Section 2.3, where here we distinguish it by the order of accuracy p. All of these derivatives exist analytically, and we could write out expressions for each of them instead of numerically differentiating. We choose to do this numerically because it is easier to apply to any form of F (r) and G(z) and does not require there being an analytic expression for them. As a result of using numerical derivatives, in addition to checking that the truncation of the sum (replacing ∞ by imax) does not affect the solution, we have to check that these approximate derivatives also do not.
Figures 5.1 - 5.2 show the density, conformal factor, and error for the case given by the functions,
! r 2 F (r) =Arr exp − , (5.3) σr A G(z) = z tanh(z + z ) + tanh(−z + z ) . 2 c c
With this form of F (r) and G(z), we adjust the width of the string by σr, the length by zc and the magnitude by Ar, Az and λ. The accuracy of the solution is controlled by imax and λ. As the number of points increases, the numerical error tends to get worse. This is a result of the division by r and while we choose functions which contain products of r in convenient places to cancel the division by r, when we compute the derivatives numerically we are still at some point using a divergent function.
In Figures 5.3-5.4 we show some examples of the solutions we obtain from the analytic method.
The specific parameters are included in the captions. Figure 5.3 shows that we can obtain elongated solutions of varying magnitude (note the range in the maximum of the conformal factors), with a small error. Notably, the minimum of the density is negative. However, the maximum is always positive and at least an order of magnitude larger. This is a result of the fact that we did not impose any constraint on the density when solving for the analytic solutions. In Figure 5.4 we show another set of solutions obtained by modifying the function G(z). Here we obtain a solution with variation along the z-dimension of the AFBS, while still having an asymptotically flat solution. This illustrates the versatility of the analytic method; even though we are constrained to separable solutions, we can obtain interesting conformal factors which would be difficult to resolve numerically.
72 ψ max =16.2952 error=9.49e−013 20
18
16
14
12
10
8
6
4
2
0 0 2 4 6 8 10 12 14 16 18 20
ρ [−5.11e+001,5.63e+003] 20
18
16
14
12
10
8
6
4
2
0 0 2 4 6 8 10 12 14 16 18 20
Figure 5.1: Solution to equation (5.1) with F (r) and G(z) as in equation (5.3). The vertical axis is the radial coordinate r, and the horizontal is z. Top shows the conformal factor and in brackets the maximum value it takes (minimum is 1). The error (norm of residue) is also shown. Bottom shows the density distribution corresponding to the conformal factor, and its range of values. Notably, the minimum density is negative, but the maximum is large. 4 4 In both, blue is the lowest value, and pink is the highest. Here λ = 5000, Ar = 2 × 10 , Az = −2 × 10 , σr = 0.5, zc = 4.8 and imax = 15 and the boundaries are at r, z = ±20.
73 −9.5 i =7 max i =9 max −10 i =11 max i =13 max i =15 −10.5 max ) 2
−11 (||Error|| 10 log −11.5
−12
−12.5 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 ∆ r
−10.5 λ =5000 λ =10000 λ =15000 −11 λ =20000 λ =25000 ) 2 −11.5 (||Error|| 10 −12 log
−12.5
−13 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 ∆ r
Figure 5.2: Solution to equation (5.1) with F (r) and G(z) as in equation (5.3) and parameters as in Figure 5.1. 4 The vertical axis is the radial coordinate r, and the horizontal is z. The parameters are: λ = 5000, Ar = 2 × 10 , 4 Az = −2 × 10 , σr = 0.5, zc = 4.8 and imax = 15 and the boundaries are at r, z = ±20. In the top figure imax is varied, and in the bottom λ is varied.
74 ψ max =1.0036 error=1.40e−014 0.5
0 0 1 2 3 4 5 6 7 8 9 10
ρ [−4.18e−002,3.43e−001] 0.5
0 0 1 2 3 4 5 6 7 8 9 10 3 3 (a) Ar = 10 , Az = −10
ψ max =1.3608 error=4.43e−014 0.5
0 0 1 2 3 4 5 6 7 8 9 10
ρ [−4.19e+000,4.65e+001] 0.5
0 0 1 2 3 4 5 6 7 8 9 10 4 4 (b) Ar = 10 , Az = −10
ψ max =37.0774 error=3.65e−012 0.5
0 0 1 2 3 4 5 6 7 8 9 10
ρ [−4.41e+002,1.27e+005] 0.5
0 0 1 2 3 4 5 6 7 8 9 10 5 5 (c) Ar = 10 , Az = −10
Figure 5.3: Solution to equation (5.1) with F (r) and G(z) as in equation (5.3). The vertical axis is 4 the radial coordinate r, and the horizontal is z. Parameters are λ = 10 , σr = 0.1, zc = 8, imax = 15 and the magnitude of Ar and Az increasing from panel (a)-(c). In all cases the boundaries (not shown) are at r, z = ±20. The colours are such that pink is the largest value and blue the smallest.
75 ψ max =55.0958 error=3.32e−012 0.5
0 0 1 2 3 4 5 6 7 8 9 10
ρ [−6.77e+002,2.86e+005] 0.5
0 0 1 2 3 4 5 6 7 8 9 10
Az 1 2 (a) G(z) = 2 tanh(z + zc) + tanh(−z + zc) ( 2 + cos(z) )
ψ max =76.4679 error=2.87e−011 0.5
0 0 1 2 3 4 5 6 7 8 9 10
ρ [−9.13e+003,1.97e+006] 0.5
0 0 1 2 3 4 5 6 7 8 9 10
Az 1 1 (b) G(z) = 2 tanh(z + zc) + tanh(−z + zc) ( 2 + 2 cos(6z))
ψ max =114.3813 error=4.96e−011 0.5
0 0 1 2 3 4 5 6 7 8 9 10
ρ [−1.60e+004,4.61e+006] 0.5
0 0 1 2 3 4 5 6 7 8 9 10
Az 1 (c) G(z) = 2 tanh(z + zc) + tanh(−z + zc) (1 + 2 cos(10z)) Figure 5.4: Solutions to equation (5.1) with F (r) as in equation (5.3). The vertical axis is the radial 4 5 coordinate r, and the horizontal is z. Parameters are λ = 10 , σr = 0.1, zc = 8, imax = 15, Ar = 10 , 5 Az = 10 . Panels (a)-(c) show solutions for different G(z). In all cases the boundaries (not shown) are at r, z = ±20. The colours are such that pink is the largest value and blue the smallest.
76 5.1.2 Numerical Solutions
Here we show the results of solving for the conformal factor numerically using the methods in
Section 3.2.2. In principle using a numerical method allows us to use a generic density distribution, unlike in the analytic solutions. However, as discussed in Section 3.2.2, for the numerical method we have assumed that the density divided by the square of the conformal factor is separable. Still, these solutions are of a different form than the analytic solutions.
We specify the non-physical density distribution, ρ(r, z). Figure 5.5 shows some solutions obtained by the choice,
A 2 ρ(r, z) = tanh(z + |z |) − tanh(z − |z |) exp − r/σ , (5.4) 2 c c
th where zc determines the length of the distribution in the 5 dimension, σ determines the width in the radial direction and A is the maximum of the density.
To verify that the solution converges as expected, we monitor the residue F (Ψ˜ ) where,
F (Ψ˜ ) = D2 + 2D1 R + PTD2 P Ψ˜ + ρ˜ Ψ˜ . nr nr nz
As the solution Ψ˜ approaches the true solution ψ∗ then we expect F (Ψ˜ ) → 0.
The magnitude of F (Ψ˜ ) depends on the grid spacing (determined by the boundaries and nr, nz above) and the error tolerances specified in the algorithm. If we are using pth order accurate finite difference equations, then F (Ψ˜ ) depends on the grid spacing ∆ as,
F (Ψ˜ , ∆) = f Ψ˜ ∆p, where, for simplicity, we adopt the spacing ∆ is equal in r and z. Because the equation is nonlinear the residual is proportional to an (unknown1) function of the solution, f Ψ˜ . We can estimate the order p by comparing F (Ψ˜ , ∆) at different resolutions,
˜ ˜ log2 F Ψ, ∆ − log2 F Ψ, ∆/2 = p.
1The terms with derivatives are linear and so the error is additive, where as for the density term the error is proportional to ρ˜ Ψ˜ ◦Ψ˜ . The overall dependence on the spacing is a function of the solution
77 When we reduce the step size from ∆ to ∆/2 the error tolerance is also adjusted. As discussed in Chapter 3, this is because we imagine the tolerance set to be relative to the actual error, which is proportional to ∆p. When we reduce ∆, we also reduce the tolerance in order to ensure we are solving to the same accuracy at each step size.
2 We compare the L2-norm of the log of the residue between solutions obtained by reducing a given step size by 1/2. While it may be cleaner to compute the element-wise ratio of the two solutions, if the solution is converging point-wise it will do so overall as well. To account for the different grid sizes we normalise by the size of the domain, finding the error as,
v n n u 1 Xr Xz |Error| = u F 2 . 2 t(n − 1)(n − 1) i,j r z i=1 j=1
Figure 5.5 shows the log of this error found for different step sizes ∆r. While it is not always the case3 that ∆r 6= ∆z, and for both ∆r and ∆z a fine and a coarse spacing are used, when ∆r
(fine or coarse) is halved, so are both ∆z. Figure 5.5 shows the fine ∆r. The tolerances are set at i = k = 0.001, multiplied by the maximum of the density, for the largest spacing, and this is reduced by a factor of 24 each time the spacing is halved. Making the tolerance proportional to the maximum of the density is done because we are taking absolute error, and with a larger conformal factor (owing to a larger density) we expect the absolute error to be larger. It affects nothing in the results shown, but helps us when solving with an arbitrary density as we do not have to wait a longer time because of having used a different density.
2 The l2 norm is used, but the particular choice of norm does not matter as if the solution converges for the l2norm, it will converge for any lp norm [9]. 3This depends on the boundaries and the cutoff which separates the coarse and fine spacing, and for the AFBS it often makes sense to have a finer spacing and/or larger boundaries in the z direction than in the r direction.
78 −12 13 −14
−16
) −18 20 2
−20 log2(|Error| −22 30
−24 39 −26
52 −28 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 ∆ r
ψ max =1.51, |Error| =4.03e−09 2 2 1.4
1 1.2
0 0 2 4 6 8 10
ρ max =2.56, min=3.08e−23 2 2 1.5 1 1 0.5 0 0 2 4 6 8 10
Figure 5.5: Solutions for the conformal factor found using the numerical method and density as in equation (5.4) with A = 2, σ = 1 and zc = 6.4. The vertical axis is the radial coordinate r, and the horizontal is z. The boundaries (not shown) are at r = ±6, z = ±16. Fourth order accurate derivatives are used. In the top figure, the points are labelled with the total number of iterations required to attain the tolerance. The colours are such that pink is the largest value and blue the smallest.
5.2 Physical properties
Figure 5.6 shows an example of finding the ADM mass. The horizontal axis shows the different values of R that the integral in equation (4.5) is evaluated at. To take the limit R → ∞ we fit the value for some of the larger R and take the limit of the fitted expression. As shown, as the spacing gets smaller the curves get closer together suggesting we are getting a more precise value for the
79 ADM mass. Because we have not taken R very large the extrapolated values of the mass are likely with more error than shown (this the standard error of the fit, which does not account for the error of the infinite missing data points). To get an accurate value for the ADM mass we can use a much larger domain. In that case we can see very clearly that the mass approaches a limiting value.
Figure 5.7 shows the conformal factor, density and Kretchmann scalar obtained using the analytic method. Figure 5.8 shows the location of the horizon (top) and the residue of the conformal factor evaluated on the horizon. The location of the apparent horizon is converging as the step size is reduced. Since the conformal factor is obtained analytically, this change is not due to an improvement in the precision of the conformal factor (which is evident from the residue along the horizon), but an improvement in the solution for the apparent horizon which is found numerically. Figures 5.9 show the Kretchmann scalar evaluated on the horizon, unscaled (top) and scaled by the radius of the horizon (bottom). Keeping in mind that there is negligible error coming from the conformal factor, there is also very little coming from the calculation of the Kretchmann scalar. Here the fact that the values obtained on the horizon change substantially as the grid spacing varies is a result of the change in the estimated horizon.
Figures 5.10 and 5.11 show the same, but for a conformal factor obtained numerically. Here, the error of the conformal factor gets smaller as the resolution is increased, as expected.
80 0.7 ∆ =0.0833 0.68 M=0.314±0.006 ∆ =0.0417 0.66 M=0.286±0.003 ∆ =0.0208 0.64 M=0.285±0.002 ∆ 0.62 =0.0104 M=0.289±0.001 0.6
ADM mass 0.58
0.56
0.54
0.52
0.5 10 10.5 11 11.5 12 12.5 13 13.5 14 radius (R)
Figure 5.6: Example of estimating the ADM mass. For different resolutions, the mass found as a function of R is plotted as well as the fitted values. As the resolution is increased, the values converge.
81 ψ max =2.2437 error=2.23e−13 4
2
0
−2
−4 −10 −5 0 5 10
ρ [−4.53e−01,1.67e+01] 4
2
0
−2
−4 −10 −5 0 5 10 Curvature scalar [0.00e+00,4.83e+00] 4
2
0
−2
−4 −10 −5 0 5 10
Figure 5.7: Solutions to equation (5.1) with F (r) and G(z) as in equation (5.3). The vertical axis is the radial 4 4 3 coordinate r, and the horizontal is z. Parameters are λ = 10 , σr = 1, zc98, imax = 15, Ar = 2.5×10 , Az = −5×10 . (a) conformal factor (b) density and (c) Kretchmann scalar.-(c) show solutions for different G(z). In all cases the boundaries (not shown) are at r, z = ±20. The colours are such that pink is the largest value and blue the smallest.
82 3.2 ∆ =0.201 3 ∆ =0.134 ∆ =0.1 2.8 ∆ =0.0802 ∆ =0.0668 2.6 ∆ =0.0572 ∆ =0.0501 ) z , r 2.4 ( h
, n o
z 2.2 i r o h t
n 2 a r a p p 1.8 A
1.6
1.4 −10 −8 −6 −4 −2 0 2 4 6 8 10 z
−12 ∆ =0.201 −12.5 ∆ =0.134 ∆ =0.1 −13 ∆ =0.0802 ∆ =0.0668 −13.5 ∆ =0.0572 ∆ =0.0501 −14
−14.5
−15 Error on horizon, log(res) −15.5
−16
−16.5 −10 −8 −6 −4 −2 0 2 4 6 8 10 z
Figure 5.8: Apparent horizon (a) and residue evaluated on the horizon (b) for the solutions in figure 5.7. The colours are such that pink is the largest value and blue the smallest.
83 1 ∆ =0.201 0.9 ∆ =0.134 ∆ =0.1 0.8 ∆ =0.0802 ∆ =0.0668 0.7 ∆ =0.0572 ∆ 0.6 =0.0501
0.5
0.4
Kretchmann scalar, K 0.3
0.2
0.1
0 −10 −8 −6 −4 −2 0 2 4 6 8 10 z
30 ∆ =0.201 ∆ =0.134 25 ∆ =0.1 4 ∆ =0.0802 ∆ =0.0668 20 ∆ =0.0572 ∆ =0.0501
15
10 Kretchmann scalar rescaled, K * r 5
0 −10 −8 −6 −4 −2 0 2 4 6 8 10 z
Figure 5.9: Kretchmann scalar (a) and Kretchmann scalar rescaled (b) evaluated on the horizon for the solutions in Figure 5.7 and horizon in Figure 5.8. The colours are such that pink is the largest value and blue the smallest.
84 ψ max =1.4689 error=2.33e−08 2
1.5
1
0.5
0 0 1 2 3 4 5 6 7 8 ρ(r,z) 4
12 3 10 8 r 2 6
1 4 2 0 0 1 2 3 4 5 6 7 8 z
Figure 5.10: Solutions for the conformal factor (a) and unphysical density (b) found using the numerical method as in equation (5.4) with A = 15, σ = 0.3.. The boundaries (not shown) are at r = ±15, z = ±24. Fourth order accurate derivatives are used. The colours are such that pink is the largest value and blue the smallest.
85 1.4 ∆ =0.0833 ∆ =0.0417 1.2 ∆ =0.0208 ∆ =0.0104
) 1 z , r ( h
, n
o 0.8 z i r o h t n 0.6 a r a p p A 0.4
0.2
0 −8 −6 −4 −2 0 2 4 6 8 z
−2 ∆ =0.0833 ∆ =0.0417 −4 ∆ =0.0208 ∆ =0.0104 −6
−8
−10
Error on horizon, log(res) −12
−14
−16 −8 −6 −4 −2 0 2 4 6 8 z
Figure 5.11: Apparent horizon (a) and residue of the conformal factor evaluated on the horizon (b) for the solutions in Figure 5.10. The colours are such that pink is the largest value and blue the smallest.
86 5.3 Concluding Remarks
We wanted to find suitable initial data for an asymptotically flat black string. Both the shape of the horizons and the curvature scalar suggest we have succeeded at this. The apparent horizons found are elongated and the curvature scalar appears to be constant along the interior away from the endpoints.
Ultimately we would like to have found initial data which would be subject to the Gregory-
Laflamme instability. This will depend on the length of the string – the proper distance between the apparent horizon at each end of the AFBS. For any string length, a non-static AFBS would contract at a maximum speed of the speed of light c. If the string is above a critical value of length, it will also be subject to the Gregory-Laflamme instability. The contraction will take a time proportional to L/c, and the instability will depend exponentially on time. From this we can infer that the string can be made sufficiently long so that the instability and bifurcation would proceed the contraction.
Our results show that there appears to be no impediment to solving for initial data for an ar- bitrarily long black string. Black strings with higher length to radius ratio would require more computational time, but otherwise could be found as we did here.
87 Bibliography
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89 Appendix A
Numerical Methods
A.1 Gauss-Seidel Newton
Here x = Ψ˜ , A(x) = J(x)T J(x), and f(x) = J(x)T f(x). Following [ref], we solve for x = Ψ˜ iteratively,
xk+1 = xk − h, A(xk)h = f(xk).
To avoid inverting A(xk), it is decomposed as,
A(xk) = Bk − Ck,
A(xk)h = f(xk),
(Bk − Ck)h = f(xk),
−1 −1 h = Bk Ckh + Bk f(xk).
This relation is used to update h iteratively,
A+1 −1 A −1 h = Bk Ckh + Bk f(xk).
90 −1 Then denoting Hk = Bk Ck, the iterations of h become,
1 0 −1 h =Hkh + Bk f(xk),
2 1 −1 h =Hkh + Bk f(xk),
2 0 −1 −1 2 0 −1 −1 h =Hk Hkh + Bk f(xk) + Bk f(xk) = Hk h + HkBk f(xk) + Bk f(xk),
3 3 0 2 −1 −1 −1 h =Hk h + Hk Bk f(xk) + HkBk f(xk) + Bk f(xk),
etc.
−1 0 Since h = A(xk) f(xk) and the solution has f(xk) = 0, then h = 0 is a good initial guess. This reduces the above expression, for mk steps, to,
mk mk X A−1 −1 h = Hk Bk f(xk). A=1
Alternatively h is iterated until the changes are sufficiently small. For successive over-relaxation, the step h is scaled by a relaxation parameter 0 < w < 2 which can speed up the convergence as compared with Gauss-Seidel (w = 1), so that the iteration for x is,
mk xk+1 = xk − wkh .
Separating A(xk) = Dk −Lk −Uk, such that Lk and Uk each appear only in Bk or Ck guarantees that
−1 Bk is easily inverted. One way to define Bk and Ck such that A(xk) = Bk − Ck = Dk − Lk − Uk,
m which satisfies xk+1 = xk − wh k is,
−1 Bk = w [Dk − wLk],
−1 Ck = w [Dk(1 − w)] + Uk.
Defining these, gives,
mk X A−1 −1 −1 xk+1 = xk − w Hk [Dk − wLk] f(xk),Hk = Bk Ck. (A.1) A=1
91 A.2 Linear Least Squares Regression
To estimate the ADM mass, we integrate a function at a fixed radius R with the true mass being the value obtained as R → ∞. We can not take R → ∞ and even taking very large R is unpleasant
(in the sense that it requires a larger grid, and therefore takes more time). For this reason we look at the values obtained for the ADM mass at a series of fixed R and try to estimate the value as
R → ∞. We fit the data to a function,
c c c c ψ = c + 1 + 2 + 3 + ... + n . 0 r r2 r3 rn
1 For an equation of degree n (powers up to rn ) with m data points in the r direction, the data can be fit using linear least squares by minimising the residual of the overdetermined m × n system,
Ac = y,
where the vector c = (c0, c1, c2, ..., cn) contains the n unknown coefficients, y = (y0, y1, y2, ..., yn) is a vector of the m data points, ie. each discrete point from the solution ψ, and A is the Vandermonde
1 matrix containing the independent variables xi = , i = 0..m: ri
1 x x2 ... xn 0 0 0 2 n 1 x1 x1 ... x1 A = 1 x x2 ... xn 2 2 2 ...... 2 n 1 xm xm ... xm
The residual vector r = y − Ac is minimized for c satisfying the normal equations AT Ac = AT y.
To avoid costly matrix inversion, the above is solved as follows: AT A is symmetric and positive definite and so it has a Cholesky factorization AT A = LLT for a lower triangular matrix L. Using the Cholesky factorization the normal equations become Lq = AT y, q = LT c, such that q can be solved for using substitution (as L is lower triangular), and then q can be used as the right hand side vector to solve LT c = q for the desired coefficient vector c using substitution (as LT is upper triangular). The algorithm used for the Cholesky factorization is from [9].
92 A.3 Interpolation
Large arrays are used to represent solutions for the conformal factor and shape function for the horizon so that we can find solutions which more closely approximate the continuous functions. Once a solution is obtained, it is easier to work with smaller arrays — either for visualising the solutions or for calculating other things which do not depend so much on the the spacing of the grid.For this the arrays are resized using Lagrangian interpolation.
A vector V0 defined at points x0 is interpolated to a vector V at points x as,
n X x − x0(k) V (i) = V (j)Πn . 0 k=1,k6=j x (j) − x (k) j=1 0 0
To interpolate an arrray this is repeated along each dimension. The size of V and x has to be chosen as to correspond to a division of dx0 by an integer. Therefore along each dimension being
interpolated, we choose to interpolate from Nx0 points to Nx where,
Nx = (Nx0 − 1)/(N ∗ −1),
for some integer Nx. We do this by setting a minimum value for N∗ and picking the first value of
Nx (for which N∗ > N ∗m in) that is an integer.
93