Generating spiky solutions of Einstein field equations with the Stephani transformation
GGenerating spiky solutions of Einsteinfield equations with the Stephanitransformation
A thesissubmitted in partial fulfilmentof the requirements for the DegreeofDoctor of Philosophy in MathematicsatThe University of Waikato by Muhammad Zubair Ali Moughal a r X i v : . [ g r- q c ] F e b bstract The Geroch/Stephani transformation is a solution-generating transformation,and may generate spiky solutions. The spikes in solutions generated so far areeither early-time permanent spikes or transient spikes. We want to generatea solution with a late-time permanent spike. We achieve this by applyingthe Stephani transformation with the rotational Killing vector field of thelocally rotationally symmetric Jacobs solution. The late-time permanent spikeoccurs along the cylindrical axis. The generated solution also features a richvariety of transient structures. We introduce a new technique to analyse thesestructures. Our findings lead us to discover a transient behaviour, which wecall the overshoot transition. cknowledgements
In the name of Allah, The Most Gracious and The Most Merciful, who pro-vided me the opportunity to unveil the concealed realities in the world ofMathematics.In the first place, I owe my deepest gratitude to my supervisor, Dr. WoeiChet Lim. It would have been next to impossible to write this dissertationwithout the guidance and valuable input/feedback of my respected supervisor.He is a source of true inspiration as he has always supported and guided meduring my stay in New Zealand – especially his financial and personal supportfor providing a grant to present my work at the 22nd International Confer-ence on General Relativity and Gravitation (GR22) and the 10th AustralasianConference on General Relativity and Gravitation (ACGRG10).I deeply acknowledge the role of the Department of Mathematics at Uni-versity of Waikato which facilitated and provided learning and a healthy en-vironment to complete my thesis. I am also thankful to the other PhD labstudents Ejaz, Liam, Fahim, Hamish, Chris and Nick for a healthy workingenvironment.My acknowledgment will never be complete without the special mention ofmy friends: Bilal, Hassam, Humair, Irfan, Atta, Irfan Habib, Shaheer, Mairajand many more (it is not feasible to name all here). Thanks for being aroundand sharing several good times together during my stay in the University andNew Zealand.I owe a special debt to my affectionate and lovely parents. The credit, formy enjoying this status in my life, goes to my parents. I donot have wordsto convey my cordial regards and thanks to my mother for her utmost efforts,sacrifices, and prayers. I am also grateful to my father for all that he did formy bright future. Besides I cannot dare to forget to mention the co-operationof my sweet sister, brothers and their naughty kids. I appreciate the moralsupport of my friends in Pakistan as well, specially Khalil, Kamran, Hassan,Faraz and Waqas.Lastly, I would like to thank the Higher Education Commission (H.E.C) ofthe government of Pakistan that provided me with the scholarship to do thisPhD. ontents k = 0 ω (cid:54) = 0 . . . . . . . . . . . . . . . . . . . . . . . . 294.3.2 Case ω = 0 . . . . . . . . . . . . . . . . . . . . . . . . 304.4 Radius of spike . . . . . . . . . . . . . . . . . . . . . . . . . . 324.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 k (cid:54) = 0 At late times ( t → ∞ ) . . . . . . . . . . . . . . . . . 405.3.2 At early times ( t →
0) . . . . . . . . . . . . . . . . . 445.4 When Σ +0 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 485.5 Radius of spike . . . . . . . . . . . . . . . . . . . . . . . . . . 535.6 Weyl scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 f . . . . . . . . . . . . . . . . . . . . . 607.1.1 Case Σ +0 = − − < Σ +0 < +0 = 0 . . . . . . . . . . . . . . . . . . . . . . . 807.1.4 Case 0 < Σ +0 < . +0 = 0 . . < Σ +0 ≤ ist of Figures + , Σ − ) plane for variousvalues of Σ +0 . A circle represents the orbit along r = 0, whichis a fixed point. At t increases, r (cid:54) = 0 orbits move away fromthese fixed points for Σ +0 >
0, and towards these fixed pointsΣ +0 <
0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.1 State space orbits projected on the (Σ + , Σ − ) plane for variousvalues of Σ +0 for ω (cid:54) = 0. A circle represents the orbit along r = 0, which is a fixed point. r (cid:54) = 0 orbits move away fromthese fixed points as t increases, mimicking the orbits of Taubsolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2 Snapshots of Ω k and Σ for ω = 1 and Σ +0 = −
1, showing thatthe inhomogeneities do not become narrow. . . . . . . . . . . . 315.1 Exponents in λ , ω and f . Dotted lines indicate the next domi-nant exponent. . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2 State space orbits projected on the (Σ + , Σ − ) plane for k = 1,Σ +0 = − , − . , ω = 1. All orbits end up at the samepoint indicated by an asterisk. . . . . . . . . . . . . . . . . . . 405.3 State space orbits for k = 1, Σ +0 = 0 .
25 and ω = −
1. . . . . . 415.4 State space orbits for k = 1, Σ +0 = 0 . ω = −
1. . . . . . 425.5 State space orbits for k = 1, Σ +0 = 0 .
75 and ω = −
1. . . . . . 435.6 State space orbits for k = 1, Σ +0 = 1 and ω = −
1. . . . . . . 445.7 State space orbits for k = 1, Σ +0 = − ω = 1. . . . . . . 45ii5.8 State space orbits for k = 1, Σ +0 = − . ω = 1. . . . . . 465.9 State space orbits for k = 1, Σ +0 = 0 .
25 and ω = −
1. . . . . . 475.10 State space orbits for k = 1 and ω = −
1. . . . . . . . . . . . . 485.11 State space orbits for k = 1 and Σ +0 = 0. . . . . . . . . . . . . 505.12 Plots of Σ + for k = 1, ω = 1 and Σ +0 = 0, showing a transientspike along r = 0. . . . . . . . . . . . . . . . . . . . . . . . . . 515.13 Plots of Σ + for k = 1, ω = 0 and Σ +0 = 0, showing a transientspike along r = 0. . . . . . . . . . . . . . . . . . . . . . . . . . 525.14 Weyl scalars when t = 0 . k = 1, ω = 1 and Σ +0 = − .Compare with Figure 5.8. This shows that the spike along r = (cid:113) − ω k Σ +0 is real, while the spike along r = 0 is a coordinate effectfor the interval − ≤ Σ +0 < k = 1, ω = − +0 = .We see that curve is not narrowing as time increases. i.e it isnot a spike. Compare with Figures 5.3 − r = 0 is not real for the interval 0 < Σ +0 ≤ . t of the terms in (7.1) against Σ +0 . . . . . . . . . . . 617.2 Transition time t (1&4)(2&3) as a function of r for Σ +0 = − k = 10 and ω = 10 . t (1&4)(2&3) has a global minimum at r = (cid:113) ω kk +1 = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 637.3 Transition time t (1&4)(2&3) as a function of r for Σ +0 = − k = 10 and ω = − . t (1&4)(2&3) has a global minimum at r = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647.4 f against ln t and r for Σ +0 = − k = 10 and ω = 10 .
1. Thetransition time has a global minimum at r = 1. . . . . . . . . 647.5 f against ln t and r for Σ +0 = − k = 10 and ω = − .
1. Thetransition time has a global minimum at r = 0. . . . . . . . . 65iii7.6 Qualitative plot of the log of each term squared against ln t ,showing 4 dominant equilibrium states, for any value of Σ +0 satisfying − < Σ +0 <
0. . . . . . . . . . . . . . . . . . . . . . 677.7 Blue line is the plot of t − t and red line is t − t , whenΣ +0 = − . k = 10, ω = 5. The blue line is positive for asmall interval around r = 1 and for r > . r < . f against ln t and r for Σ +0 = − . k = 10 and ω = 5 for theinterval 0 . < r < . r (cid:54) = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.9 f against ln t and r for Σ +0 = − . k = 10 and ω = 5 for theinterval 100 < r < t (green), t (blue) and t (red)against r for Σ +0 = − . k = 10 and ω = 5 for the interval100 < r < t ,showing 3 dominant equilibrium states, for any value of Σ +0 satisfying − < Σ +0 <
0. . . . . . . . . . . . . . . . . . . . . . 717.12 Blue line is the plot of t − t and red line is t − t , whenΣ +0 = − . k = 10, ω = 5. The blue line is positive for asmall interval 0 ≤ r < . . < r < . r < . f against ln t and r for Σ +0 = − . k = 10 and ω = 5 forthe interval 0 ≤ r < . r greater than 0 . f against ln t and r for Σ +0 = − . k = 10 and ω = 5 for theinterval 1 < r < r whichfade away to two visible states as the transition times becomecloser together as r increases. . . . . . . . . . . . . . . . . . . 737.15 The log of transition times t (blue), t (green) and t (red)against r for Σ +0 = − . k = 10 and ω = 5 for the interval1 < r < t becomes closerto t as r increases. . . . . . . . . . . . . . . . . . . . . . . . 747.16 Qualitative plot of the log of each term squared against ln t ,showing 3 dominant equilibrium states, for any value of Σ +0 satisfying − < Σ +0 <
0. . . . . . . . . . . . . . . . . . . . . . 757.17 Blue line is the plot of t − t and red line is t − t whenΣ +0 = − . k = 10, ω = 5. The blue line is positive for asmall interval 0 . ≤ r < . r > . r > . f against ln t and r for Σ +0 = − . k = 10 and ω = 5 forthe interval 0 ≤ r < r > . r < . +0 = − . k = 10 and ω = 5 showing the different scenarios along eachfixed r . Each cell is labelled with the index of the dominant term. 777.20 Qualitative plot of the log of each term squared against ln t ,showing 2 dominant equilibrium states, for any value of Σ +0 satisfying − < Σ +0 <
0. . . . . . . . . . . . . . . . . . . . . . 787.21 Blue line is the plot of t − t and red line is t − t , whenΣ +0 = − . k = 0 . ω = 500. The blue line is positive fora small interval 0 ≤ r < . r except 70 . < r < . f against ln t and r for Σ +0 = − . k = 0 . ω = 500,showing 2 distinct states for the interval 0 ≤ r <
10. . . . . . . 797.23 Blue line is the plot of t (1&2)3 and red line is t , when k = 0 . ω = 2. Figure shows that for r > . t < t (1&2)3 and for r < . t > t (1&2)3 . . . . . . . . . . . . . . . . 817.24 f against ln t and r for k = 0 . ω = 2 for the interval0 ≤ r <
10, showing scenario (7.33) for r > . r < . t ,showing 4 dominant equilibrium states, for any value of Σ +0 satisfying 0 < Σ +0 < .
5. . . . . . . . . . . . . . . . . . . . . . 847.26 Blue line is the plot of t − t and red line is t − t , whenΣ +0 = 0 . k = 15, ω = 6. The blue line is positive for ainterval r < . r > . . < r < . f against ln t and r for Σ +0 = 0 . k = 15 and ω = 6 for theinterval 7 < r <
11, showing 2 visible distinct states instead of4, because the transition times are too close together. . . . . . 857.28 The log of transition times t (green), t (red) and t (blue)against r for Σ +0 = 0 . k = 15, ω = 6 for the interval5 < r <
15, showing that the transition times are close together. 857.29 Qualitative plot of the log of each term squared against ln t ,showing 3 dominant equilibrium states, for any value of Σ +0 satisfying − < Σ +0 <
0. . . . . . . . . . . . . . . . . . . . . . 877.30 Blue line is the plot of t − t and red line is t − t , whenΣ +0 = 0 . k = 15 and ω = 6. The blue line is positivefor a small interval r < . r < . ≤ r < . f against ln t and r for Σ +0 = 0 . k = 15 and ω = 6 for theinterval 0 ≤ r < . r <
4. 88i7.32 Qualitative plot of the log of each term squared against ln t ,showing 3 dominant equilibrium states, for any value of Σ +0 satisfying 0 < Σ +0 < .
5. . . . . . . . . . . . . . . . . . . . . . 897.33 Blue line is the plot of t < t and red line is t < t whenΣ +0 = 0 . k = 15 and ω = 6. The blue line for a smallinterval r > . r > . r > . f against ln t and r for Σ +0 = 0 . k = 15 and ω = 6 ,showing 3 distinct states for r > . r < . +0 = 0 . k = 15 and ω = 6 showing the different scenarios along eachfixed r . Each cell is labelled with the index of the dominant term. 907.36 Qualitative plot of the log of each term squared against ln t ,showing 3 dominant equilibrium states, for any value of Σ +0 satisfying 0 < Σ +0 < .
5. . . . . . . . . . . . . . . . . . . . . . 927.37 Blue line is the plot of t − t and red line is t − t , whenΣ +0 = 0 . k = 10 and ω = 200. The blue line is positive fora interval r > . r < . . < r < . f against ln t and r for Σ +0 = 0 . k = 10 and ω = 200 forthe interval 0 ≤ r <
10, showing 2 distinct states. . . . . . . . 937.39 Transition time t (2&4)(1&3) as a function of r for Σ +0 = 0 . k = 2and ω = 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947.40 Transition time t (2&4)(1&3) as a function of r for Σ +0 = 0 . k = − ω = 15. . . . . . . . . . . . . . . . . . . . . . . . 947.41 f against ln t and r for Σ +0 = 0 . k = 2 and ω = 15. Thetransition time has a local maximum at r = 2. . . . . . . . . . 957.42 f against ln t and r for Σ +0 = 0 . k = − ω = 15. Thetransition time has a local minimum at r = 4. . . . . . . . . . 95ii7.43 Qualitative plot of the log of each term squared against ln t ,showing 4 dominant equilibrium states, for any value of Σ +0 satisfying 0 . < Σ +0 ≤
1. . . . . . . . . . . . . . . . . . . . . . 987.44 Blue line is the plot of t − t and red line is t − t , forΣ +0 = 0 . k = 0 . ω = 10. The blue line is positive forall the values of r . The red line is positive for r < . f against ln t and r for Σ +0 = 0 . k = 0 . ω = 10 for theinterval 0 < r < .
3, showing 4 distinct states. At r = 0, wehave a permanent spike at late times. . . . . . . . . . . . . . . 997.46 Qualitative plot of the log of each term squared against ln t ,showing 3 dominant equilibrium states, for any value of Σ +0 satisfying 0 . < Σ +0 ≤
1. . . . . . . . . . . . . . . . . . . . . . 1007.47 Red line is the plot of t − t and blue line is t − t , forΣ +0 = 0 . k = 0 . ω = 10. The blue line is negativefor a small interval r < . r . Together they give the interval r > . f against ln t and r for Σ +0 = 0 . k = 0 . ω = 10 for theinterval 0 ≤ r < r > . t ,showing 3 dominant equilibrium states, for any value of Σ +0 satisfying 0 < Σ +0 ≤
1. . . . . . . . . . . . . . . . . . . . . . . 1037.50 Blue line is the plot of t − t and red line is t − t , forΣ +0 = 0 . k = 250 and ω = 0 .
1. The red line is positive for ainterval 0 ≤ r < . ≤ r < . ≤ r < . f against ln t and r for Σ +0 = 0 . k = 250 and ω = 0 . ≤ r <
10, showing 3 distinct states. From r greaterthan 9 . t ,showing 2 dominant equilibrium states, for any value of Σ +0 satisfying 0 . < Σ +0 ≤
1. . . . . . . . . . . . . . . . . . . . . . 1057.53 Red line is the plot of t − t and blue line is t − t , forΣ +0 = 0 . k = 2 and ω = −
19. The blue line is positive forsmall interval 3 . < r < . r > . . < r < . f against ln t and r for Σ +0 = 0 . k = 2 and ω = −
19 for theinterval 0 ≤ r <
10, showing 2 distinct states for 3 . < r < . f against ln t and r for Σ +0 = 0, k = 0 . ω = − r (cid:46)
1, around ln t ≈ . r (cid:46) − (cid:46) ln t (cid:46)
3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.56 Plot of the cells and transition times in the example Σ +0 = 0, k = 0 . ω = − r . Each cell is labelled with the index of the dominantterm. An overshoot transition occurs on r (cid:46)
1, around ln t ≈ . r (cid:46) − (cid:46) ln t (cid:46)
3. . . . . . . . . . . . . . . . . . . . . . . . 1107.57 f against r for Σ +0 = 0, k = 0 . ω = −
2, showing theovershoots occurring on r (cid:46)
1. . . . . . . . . . . . . . . . . . . 1117.58 f against ln t along r = 0 for Σ +0 = 0, k = 0 . ω = − hapter 1Introduction The big bang theory postulates that the universe has always been expand-ing. Extrapolating this into the past, there was a time when the universewas incredibly dense and hot, such that even the laws of Einstein’s generalrelativity, which predicted the big bang, fail. Such time or place is calleda singularity. Similarly, after depleting their nuclear fuel, massive stars cancollapse under their own weight and become black holes. Inside a black hole,the collapse also inevitably leads to an incredibly dense and hot state wherethe laws of general relativity fail again. In the final stage before the laws fail,spacetime undergoes a kind of chaotic dynamics called Mixmaster or BKL dy-namics [1, 2, 3, 4]. Under the influence of Mixmaster dynamics, a collapsingobject is crushed or stretched at predictable and alternating speeds along itsthree dimensions. But what is really being crushed and stretched is space itself,regardless of the presence of any object there. Spikes form when adjacentparts of space experience vastly different rates of crushing and stretching.It is important to understand spikes because it occurs during a regime thattransitions into the quantum regime, and quantum gravity researchers need tounderstand what happens during the transition. In another context, spike canalso help explain the formation of large scale structure at late times.On approach to the singularity a generic solution is approximated by asequence of Kasner states, described by the Kasner solution of Einstein fieldequations (EFEs). The transition between Kasner states is approximated byanother exact solution, the Taub solution. Furthermore, this transition canbe achieved in two ways, as reflected by the sign of a curvature variable. In aspatially inhomogeneous model, this curvature variable may change sign fromone place to another. As a result, transition fails to occur in a normal wayat places where this curvature variable is zero. A different, inhomogeneous,dynamics occur instead, and this is the spike [5].What is the mechanism behind spike formation? Briefly, when a solutionis close to an unstable background solution that is represented by a saddlepoint in the state space, it generally becomes unstable. If the initial conditionis such that the solution straddles the separatrix of the saddle point, then asthe solution becomes unstable, some region of spacetime evolves one way, theother region evolves in a different way. The boundary between these two partsevolves in a way that is different from both regions (because on this boundary,its state lies exactly on the separatrix), and this creates a spiky inhomogeneityin the neighborhood of this boundary [5, 6].
It is a general feature of solutions of partial differential equations that spikesoccur [7]. As Einstein field equations (EFEs) of general relativity are a set ofpartial differential equations, spikes can arise in the solutions of these equations[5]. Spikes were first discovered in numerical simulations by Berger and Mon-crief in 1993 [8]. In their numerical study on the so-called T Gowdy model [9], where the BKL dynamics terminates at a final Kasner state, they observed T Gowdy models are orthogonally transitive G models with toroidal spatial topology.See Appendix B. the development of large spatial derivatives near the singularity, which theytermed as spiky features. These spikes are permanent spikes as their ampli-tude does not tend to zero towards singularity [10]. Furthermore, based on thework by Grubiˇsi´c and Moncrief of the same year [11], these structures wherefound to occur in the neighbourhood of isolated spatial surfaces. Toward theend of the 1990s, Berger, Moncrief and co-workers had found further numeri-cal evidence that the BKL picture seemed to be correct generically but therewere difficulties in simulating these spikes [12, 13, 14, 15]. Also, Hern in hisPhD thesis [16] resolved individual spatially spiky features to high numericalaccuracy but for a short time interval. The observed inhomogeneity of thecurvature invariants makes it clear that the spikes are physical features of thespacetime not effects of the coordinate system.In 2001 Rendall and Weaver [17] made a significant analytic step towardthe understanding of spikes. They discovered a composition of two transforma-tions that can map a spike-free solution to a solution with spike. They appliedthe solution-generating transformation and Fuchsian methods in [18, 22], toproduce asymptotic expansions for spikes. In their numerical study they findfalse and true(real) spikes. False spikes are an effect of parameterisation of themetric, not a geometrical one, while true spikes are the geometric change, whichthey check by observing highly non-uniform behaviour in curvature invariants.The work on spiky features in Gowdy spacetimes by Rendall and Weaver wasfollowed up by Garfinkle and Weaver in 2003, who used two different com-plementary numerical techniques [23]. In particular they studied the so-called(transient and recurring) high-velocity spikes and found that they eventuallyevolve into permanent low-velocity spikes. Also, Lim in his PhD thesis [5]applied the Rendall-Weaver transformation on the Wainwright-Marshman so-lution [24]. He obtained a new explicit vacuum OT G (Appendix B) solution The theory of Fuchsian equations has been applied to analyse singularities in a varietyof classes of spacetimes in general relativity. In [18, 19], Fuchsian algorithm is applied toEinstein’s equations to establish the existence of a family of solutions. Recent work can befound in [20], [21]. that develops a permanent spike.After the 1993 discovery of spikes many researchers tried to understandthe behaviour and dynamics of these structure. They put a lot of effort tosolve it through numerical solutions and analytical approximation. In thebeginning these analytical approximation were insufficient and the numericalsimulations lack resolution. In 2008 [25], Lim discovered the first exact spikesolution. Iteratively, he applied the Rendall-Weaver transformation on Kasnerseed solution. The generated spike solution admits two Killing vector fields(KVFs) and it is an orthogonally transitive (OT) G solution (see AppendixB). This exact solution is the gateway to the understanding and analysis ofthe spikes. There was a question whether numerical solutions match the exactsolution. The answer is yes – in 2009 Lim et al. [26] using a new zoomingtechnique, provided highly accurate numerical evidence.Nungesser and Lim [27] found the inhomogeneous electromagnetic spikesolution. They use the existing relation between vacuum Gowdy spacetimeand electromagnetic Gowdy spacetime to find this new explicit solution. Beyerand Hennig [28] derived a family of Gowdy-symmetric generalized Taub–NUTsolutions and found both false and true spikes.Coley and Lim [29] discussed the influence of spikes on matter that leadsto the formation of large scale structure at the early universe. They concen-trated on how spikes generate matter overdensities in a radiation fluid in aspecial class of inhomogeneous models. In 2014 Lim and Coley [30] examinedthe tilted fluid, whose tilt provided another mechanism in generating matterinhomogeneity through the divergence term.Coley and Lim [6] demonstrated the spike phenomenon by using the Lemaˆıtre-Tolman-Bondi (LTB) model. The LTB model is an exact solution that makesit easier to construct the spike. In this paper, they showed that spikes donot form in the matter density directly, it forms in the curvature as in [29].They also explained that spike can provide an alternate contribution to theformation of large scale structures in the Universe.The OT G spike solution contains permanent spikes, and there is a debatewhether in the non-OT G solution these permanent spikes are unresolved spiketransitions or are really permanent. In other words, would the yet undiscov-ered non-OT G spike solution contain permanent spikes? Numerical evidencesuggests that the permanent spikes are unresolved spike transitions. Heinzle et al. [31] described how BKL and spike oscillations arise from concatenationsof exact solutions, suggesting the existence of hidden symmetries and showingthat the results of BKL are part of a greater picture. Woei Chet Lim usedGeroch’s transformation to discover the non-OT G spike solution in 2015.He applied the transformation to a Kasner seed solution, with a generic linearcombination of KVFs [35]. He showed that non-OT G spike solution alwaysresolves its spike as opposed to previous OT G solutions. This method showsa new way to generate various kinds of spikes.The above advancements were made in the particular case where spikes varyin only one direction. Spikes that vary in two or three directions are much morecomplex; sheets of spikes can intersect each other and interact. Similarly, innon-vacuum models, sheets of overdensity in the fluid can intersect in filamentsand points to make even more pronounced overdensity in the fluid – a webof large scale structures form. The space between the sheets are filled withunderdensed fluid, and the underdensity becomes more pronounced – voidsform. The search for these complex structure led researchers to apply theStephani transformation on various seed solutions. Coley and Lim generalisedthe non-OT G vacuum spike solution by applying Stephani transformationon Jacobs solution [36].Coley et al. [37, 38] found the first exact spike solution in which two spikesintersect. They applied the Stephani transformation on a family of Bianchi In 1971-72, Geroch wrote two papers on generating new exact solutions of Einsteins fieldequations by using a transformation [32, 33]. The transformation acts on a vacuum solutionof Einstein field equations that possesses at least one KVF. This KVF remains preservedin the transformed solution. The stiff fluid version of Geroch transformation was given byStephani in 1988 [34]. type V solution. These are the first G stiff fluid spike solutions. In thegenerated solution, they observed some interesting phenomena at early times.They discussed many cases and some of them have permanent spike. But themost interesting one is the intersecting spike. This is the first exact spikesolution that has an intersecting spike. In this case the intersecting planes are Y = 0 and Z = 0. Intersecting spikes epitomise a prototypical intersectingwall. The density is higher on the walls but highest at the intersection. We notethat the Universe is dominated by bubbles of large voids surrounded by denserwalls [39]. The existence of non-linear structures at early times in the universemay support the large scale observational anomaly [40]. Another interestingresult in one of the cases of this paper is two phenomena at the same time.i.e. it has spike crossing at early times with a close-to-FL background. Thesespikes form at early times. With the exception of the LTB models [6], the exact solutions which are dis-cussed in the previous section have spikes at early times. LTB models aresilent , so our main goal is to find spikes at late times in non-silent models.For this we will use seed solutions that have a rotational KVF. Before thisno one used the rotational KVF in a transformation. i.e. all the KVFs usedare translational. We will apply the Stephani transformation on the LRS Ja-cobs solution. Our second goal is to develop a new technique to carry outintermediate time analysis of inhomogeneous structures. The thesis consists of two main results. The first is the generation of new spikysolutions and their properties using existing method of analysis (Chapter 3-6). In silent model there is no exchange of information between different fluid element eitherby sound waves (p=0) or gravitational waves ( H ab = 0) [41, Chapter 13]. The second is the development of a new method of analysis, and its applicationto the spiky solution (Chapter 7).In Chapter 2, we review some background material. We write the generalmetric in Iwasawa frame variables. We describe the Geroch/Stephani trans-formation with a KVF adopted to the Iwasawa frame, and give the formulasfor the transformation of the Iwasawa frame variables.In Chapter 3, we choose a seed solution, the LRS Jacobs solution and set itup for the Geroch/Stephani transformation. The linear combination of KVFsintroduces a parameter k . The solution is cast in cylindrical coordinates, andwe take note of its false spikes.In Chapters 4 and 5, we apply the Stephani transformation to the seedsolution for the cases k = 0 and k (cid:54) = 0 respectively. We analyse the dynamicsof the generated solution at early and late times . The case k = 0 has a late-time permanent spike forming along the rotational axis. The case k (cid:54) = 0 hasboth true and false spikes.We develop a heuristic for permanent spike in Chapter 6, for an arbitrarymetric. We define a general way of finding a permanent spike and compare itto previous results.In Chapter 7 we develop a new technique to explore the dynamics of the k (cid:54) = 0 case and revise the description of transient spikes. We also discover anddescribe the overshoot transition.In the concluding Chapter 8 we summarise the new results in this thesis,and remark on future research. Early time means the time towards Big bang and late times means the time away fromBig bang. hapter 2Background material
Einstein’s field equations (EFEs) are 16 coupled nonlinear partial differentialequations relating a set of symmetric tensors that explain the gravitationaleffects. In general relativity these gravitational effects are produced by a givenmass distribution. These field equations were presented by Einstein in 1915.Mathematically they are written as R ab − g ab R + g ab Λ = 8 πGc T ab . (2.1)where g ab is the metric tensor, R ab is the Ricci curvature tensor, R is theRicci curvature scalar, Λ is the cosmological constant, and T ab is the stress–energy tensor. Because of the symmetry of T ab and R ab , the genuine number ofequations decreases to 10, while there are Bianchi identities (four differentialidentities) satisfied by R ab that are one for each coordinate, so it reduces thenumber of independent equation’s to 6. Einstein felt cosmological constantdesirable at that time but Hubble’s observation of the expansion of the uni-verse made him reject the cosmological constant. But, recent astronomicalobservations suggest it strongly and consider that it is small but not zero butwe shall set Λ = 0. Also, in gravitational units we take 8 πG = c = 1. Soequation (2 .
1) becomes R ab − g ab R = T ab . (2.2)0In (2 . , the Ricci curvature tensor is obtained by contracting the Riemanncurvature tensor. So first we have to write Riemann curvature tensor, whichis R labk = ∂∂x b Γ lak − ∂∂x k Γ lab + Γ lbs Γ sak − Γ lks Γ sab , (2.3)where Γ lbs = 12 g lo (cid:18) ∂∂x s g ob + ∂∂x b g os − ∂∂x o g bs (cid:19) . (2.4)are the Christoffel symbols.So Ricci curvature tensor is R ab = R lalb = ∂∂x l Γ lab − ∂∂x b Γ lal + Γ lab Γ sls − Γ sal Γ lbs , (2.5)and Ricci curvature scalar is R = g ib R ib . (2.6)Equation (2 .
2) is also written as G ab = 8 πT ab . (2.7)where G ab = R aν − g ab R (2.8)is the Einstein tensor. The stress–energy tensor T ab for a perfect fluid withrespect to timelike vector field u T ab = ρu a u b + p ( g ab + u a u b ) (2.9)with energy density ρ >
0, pressure p and (unit timelike) fluid 4-vector u . Weassume that the equation of state of the perfect fluid is of the form p = ( γ − ρ ,where 0 ≤ γ ≤ γ = 1 (dust) and γ = 4 / ρ = Λ and p = − Λ, i.e γ = 0; and the value γ = 2 corresponds to a stiff fluid.A cosmological model ( M , g , u ) is determined by a Lorentzian metric g defined on a manifold M , and a family of fundamental observers, whose con-gruence of worldlines is represented by the unit timelike vector field u , which1we often identify with the matter 4-velocity. The dynamics of the model isgoverned by EFEs (2.1) with suitable matter content (2.9). It is helpful toclassify cosmological solutions of the EFEs using the dimension of orbits ofthe symmetry group admitted by the metric (see Appendix B). This classifica-tion scheme forms a hierarchy of cosmological models of increasing complexity(that can be found in Section 1.2.2 of [41]). In this thesis, we are interested inBianchi cosmologies. A Bianchi cosmological model ( M , g , u ) is a model whose metric admit athree– dimensional group of isometries acting simply transitively on space-like hypersurfaces, which are hypersurfaces of homogeneity in spacetime. ABianchi cosmology thus admits a Lie algebra of KVFs with basis ξξξ a , α = 1 , , , and structure constants C µαβ : [ ξξξ α , ξξξ β ] = C µαβ ξξξ µ (2.10)The ξξξ a are tangent to the group orbits, which are called the hypersurface ofhomogeneity. The Bianchi cosmology can be classified by classifying the Liealgebras of KVFs, and hence the associated isometry of the group G . Bianchicosmologies are classified [42] in Table 2.1. In the orthonormal frame approach one does not use the metric g directly(as done in the metric approach), but chooses at each point of the spacetimemanifold ( M , g ) a set of four linearly independent 1-forms { ωωω a } such that theline element can be locally expressed as ds = η ab ωωω a ωωω b , (2.11) An isometry of a manifold ( M , g ) is a mapping of M into itself that leaves the metric g invariant. n αβ IX + + +VIII + + − VII VII h + − VI h + + 0II IV + 0 0I V 0 0 0where η ab = diag ( − , , , e a are thenmutually orthogonal and of unit length – they form an orthonormal basis, with e being timelike (and thus defining a timelike congruence). The gravitationalfield is described by the commutation functions γ cab of the orthonormal frame,defined by [ e a , e b ] = γ cab e c . (2.12)The first step is to perform a 1+3 decomposition of the commutation functionsas follows: [ e a , e b ] = ˙ u α e − [ Hδ αβ + σ αβ − (cid:15) αβγ ( ω γ − Ω γ )] e β , (2.13)[ e a , e b ] = − (cid:15) αβµ ω µ e + [ (cid:15) αβν η µν + a α δ βµ − a β δ αµ ] e µ (2.14)The variables in (2.13) and (2.14) have physical or geometrical meanings, asfollows. The variable H is the Hubble scalar, σ αβ the rate of shear tensor,˙ u α the acceleration vector, and ω α the rate of vorticity vector of the timelikecongruence defined by e , while Ω α is the angular velocity of the spatial frame e α with respect to a nonrotating frame (Ω α = 0). We shall thus refer to n αβ and a α as the spatial curvature variables. Collectively, the variables above3describe the gravitational field. We shall refer to them as the gravitationalfield variables, and denote them by the state vector X grav = ( H, σ αβ , ˙ u α , Ω α , n αβ , a α ) , (2.15)The matter content of a cosmological model is described by the stress energytensor T ab , which is decomposed into irreducible parts with respect to e inthe following way (let e = u below): T ab = ρu a u b + 2 q ( a u b ) + ph ab + π ab , (2.16)where q a u b = 0 , π ab u b = 0 , π aa = 0 , π ab = π ba , (2.17)and h ab = g ab + u a u b is the projection tensor which locally projects into the3-space orthogonal to u . Since we are using an orthonormal frame, we have g ab = η ab , u a = (1 , , , q = 0 , π a = 0. The variables ( ρ, p, q α , π αβ )have physical meanings: ρ is the energy density, p is the (isotropic) pressure, q α is the energy flux density and π αβ is the anisotropic pressure (see, for example,van Elst Uggla 1997 [73, page 2677]). We shall refer to these variables as thematter variables, and denote them by the state vector X matter = ( ρ, q α , p, π αβ ) (2.18)The dynamics of the variables in (2.15) and (2.18) is described by the EFEs,the Jacobi identities (using e a ) and the contracted Bianchi identities respec-tively. The evolution of p and π αβ has to be specified by giving an equation ofstate for the matter content (e.g. perfect fluid). The variables ˙ u α and Ω α cor-respond to the temporal and spatial gauge freedom respectively. The variablesin (2.15) and (2.18) are scale-dependent and dimensional, and are unsuitablefor describing the asymptotic behaviour of cosmological models near the ini-tial singularity, since they typically diverge. It is thus essential to introducescale-invariant (dimensionless) variables, which one hopes will be bounded asthe initial singularity is approached. So we use the Hubble-normalised gravi-4tational and matter variables respectively as follows:(Σ αβ , ˙ U α , R α , N αβ , A α ) = ( σ αβ , ˙ u α , Ω α , n αβ , a α ) /H, (2.19)(Ω , Q α , P, Π αβ , Ω Λ ) = ( ρ, q α , p, π αβ , Λ) / (3 H ) . (2.20) Assume zero vorticity (zero shift). The spatial metric components are givenby the formula g αβ = e ai e bj δ ij , where α , β , i , j = 1 , ,
3. The Iwasawa frameis a choice of orthonormal frame that makes e αi (and equivalently e αi ) uppertriangular, as follows. The frame coefficients e αi simplify from 9 componentsto 6 components, represented by b , b , b , n , n and n . e αi = e e e e e e e e e = e − b e − b
00 0 e − b n n n = e − b e − b n e − b n e − b e − b n e − b (2.21) e αi = e e e e e e e e e = − n n n − n − n e b e b
00 0 e b = e b − e b n e b ( n n − n )0 e b − e b n e b (2.22)The frame derivative operators e = N − ∂ , e α = e αi ∂ i in the Iwasawa5frame are [43] e = 1 N ∂ (2.23) e = e b ∂ (2.24) e = e b [ − n ∂ + ∂ ] (2.25) e = e b [( n n − n ) ∂ − n ∂ + ∂ ] . (2.26) In the Iwasawa frame, the metric components in terms of the b ’s and n ’s aregiven by g = − N (2.27) g = e − b , g = e − b n , g = e − b n (2.28) g = e − b + e − b n , g = e − b n n + e − b n (2.29) g = e − b + e − b n + e − b n . (2.30)If the metric is given, we can compute the b ’s and n ’s as follows. b = − ln g (2.31) n = g g (2.32) n = g g (2.33) b = − ln( g − g n ) (2.34) n = ( g − g n ) e b (2.35) b = − ln( g − g n − e − b n ) . (2.36)The determinant g of the metric satisfies √− g = N e − b − b − b . (2.37) Consider a solution g ab of the vacuum Einstein’s field equations with a KVF ξ a .The Geroch transformation [32, 33] (see also [44, Section 10.3]) is an algorithm6for generating new solutions, by exploiting the KVF ξ a . The algorithm involvessolving the following PDEs ∇ a ω = ε abcd ξ b ∇ c ξ d , (2.38) ∇ [ a α b ] = 12 ε abcd ∇ c ξ d , ξ a α a = ω, (2.39) ∇ [ a β b ] = 2 λ ∇ a ξ b + ωε abcd ∇ c ξ d , ξ a β a = λ + ω − ω , α a and β a , where λ = ξ a ξ a , ∇ a is the covariant derivative and ε abcd isthe totally antisymmetric permutation tensor, with ε = √− g [41].Next, define ˜ λ and η a as˜ λ = λ (cid:104) (cos θ − ω sin θ ) + λ sin θ (cid:105) − , (2.41) η a = ˜ λ − ξ a + 2 α a cos θ sin θ − β a sin θ, (2.42)for any constant θ . Then the new metric is given by˜ g ab = λ ˜ λ ( g ab − λ − ξ a ξ b ) + ˜ λη a η b . (2.43)This new metric is again a solution of the vacuum Einstein’s field equationswith the same KVF. θ = 0 gives the trivial transformation ¯ g ab = g ab .Notice from (2.42) that α a appears in the new metric only through η a , andif θ is chosen to be π/ α a does not appear at all. We shall exploit thissimplification. In this case the new metric simplifies to˜ g ab = ( λ + ω ) g ab + λλ + ω β a β b − ξ a β b − β a ξ b . (2.44)Stephani [34] generalised the Geroch transformation to the case of comovingstiff fluid if the KVF is spacelike (and to the case of perfect fluid with equationof state p = − ρ/ ρ = ρλ + ω . (2.45)7 Before applying the Geroch transformation or the Stephani transformation,we set up the coordinates such that the KVF to be used has the form ξ a = (0 , , , , (2.46)to adapt to the Iwasawa frame for simplicity. If the seed metric in thesecoordinates has the general form g ab = − N g g g g g g g g g , (2.47)then the generated metric has the form˜ g ab = − F N λ g − β ˜ λ g − β ˜ λ g F g − g β + β ˜ λ F g − g β − g β + β β ˜ λ g ˜ g F g − g β + β ˜ λ , (2.48)where ˜ λ = λF , F = ω + λ , λ = ξ a ξ a = g = e − b . (2.49)The twist of the KVF, ω has gradient ω a = ( ω , ω , ω , ω ) , (2.50)whose components satisfy ω = − N e b + b λ / ∂ n (2.51) ω = 0 (2.52) ω = − N − e − b + b λ / ( n ∂ n − ∂ n ) (2.53) ω = − N − e b + b λ / ( e − b ∂ n + e − b n ∂ n − e − b n ∂ n ) . (2.54)8The covector β a = (0 , F − , β , β ) (2.55)satisfies the following partial differential equations: ∂ β = 2 λn ∂ λ + 2 λ ∂ n + 2 ωλ − e b √− g∂ λ − ωN λ ( √− g ) − n ∂ n (2.56) ∂ β = 2 λn ∂ λ + 2 λ ∂ n + 2 ωλ − e b √− gn ∂ λ − ωN λ ( √− g ) − n ∂ n (2.57) ∂ β − ∂ β = 2 λn ∂ λ + 2 λ ∂ n + 2 ωλ − N − √− g∂ λ − ωe − b λ ( √− g ) − n ∂ n + N − e − b + b ωλ / [ − n n ∂ n + n n ∂ n − n n ∂ n + n ∂ n ] . (2.58)Expressing the metric ˜ g ab in (2.48) in ˜ N , ˜ b ’s and ˜ n ’s gives˜ N = N √ F (2.59)˜ b = b + 12 ln F (2.60)˜ b = b −
12 ln F (2.61)˜ b = b −
12 ln F (2.62)˜ n = n F − β (2.63)˜ n = n F − β (2.64)˜ n = n . (2.65)In simpler cases, if the seed metric has the form g ab = − N g g g g
00 0 0 g , (2.66)9i.e. if n = 0 = n , then the generated metric has the form˜ g ab = − F N λ g − β ˜ λ
00 ˜ g F g − g β + β ˜ λ
00 0 0
F g . (2.67)The twist of the KVF, ω has gradient ω a = ( − N e b + b λ / ∂ n , , , − N − e b − b λ / ∂ n ) , (2.68)The covector β a = (0 , F − , β ,
0) (2.69)satisfies the following partial differential equations: ∂ β = 2 λn ∂ λ + 2 λ ∂ n + 2 ωλ − e b √− g∂ λ − ωN λ ( √− g ) − n ∂ n (2.70) ∂ β = 2 λn ∂ λ + 2 λ ∂ n + 2 ωλ − N − √− g∂ λ − ωe − b λ ( √− g ) − n ∂ n , (2.71)Expressing the metric ˜ g ab in (2.48) in b ’s and n ’s gives˜ N = N √ F (2.72)˜ b = b + 12 ln F (2.73)˜ b = b −
12 ln F (2.74)˜ b = b −
12 ln F (2.75)˜ n = n F − β (2.76)˜ n = 0 (2.77)˜ n = 0 . (2.78) hapter 3The seed solution As discussed in the introduction, past applications of Geroch/Stephani trans-formation used translational KVFs . In this thesis, we will use rotationalKVFs . Stephani transformation requires the matter to be a stiff fluid, so westart by looking at locally rotationally symmetric (LRS) solutions [45, page22] with a stiff fluid. The simplest such solution is the flat FLRW solutions[45, page 53], but it does not generate as much structure as the next simplestsolution, the LRS Jacobs (Bianchi type I) solution, which we shall use as theseed solution.The Jacobs solution [45, page 189] is given by the line element ds = − dt + t p dx + t p dy + t p dz , (3.1)where the coordinates are ( t, x, y, z ), and p = 13 (1 + Σ +0 + √ − ) , (3.2) p = 13 (1 + Σ +0 − √ − ) , (3.3) p = 13 (1 − +0 ) . (3.4)The non-zero Hubble-normalised shear components are Σ +0 and Σ − , and theyare constant, with Σ + Σ − ≤
1. The stiff fluid has pressure p and density ρ We had Cartesian coordinates mostly and the KVFs are translational. A rotational KVF is a KVF that is present in axis-symmetric solutions. The length ofa rotational KVF vanishes at the axis of rotation. p = ρ = 1 − Σ − Σ − t . (3.5)To impose the LRS condition; it is simplest to set Σ − = 0, so the parameterΣ +0 takes values from − +0 = − p , p , p ) = (0 , ,
1) (also known as the Taub form offlat spacetime); Σ +0 = 1 gives the LRS Kasner solution with ( p , p , p ) =( , , − ); Σ +0 = 0 gives the flat FLRW solution with stiff fluid.The LRS Jacobs solution admits four KVFs, namely ∂ x , ∂ y , ∂ z , − y∂ x + x∂ y , (3.6)where the fourth one is rotational.We intend to apply the Stephani transformation with the general linear com-bination of the KVFs: c ∂ x + c ∂ y + c ∂ z + c ( − y∂ x + x∂ y ) = ( c − c y ) ∂ x + ( c + c x ) ∂ y + c ∂ z (3.7)Observe that c and c can be eliminated without loss of generality by atranslation in x and y directions. We set c = 1 and c = k , so the KVF reads − y∂ x + x∂ y + k∂ z . (3.8)This KVF forms an Abelian OT G group with exactly one other KVF (namelya linear combination of ∂ z and − y∂ x + x∂ y ). By Geroch’s theorem [33, Ap-pendix B], the generated metric will admit an Abelian OT G group.There is a rotational symmetry about the z -axis, so we adopt cylindricalcoordinates ( r, ψ, z ), but we want to arrange the coordinates in the followingorder: ( t, ψ, z, r ), due to the way we adapt the orthonormal frame to thecoordinates. In these coordinates, the KVF reads ∂ ψ + k∂ z . (3.9)We want to simplify the KVF to just ∂ ψ for the application of the Stephanitransformation, so we make a further change of coordinates, by introducing Z = z − kψ. (3.10)2Then, in the coordinates ( t, ψ, Z, r ), the KVF is simply ∂ ψ , but the line elementnow reads ds = − dt + ( k t p + r t p ) dψ + 2 kt p dψdZ + t p dZ + t p dr . (3.11)This shall be the seed solution to which we apply the Stephani transformation.It has the simple form (2.66). For later convenience we list the b ’s and n ’s ofthis line element and define the squared norm of the KVF below. λ = k t p + r t p , (3.12) N = 1 (3.13) b = −
12 ln λ (3.14) b = −
12 ln r t p +2 p λ (3.15) b = −
12 ln( t p ) (3.16) n = kt p λ (3.17) n = 0 = n . (3.18)Observe that b + b + b = − ln( rt ) . (3.19)The ψ - Z area element A = e − b − b = rt p + p (3.20)and volume element V = e − b − b − b = rt (3.21)are always expanding.Here we list the dynamical variables of the seed solution using the formulasin Appendix A. To write the expressions in a more compact form, we list severalintermediate expressions, particularly the partial derivatives of the essentialvariable. ∂ λ = 2 p k t p − + 2 p r t p − (3.22) ∂ λ = 2 rt p (3.23)3So the expressions are H = 13 t (3.24)Θ = 12 (cid:18) ∂ λλ (cid:19) (3.25)Θ = 12 (cid:18) p + 2 p t − ∂ λλ (cid:19) (3.26)Θ = 12 (cid:18) p t (cid:19) (3.27)Θ = 12 kt − p + p r (cid:18) − ∂ λλ + 2 p t (cid:19) (3.28)˙ u = 0 (3.29) n = kt − p + p r (cid:18) − ∂ λλ (cid:19) (3.30) n = 12 t − p (cid:18) r − ∂ λλ (cid:19) (3.31) a = − t − p r (3.32)Σ = 32 (cid:18) t∂ λλ (cid:19) − = 32 (cid:18) p + 2 p − t∂ λλ (cid:19) − = 3 p − = 32 kt − p + p r (cid:18) − ∂ λλ + 2 p t (cid:19) . (3.36)Define Hubble-normalised expansion shear components Σ + and Σ − asΣ + = −
12 Σ (3.37)Σ − = Σ − Σ √ , (3.38)which gives Σ + = −
12 Σ +0 (3.39)Σ − = √ (cid:18) t (ln λ ) t − − Σ +0 (cid:19) . (3.40)Figure 3.1 shows that state space orbits projected on the (Σ + , Σ − ) plane forvarious values of Σ +0 . The r = 0 orbits are fixed points. r (cid:54) = 0 orbits moveaway from these fixed points as t increases but only in Σ − direction. The4Figure 3.1: State space orbits projected on the (Σ + , Σ − ) plane for variousvalues of Σ +0 . A circle represents the orbit along r = 0, which is a fixed point.At t increases, r (cid:54) = 0 orbits move away from these fixed points for Σ +0 > +0 < r = 0 if k = 0 (coordinate singularity). It is straight-forward to analyse l = t (ln λ ) t . If k = 0 then l = 2 p . If k (cid:54) = 0, then l = 2 p at r = 0. For r (cid:54) = 0 write l = p (1 − τ ) + p (1 + τ ) , τ = tanh(Σ +0 (ln t ) + ln | k | − ln r ) . (3.41)For Σ +0 > l goes from 2 p to 2 p as t goes from 0 (early times) to ∞ (latetimes). For Σ +0 < l goes from 2 p to 2 p . For Σ +0 = 0, l = . So l has a simple sigmoid transitional dynamics. It has a discontinuous limit along r = 0 (at late times for Σ +0 >
0, at early times for Σ +0 < l we define the transition time to be τ = 0, that is whenΣ +0 ln t + ln | k | − ln r = 0 , (3.42)or equivalently when the two competing terms in λ are equal. Solving for t gives the transition time t = (cid:18) r | k | (cid:19) − . (3.43)From (3.41), observe that Σ +0 appears as the coefficient of ln t in the tanhfunction. So smaller | Σ +0 | leads to milder slope/longer transition duration.5To summarise, the seed solution (3.11) has permanent false spikes in the case k (cid:54) = 0: . At r = 0 at late times for 0 < Σ +0 ≤ . At r = 0 at early times for − ≤ Σ +0 < k = 0, the seed solution has no false spikes.In the next two chapters we apply the Stephani transformation on this seedsolution. hapter 4Generated Solution, k = 0 In this chapter, we apply Stephani transformation on the seed solution (3.11)for the case k = 0 (that is, using only the rotational KVF) and discuss thedynamics of the generated solution. Computation is carried out using Maple. We now carry out the Stephani transformation with the rotational KVF ∂ ψ .First we find the squared norm of the KVF, λ = r t p , (4.1)and the twist of the KVF, ω = ω . (4.2)The combination λ + ω will appear frequently, so for brevity we introduce F = λ + ω . (4.3)The next step is to find a particular solution β a for the constrained system ∇ [ a β b ] = 2 λ ∇ a ξ b + ω(cid:15) abcd ∇ c ξ d , ξ a β a = F − . (4.4)We obtain β a = (0 , F − , β , , (4.5)7where β = 2 p ω r + 4 ω p t p (4.6)is found by integration.The generated metric is then given through b ’s and n ’s by the formulas(2.72)–(2.78), dropping tildes for brevity. N = F / (4.7) b = −
12 ln λF (4.8) b = −
12 ln
F r t p +2 p λ (4.9) b = −
12 ln(
F t p ) (4.10) n = − β (4.11) n = 0 (4.12) n = 0 . (4.13)Observe that b + b + b = −
12 ln(
F r t p +2 p ) = −
12 ln(
F r t ) . (4.14) Here we list the dynamical variables of the generated solution using the for-mulas in Appendix A. To write the expressions in a more compact form, welist several intermediate expressions, particularly the partial derivatives of theessential variables. ∂ λ = 2 p r t p − (4.15) ∂ λ = 2 rt p (4.16) ∂ ω = 0 (4.17) ∂ ω = 0 (4.18) ∂ F = 2 λ∂ λ + 2 ω∂ ω (4.19)8 ∂ F = 2 λ∂ λ + 2 ω∂ ω (4.20) ∂ β = 4 ωt p (4.21) ∂ β = 4 p ωr. (4.22)Now using the above convention, we get H = 16 F − / (cid:18) ∂ FF + 2 t (cid:19) (4.23)Θ = 12 F − / (cid:18) ∂ λλ − ∂ FF (cid:19) (4.24)Θ = 12 F − / (cid:18) ∂ FF − ∂ λλ + 2 p + 2 p t (cid:19) (4.25)Θ = 12 F − / (cid:18) ∂ FF + 2 p t (cid:19) (4.26)Θ = λt − p − p rF / ∂ β (4.27)˙ u = − F − / t − p (cid:18) ∂ FF (cid:19) (4.28) n = − λt − p − p rF / ∂ β (4.29) n = 12 F − / t − p (cid:18) ∂ FF − ∂ λλ + 1 r (cid:19) (4.30) a = − F − / t − p r . (4.31) The state space orbits of a solution, projected onto the (Σ + , Σ − ) plane, canprovide some insight into the dynamics of the solution. Recall that (Σ + , Σ − )are defined in terms of the diagonal components of the Hubble-normalisedexpansion shear as Σ + = −
12 Σ , (4.32)Σ − = Σ − Σ √ . (4.33)This reduces to Σ + = − Σ +0 + f f , Σ − = √ +0 − f )2 + f , (4.34)9where f = t (ln F ) t = 4 p r t p r t p + ω , p = 13 (1 + Σ +0 ) . (4.35)We carry out the asymptotic analysis of f , similar to what we did for l inChapter 3. ω (cid:54) = 0 Along r = 0, we have f = 0 and(Σ + , Σ − ) = ( −
12 Σ +0 , √
32 Σ +0 ) . (4.36)Along r (cid:54) = 0, provided that p (cid:54) = 0, we have f → t → p as t → ∞ , (4.37)and (Σ + , Σ − ) → ( − Σ +0 , √ Σ +0 ) as t → − (7Σ +0 +4)2(2Σ +0 +5) , − √ +0 )2(2Σ +0 +5) ) as t → ∞ . (4.38)There is a permanent spike at r = 0 at late times, for p (cid:54) = 0 ⇔ Σ +0 > − + , Σ − ) plane forvarious values of Σ +0 for ω (cid:54) = 0. The r = 0 orbits are fixed points, while the r (cid:54) = 0 orbits move away from these fixed points as t increases, mimicking theorbits of Taub solutions [45, page 136].The exceptional case is when Σ +0 = −
1, where (Σ + , Σ − ) = ( , − √ ) forall t and r . But this is not a Kasner solution. Recall that the timelike KVF ispreserved. The Weyl scalars (see Appendix C) are time-independent : CC = − (192( − r + 15 ω r − ω r + ω ))( r + ω ) (4.39) CCs = − ω r ( − r + 3 ω )( − r + ω )( r + ω ) (4.40) CCC = − r ( − r + 3 ω )( − r + 33 ω r − ω r + 3 ω )( r + ω ) (4.41) CCCs = 768 ω ( − r + ω )( − r + 27 ω r − ω r + ω )( r + ω ) . (4.42)0Figure 4.1: State space orbits projected on the (Σ + , Σ − ) plane for variousvalues of Σ +0 for ω (cid:54) = 0. A circle represents the orbit along r = 0, which isa fixed point. r (cid:54) = 0 orbits move away from these fixed points as t increases,mimicking the orbits of Taub solutions.The inhomogeneities do not become narrow, so there are no spikes. Figure4.2 shows the snapshots of Ω k and Σ , whose amplitude increases indefinitely,but whose width is constant. ω = 0 In this case (4.35) reduces to f = 4 p , and (4.34) simplifies toΣ + = − +0 + 42(2Σ +0 + 5) , Σ − = − √ +0 + 4)2(2Σ +0 + 5) . (4.43)From the fluid density ρ = 1 − Σ r t p (4.44)we conclude that for − < Σ +0 < r = 0.For the vacuum cases Σ +0 = ±
1, we can reach the same conclusion by1 (a) Ω k (b) Σ Figure 4.2: Snapshots of Ω k and Σ for ω = 1 and Σ +0 = −
1, showing thatthe inhomogeneities do not become narrow.2examining the Weyl scalars. For Σ +0 = − CC = 192 r (4.45) CCs = 0 (4.46)
CCC = − r (4.47) CCCs = 0 . (4.48)And for Σ +0 = 1 we have CC = 64(7 r − r t / + 27 t / )9 r t / (4.49) CCs = 0 (4.50)
CCC = 256( r t / − r t / + 9 r t − t / )3 r t / (4.51) CCCs = 0 . (4.52)Because there is a physical singularity at r = 0, we should not say that a spikeoccurs there. Following [25], we define the coordinate radius of spike (or inhomogeneities)to be the value of r where λ = | ω | , (4.53)which gives coordinate radius = (cid:112) | ω | t − (1+Σ +0 ) . (4.54)For Σ +0 = − +0 > −
1, the coordinate radius tends to zero as t → ∞ , indicatingthe formation of permanent spike along r = 0 at late times. This solution is equivalent to a Kasner solution (in the broader sense), or a Levi-Civitasolution [44, pages 197 and 343]. (cid:90) t N √ g dt (4.55)= (cid:90) t t − (1+Σ +0 ) dt (4.56)= 32 − Σ +0 t (2 − Σ +0 ) (4.57)Their ratio iscoordinate radius of spikecoordinate radius of particle horizon = (cid:18) − Σ +0 (cid:19) (cid:112) | ω | t − . (4.58)The physical radius of the spike is s = (cid:90) coordinate radius of spike0 √ g rr dr. (4.59)where √ g rr = √ F t p . (4.60)Let u = rt + Σ +0 | ω | and du = t + Σ +0 | ω | dr (4.61)and the limits are u = 0 and 1. Then s = | ω | / (cid:90) √ u du. (4.62)i.e. the physical radius of the spike is constant. As the universe expands, incomparison the spike becomes narrower and narrower. We set out to generate a solution with a permanent spike that forms at latetimes. We have achieved this by applying the Stephani transformation onthe LRS Jacobs solution, using the rotational KVF. The generated solutionis cylindrically symmetric, and has a spike along its rotational axis for the4case Σ +0 > − ω (cid:54) = 0. This is the first generated solution with a late-time permanent spike, and the first generated solution with a spike along aline. The spike produces an overdensity along the axis at late times, which isconducive to large scale structure formation. Thus the generated solution canserve as a prototypical model for formation of galactic filaments along web-likestrings. Such features can also be achieved through silent LTB and Szekeres models [6] withoutusing solution-generating transformations. hapter 5Generated solutions, k (cid:54) = 0 In the previous chapter we have discussed the dynamics of generated solutionfor the case k = 0. We have seen the spike at late times.In this chapter we are going to generalise the result to the case k (cid:54) = 0 (that is,using the general linear combination of KVFs). We now carry out the Stephani transformation with the general KVF ∂ ψ . Firstwe find the squared norm of the KVF, λ = k t p + r t p , (5.1)and the twist of the KVF, ω = 2 k p t p + k Σ +0 r + ω . (5.2)The combination λ + ω will appear frequently, so for brevity we introduce F = λ + ω . (5.3)The next step is to find a particular solution β a for the constrained system ∇ [ a β b ] = 2 λ ∇ a ξ b + ω(cid:15) abcd ∇ c ξ d , ξ a β a = F − . (5.4)We obtain β a = (0 , F − , β , , (5.5)6where β = 2 p ω r + p Σ +0 kr + (cid:18) ω + 2 k (1 − p ) r p (cid:19) t p + k t p + 4 k (1 + p ) t p (5.6)The generated metric is then given through b ’s and n ’s by the formulas (2.72)–(2.78), dropping tildes for brevity. N = F / (5.7) b = −
12 ln λF (5.8) b = −
12 ln
F r t p +2 p λ (5.9) b = −
12 ln(
F t p ) (5.10) n = F kt p λ − β (5.11) n = 0 (5.12) n = 0 . (5.13)Its ψ - Z area element A = rt p + p (5.14)is the same as the seed solution’s, and is always expanding. Its volume element V = rt √ F (5.15)is different from the seed solution’s and is not always expanding. This meansthe Hubble scalar H can become negative for some parameter values, andHubble-normalised variables would blow up. In this case we use β -normalisation,which is based on the ever-expanding area element [46]. Here we list the dynamical variables of the generated solution using the for-mulas in Appendix A. To write the expressions in a more compact form, welist several intermediate expressions, particularly the partial derivatives of the7essential variables. ∂ λ = 2 p k t p − + 2 p r t p − (5.16) ∂ λ = 2 rt p (5.17) ∂ ω = 2 kt p (5.18) ∂ ω = 2 k (3 p − r (5.19) ∂ F = 2 λ∂ λ + 2 ω∂ ω (5.20) ∂ F = 2 λ∂ λ + 2 ω∂ ω (5.21) ∂ β = 4 kp λt p − + 4 ωt p (5.22) ∂ β = 4 p ωr. (5.23)By using the above convention, we get this list of dynamical variables. H = 16 F − / (cid:18) ∂ FF + 2 t (cid:19) (5.24)Θ = 12 F − / (cid:18) ∂ λλ − ∂ FF (cid:19) (5.25)Θ = 12 F − / (cid:18) ∂ FF − ∂ λλ + 2 p + 2 p t (cid:19) (5.26)Θ = 12 F − / (cid:18) ∂ FF + 2 p t (cid:19) (5.27)Θ = 12 kF − / t − p + p r (cid:18) ∂ FF − ∂ λλ + 2 p t (cid:19) − λt − p − p rF / ∂ β (5.28)˙ u = − F − / t − p (cid:18) ∂ FF (cid:19) (5.29) n = kF − / t − p + p r (cid:18) ∂ FF − ∂ λλ (cid:19) − λt − p − p rF / ∂ β (5.30) n = 12 F − / t − p (cid:18) ∂ FF − ∂ λλ + 1 r (cid:19) (5.31) a = − F − / t − p r . (5.32) Now we find Σ + and Σ − by using the equations (4.32), (4.33). We haveΣ + = − Σ +0 + f f , Σ − = √ l − f − (2 − Σ +0 ))2 + f (5.33)8 (a) λ exponents (b) ω exponents(c) Dominant exponents in f at latetimes (d) Dominant exponents in f at earlytimes Figure 5.1: Exponents in λ , ω and f . Dotted lines indicate the next dominantexponent.where f = t (ln F ) t , l = t (ln λ ) t . The β -normalised (Σ + , Σ − ) areΣ + = − Σ +0 + f − Σ +0 Σ − = √ l − f )2 − Σ +0 − √ . (5.34)We have analysed the asymptotic dynamics of l in Chapter 3. Recall that l has a false spike at r = 0 at late times for 0 < Σ +0 ≤
1, and at early times for − ≤ Σ +0 <
0. The asymptotic dynamics of f can be analysed as follows. f consists of terms involving λ and terms involving ω . λ contains two differentpower terms, t p and t p ; while ω contains t p and constant. At late times,the term with the biggest exponent dominates; at early times the term withthe smallest exponent dominates. Because f has the form9 f = t ( λ ) t + t ( ω ) t λ + ω , (5.35)and λ and ω are sums of power of t , the asymptotic limits of f are determinedby coefficients of the dominant terms in the general case. When there are twoequally dominant terms, care is taken to include all equally dominant terms.It turns out that lim t →∞ f = p for ≤ Σ +0 ≤ p ) for − ≤ Σ +0 ≤ (5.36)and lim t → f = p for ≤ Σ +0 ≤
10 for − ≤ Σ +0 ≤ . (5.37)In some special cases, the coefficient of a dominant term can become zero.Then we find the next dominant term. These special cases are: r = 0, whichkills the t p term in λ , r = (cid:113) − ω k Σ +0 , which kills the constant term in ω , and k = 0, which kills the t p term. The case k = 0 has been dealt with in Chapter4. Figure 5.1 shows the exponents in λ , ω and f .Along r = 0, lim t →∞ f = 2(1 + p ) for all − ≤ Σ +0 ≤ . (5.38)Along r = (cid:113) − ω k Σ +0 , lim t → f = p for 0 ≤ Σ +0 ≤ p for − ≤ Σ +0 ≤ ω = 0 = r , f is equal to 4.In summary, f has a spike at r = 0 at late times for < Σ +0 ≤
1, a spike at r = 0 at early times for Σ +0 = − ω = 0, and a spike at r = (cid:113) − ω k Σ +0 (cid:54) = 0 atearly times for − < Σ +0 < ω k >
0, or 0 < Σ +0 < , ω k < . Combining the effects of f and l , we see that (Σ + , Σ − ) has a spike at r = 0 atlate times for 0 < Σ +0 ≤
1, a spike at r = 0 at early times for − < Σ +0 < r = (cid:113) − ω k Σ +0 (cid:54) = 0 at early times for − < Σ +0 < ω k >
0, or0 < Σ +0 < , ω k <
0. Later we will see that the spike caused by l is not real.0 (a) Σ +0 = − +0 = − . +0 = 0 Figure 5.2: State space orbits projected on the (Σ + , Σ − ) plane for k = 1,Σ +0 = − , − . , ω = 1. All orbits end up at the same point indicatedby an asterisk. ( t → ∞ ) For − < Σ +0 ≤
0, we do not have a spike.(Σ + , Σ − ) → (cid:32) − − Σ +0 − +0 ) , − √ − Σ +0 )2(7 − +0 ) (cid:33) . (5.40)See Figure 5.2.1 (a) Combine for all values of r (b) For r = (cid:113) − ω k Σ +0 (c) For r = 0 (d) For a large value of r Figure 5.3: State space orbits for k = 1, Σ +0 = 0 .
25 and ω = − < Σ +0 < .
5, we have a spike at r = 0.(Σ + , Σ − ) → (cid:16) − − Σ +0 − +0 ) , − √ − +0 )2(7 − +0 ) (cid:17) along r (cid:54) = 0 (cid:16) − − Σ +0 − +0 ) , − √ − Σ +0 )2(7 − +0 ) (cid:17) along r = 0 . (5.41)See Figure 5.3. Later we will see that this spike is not real.2 (a) Combine for all values of r (b) For r = (cid:113) − ω k Σ +0 (c) For r = 0 (d) For a large value of r Figure 5.4: State space orbits for k = 1, Σ +0 = 0 . ω = − +0 = 0 .
5, we have a spike at r = 0.(Σ + , Σ − ) → (cid:0) − , − √ (cid:1) along r (cid:54) = 0 (cid:0) − , − √ (cid:1) along r = 0 . (5.42)See Figure 5.4. Later we will see that this spike is not real.3 (a) Combine for all values of r (b) For r = (cid:113) − ω k Σ +0 (c) For r = 0 (d) For a large value of r Figure 5.5: State space orbits for k = 1, Σ +0 = 0 .
75 and ω = − . < Σ +0 ≤
1, we have a spike at r = 0(Σ + , Σ − ) → ( − +0 +0 ) , − √ +0 )2(5+2Σ +0 ) ) along r (cid:54) = 0( − − Σ +0 − +0 ) , − √ − Σ +0 )2(7 − +0 ) ) along r = 0 . (5.43)See Figures 5.5 and 5.6.4 (a) Combine for all values of r (b) For r = (cid:113) − ω k Σ +0 (c) For r = 0 (d) For a large value of r Figure 5.6: State space orbits for k = 1, Σ +0 = 1 and ω = − ( t → For Σ +0 = −
1, we have a spike along r = 0.(Σ + , Σ − ) → ( , √ ) along r = 0( , − √ ) along r (cid:54) = 0 . (5.44)See Figure 5.7. Later we will see that this spike is not real.5 (a) Combine for all values of r (b) For r = (cid:113) − ω k Σ +0 (c) For r = 0 (d) For a large value of r Figure 5.7: State space orbits for k = 1, Σ +0 = − ω = 1.6 (a) Combine for all values of r (b) For r = (cid:113) − ω k Σ +0 (c) For r = 0 (d) For a large value of r Figure 5.8: State space orbits for k = 1, Σ +0 = − . ω = 1.For − < Σ +0 <
0, we have a spike along r = 0, and if ω k >
0, anotherspike along r = (cid:113) − ω k Σ +0 .(Σ + , Σ − ) → (cid:16) − Σ +0 , − √ +0 (cid:17) along r = 0 (cid:16) − Σ +0 , √ +0 (cid:17) along r (cid:54) = 0 , (cid:113) − ω k Σ +0 (cid:16) − +0 +0 ) , − √ +0 )2(5+2Σ +0 ) (cid:17) along r = (cid:113) − ω k Σ +0 . (5.45)See Figure 5.8. Later we will see that the spike along r = 0 is not real.7 (a) Combine for all values of r (b) For r = (cid:113) − ω k Σ +0 (c) For r = 0 (d) For a large value of r Figure 5.9: State space orbits for k = 1, Σ +0 = 0 .
25 and ω = − < Σ +0 < .
5, we have a spike at r = (cid:113) − ω k Σ +0 if ω k < + , Σ − ) → (cid:16) − Σ +0 , − √ +0 (cid:17) along r (cid:54) = (cid:113) − ω k Σ +0 (cid:16) − − +0 − +0 ) , − √ − +0 )2(5 − +0 ) (cid:17) along r = (cid:113) − ω k Σ +0 . (5.46)See Figure 5.9.8 (a) Σ +0 = 0 . +0 = 0 . +0 = 1 Figure 5.10: State space orbits for k = 1 and ω = − . ≤ Σ +0 ≤
1, there are no spikes.(Σ + , Σ − ) → (cid:32) − − +0 − +0 ) , − √ − +0 )2(5 − +0 ) (cid:33) . (5.47)See Figure 5.10. Σ +0 = 0 When Σ +0 = 0 equation (5.33) simplifies toΣ + = − f f , Σ − = √ l − f − )2 + f (5.48)where f = t (ln F ) t , l = t (ln λ ) t .Now, for ω = 0, lim t → f = 43 for all r. (5.49)lim t →∞ f = 83 for all r. (5.50)9For ω (cid:54) = 0, lim t → f = 0 for all r. (5.51)lim t →∞ f = 83 for all r. (5.52)For all values of ω and t , we have l = .So after using these values, we get for ω = 0,lim t → (Σ + , Σ − ) = ( − / , − √ /
5) for all r. (5.53)lim t →∞ (Σ + , Σ − ) = ( − / , − √ /
7) for all r. (5.54)For ω (cid:54) = 0, lim t → (Σ + , Σ − ) = (0 ,
0) for all r. (5.55)lim t →∞ (Σ + , Σ − ) = ( − / , − √ /
7) for all r. (5.56)See Figure 5.11. While there are no permanent spikes, we see that in Figures5.12 and 5.13 that there are transient spikes . More analysis on transientspikes will be done in Chapter 7. Spike that form during transition for a short interval of time (a) ω (cid:54) = 0 (b) ω = 0(c) for all ω Figure 5.11: State space orbits for k = 1 and Σ +0 = 0.1 (a) Evolution along differentworldlines (b) Snapshots at different times(c) Figure 5.12: Plots of Σ + for k = 1, ω = 1 and Σ +0 = 0, showing a transientspike along r = 0.2 (a) Evolution along differentworldlines (b) Snapshots at different times(c) Figure 5.13: Plots of Σ + for k = 1, ω = 0 and Σ +0 = 0, showing a transientspike along r = 0.3 We define the coordinate radius of spike (or inhomogeneities) in (4.53), whichgives coordinate radius = (cid:115) (1 − p )( k t − p − ω ) − kt − p ) (1 − p )( k Σ +0 − t p ) . (5.57)For Σ +0 ≥ .
5, the coordinate radius tends to zero as t → ∞ , indicating theformation of permanent spike along r = 0 at late times. To see whether the spikes are real or merely a coordinate effects at r = 0 forthe intervals − ≤ Σ +0 < < Σ +0 ≤ . t = 0 . k = 1, ω = 1 and Σ +0 = − in Figure 5.14. Compare with Figure 5.8. This shows that the spike along r = (cid:113) − ω k Σ +0 is real, while the spike along r = 0 is a coordinate effect. Thisstrongly suggests that the spike along r = 0 (caused by l ) is a false spike forthe intervals − ≤ Σ +0 < k = 1, ω = − +0 = . We see that curve is not narrowing as time increases. i.e it isnot a spike. Compare with Figures 5.3 − r = 0 is not real. This strongly suggests that the spike along r = 0 (againcaused by l ) is a false spike for the interval 0 < Σ +0 ≤ . l is a false spikes,the Weyl scalar plots are strong evidence that the spike is a false spike. We have generated a solution that has permanent spikes that form at earlyand late times. We have done this by applying the Stephani transformation4 (a) CC (b) CCs(c) CCC (d) CCCs
Figure 5.14: Weyl scalars when t = 0 . k = 1, ω = 1 and Σ +0 = − .Compare with Figure 5.8. This shows that the spike along r = (cid:113) − ω k Σ +0 is real,while the spike along r = 0 is a coordinate effect for the interval − ≤ Σ +0 < (a) CC (b) CCs(c) CCC (d) CCCs Figure 5.15: Weyl scalars at late times, for k = 1, ω = − +0 = . Wesee that curve is not narrowing as time increases. i.e it is not a spike. Comparewith Figures 5.3 − r = 0 is not real for theinterval 0 < Σ +0 ≤ . k (cid:54) = 0 by using therotational KVF. The generated solution is cylindrically symmetric, and has areal spike along its rotational axis for the case < Σ +0 ≤ ω (cid:54) = 0, and a real spike along the surface r = (cid:113) − ω k Σ +0 at early times for thecase − < Σ +0 < kω > < Σ +0 < , kω < +0 = 0. In fact transientspikes also form in other cases. We will carry out intermediate times analysisin Chapter 7.Compared with the k = 0 case, the k (cid:54) = 0 case presents a variety of phe-nomena — false spikes, transient spikes, and a second spike along r = (cid:113) − ω k Σ +0 . hapter 6A heuristic for permanent spikes Is there a quick way to determine whether a Geroch/Stephani transformationsolution has permanent spike? If so, whether the permanent spike forms atearly or late times? We shall develop a heuristic to answer these two questions.For our experience in Chapter 4 and Chapter 5, we observed that theexpression f = t (ln F ) t = t ( λ ) t + t ( ω ) t λ + ω has discontinuous limit if and only if a permanent spike forms.In Chapter 4, we had, for − < Σ +0 ≤
1, discontinuous limitlim t →∞ f = r = 04 p along r (cid:54) = 0 , (6.1)which corresponds to a late-time permanent spike at the cylindrical axis, r = 0,Similarly, in Chapter 5, we had, for ≤ Σ +0 ≤
1, discontinuous limit.lim t →∞ f = p ) along r = 04 p along r (cid:54) = 0 . (6.2)In Chapter 5, we had, in addition, the following discontinuous limits. For − < Σ +0 < ω k > t → f = p along r = (cid:113) − ω k Σ +0 r (cid:54) = (cid:113) − ω k Σ +0 . (6.3)8For 0 < Σ +0 < and ω k < t → f = p along r = (cid:113) − ω k Σ +0 r (cid:54) = (cid:113) − ω k Σ +0 . (6.4)These correspond to an early-time permanent spike on the cylindrical shell r = (cid:113) − ω k Σ +0 .It is not necessary that we always have a discontinuous limits. If we see inthe Chapter 4 we have a continuous limit at r = 0 at early times,lim t → f = r = 00 along r (cid:54) = 0 . (6.5)and if we see in the Chapter 5, we also have the continuous limits for − ≤ Σ +0 ≤ ω k < t →∞ f = p ) along r = 02(1 + p ) along r (cid:54) = 0 . (6.6)In (6.1), the dominant term in the limit is contributed by λ = r t p , unless its coefficient r is zero.In (6.2), the dominant term is again contributed by the r t p term in λ ,unless the coefficient r is zero.In (6.3) and (6.4) the dominant term is contributed by the time-independentterm in ω , unless the term is zero.For another example, consider the OT G spike solution ([35],case n = n = 0). These we have, for | w | <
1, discontinuous limitslim τ →∞ f = −| w | + 1 along z = 00 along z (cid:54) = 0 , which corresponds to an early-time permanent spike on the plane z = 0. Thedominant term in the limit is contributed by the time-independent ω = kz, f = − (ln F ) τ in the time variable τ . k is a nonzero constant, unless z = 0.In all the above examples, permanent spike occur because f has a discon-tinuous limit, which in term is due to the fact that the dominant term has aspatially dependent coefficient that can become zero along certain worldlines.This gives a heuristic to quickly determine whether a given seed solutionleads to a generated solution with permanent spikes. The steps are:1: Compute λ , ω and f .2: Find the dominant term (for each asymptotic regime) and look at itscoefficient.3: If the coefficient vanishes along a certain worldline, then expect apermanent spike to form along the worldline.This also explains why the rotational KVF can leads to permanent spike atthe cylindrical axis – its length vanishes at the cylindrical axis. TranslationalKVFs, whose length is non-vanishing every where in space, do not have thismechanism. hapter 7Transient spike and otherinhomogeneous structures Figure 5.12 and 5.13 exhibit some interesting features of Σ + . Σ + transitionsfrom one equilibrium state to the next, at certain transition time that is spa-tially dependent. Each equlibrium state is coordinate independent. Recallfrom (5.33) that the dynamics of Σ + is solely due to the dynamics of f . It istherefore important to take a closer look at f . In this chapter, we shall employa method of analysis that has not been used to analyse spike solutions before. f Recall that F = λ + ω and λ = r t p + k t p and ω = k p t p + k Σ +0 r + ω .Observe that λ and ω are sums of powers of t . There are four different powers,so we group them into 4 terms on the basis of the power of t : T = r t p , T = k t p , T = 2 k p t p , T = k Σ +0 r + ω . (7.1)Figure 7.1 plots the power of t of each term in (7.1) against the parameterΣ +0 . In general, the four powers are distinct, except for 3 special values ofΣ +0 . For Σ +0 = −
1, there are two distinct powers; for Σ +0 = 0, three distinctpowers; and for Σ +0 = , two distinct powers. The term with the largest powerof t dominates at late times; the term with smallest power of t dominates at1Figure 7.1: Power of t of the terms in (7.1) against Σ +0 .early times; and the terms with intermediate power of t may or may notdominate for a finite time interval, depending on how big their coefficient is.Expressed in terms of T , T , T , T , f = 2( T + T )(2 p T + 2 p T ) + 2( T + T )(1 + p ) T ( T + T ) + ( T + T ) . (7.2)Observe that f ≈ p when T dominates4 p when T dominates2(1 + p ) when T dominates0 when T dominates . (7.3)That is, f is approximately twice the value of the power of the dominant term.Furthermore the powers depend only on the parameter Σ +0 . Its independenceof coordinates gives the graph of f a cascading appearance. An equilibriumstate corresponds to a dominant term. Therefore, there are up to 4 distinctequilibrium states for general Σ +0 ; 3 for Σ +0 = 0 and 2 for Σ +0 = − +0 = 1 /
2. The value of f at successive equilibrium states is strictly increasingin time. Among the four values, 4 p is negative for 0 . < Σ +0 ≤
1, with a2minmum value of − at Σ +0 = 1, which is still greater that −
2, so the Hubblescalar H is positive at each equilibrium state. But we will see later that f canbecome less than − T = T for t yields the transition time t = (cid:18) k r (cid:19) . (7.4)Comparing the transition times will determine how many transitions an ob-server with fixed r undergoes. The coefficients of T and T have spatial de-pendence. They can even vanish for certain worldline ( r = 0 for T , and r = (cid:113) − ω k Σ +0 for T , provided that ω k Σ +0 ≤ t .The transition time between two dominant terms can be regarded as roughlythe boundary between the two corresponding equilibrium states. We say“roughly” because the transition is a smooth, continuous process, so thereis no sharp boundary. If a transition time has spatial dependence, it also givesthe spatial location of the boundary at a fixed time. The spacetime is parti-tioned into regions of equlibrium states, separated by transition times. Whenviewed at a fixed time, we can regard space as being partitioned into cells ofequilibrium states, separated by walls (around which spatial gradient is large).If two walls are near each other, we see a narrow cell. The neighbourhood ofthe narrow cell shall be called a spike if certain additional conditions are met.We will discuss these conditions later in Section 7.2.3Figure 7.2: Transition time t (1&4)(2&3) as a function of r for Σ +0 = − k = 10and ω = 10 . t (1&4)(2&3) has a global minimum at r = (cid:113) ω kk +1 = 1. Σ +0 = − For Σ +0 = −
1, we have T = r , T = k t , T = kt , T = − kr + ω . There are only two distinct powers of t , with T and T dominating at earlytimes, and T and T dominating at late times. This conclusion can also bearrived at by inspecting Figure 7.1. We denote the sequence of dominantequilibrium states as T & T −→ T & T . (7.5)Solving the equation T + T = T + T for t yields the transition time t (1&4)(2&3) = (cid:18) ( ω − kr ) + r k ( k + 1) (cid:19) . (7.6)We now analyse the behaviour of t (1&4)(2&3) as a function of r . Observethat lim r →∞ t (1&4)(2&3) = ∞ . If ω k >
0, then t (1&4)(2&3) has a global minimum at4Figure 7.3: Transition time t (1&4)(2&3) as a function of r for Σ +0 = − k = 10and ω = − . t (1&4)(2&3) has a global minimum at r = 0.Figure 7.4: f against ln t and r for Σ +0 = − k = 10 and ω = 10 .
1. Thetransition time has a global minimum at r = 1.5Figure 7.5: f against ln t and r for Σ +0 = − k = 10 and ω = − .
1. Thetransition time has a global minimum at r = 0. r = (cid:113) ω kk +1 , otherwise it has a global minimum at r = 0. For example, if wehave Σ +0 = − k = 10 and ω = 10 .
1, then t (1&4)(2&3) has a global minimumat r = (cid:113) ω kk +1 = 1 (see Figure 7.2). But if we have Σ +0 = − k = 10 and ω = − .
1, then t (1&4)(2&3) has a global minimum at r = 0 (see Figure 7.3).We plot f against ln t and r for these examples in Figures 7.4 and 7.5.Note that, we do not have spikes in this case, even through f looks spikyaround r = 1 during transition in Figure 7.4. But if ω = 0 then we havepermanent spike at early time (see Section 5.3). − < Σ +0 < For the case − < Σ +0 <
0, Figure 7.1 gives the ordering T , T , T , T ,in increasing power of t . We have up to 4 distinct equilibrium states, andalong general worldlines there are 4 possible sequences of dominant equilibriumstates, which we shall refer to as scenarios:1. T −→ T −→ T −→ T T −→ T −→ T T −→ T −→ T T −→ T .There are two special worldlines. The first one is r = 0, where T vanishes.The possible scenarios along this worldline are:1. T −→ T −→ T T −→ T ,which are qualitatively the same as scenarios 2 and 4 above. The second specialworldline is r = (cid:113) − ω k Σ +0 , where T vanishes, giving an early-time permanentspike. The possible scenarios along this worldline are:1. T −→ T −→ T T −→ T .The two special worldlines coincide if ω = 0. In this case the only possiblescenarios along this worldline is T −→ T . We now introduce a useful diagram. From (7.1), we see that the logarithmof the square of each term is a linear function of ln t . Figure 7.6 shows aqualitative plot of the log of each term squared against ln t , for the scenario T −→ T −→ T −→ T . It is clear from the diagram that the transition times t = (cid:18) | k Σ +0 r + ω | r (cid:19) p , t = (cid:18) k r (cid:19) , t = (cid:18) | k | (2 − Σ +0 )3 (cid:19) p (7.7)must satisfy the condition t < t < t . (7.8)7Figure 7.6: Qualitative plot of the log of each term squared against ln t , show-ing 4 dominant equilibrium states, for any value of Σ +0 satisfying − < Σ +0 < t < t implies | k Σ +0 r + ω | < (cid:18) | k | p r p (cid:19) , (7.9)which gives one or more intervals of r . t < t gives an upper bound on r : r < | k | (cid:18) | k | (2 − Σ +0 ) (cid:19) Σ+02 p . (7.10)So the condition (7.8) restricts r to one or more intervals. As a concreteexample, take Σ +0 = − . k = 10 and ω = 5. (7.9) can be solved numericallyto give the intervals0 . < r < . . < r. (7.11)Note that r = 1 is the second special worldline, so it must be excluded fromthis scenario. (7.10) gives r < . . < r < , < r < . . < r < . . (7.12)8 (a) For 0 ≤ r ≤
250 (b) For 0 . ≤ r ≤ . Figure 7.7: Blue line is the plot of t − t and red line is t − t , whenΣ +0 = − . k = 10, ω = 5. The blue line is positive for a small intervalaround r = 1 and for r > . r < . f against ln t and r on a small interval around r = 1in Figure 7.8, showing 4 distinct states along r (cid:54) = 1. Along r = 1, thereis a permanent spike at early times. We plot f against ln t and r for theinterval 100 < r <
250 in Figure 7.9, showing 2 visible distinct states becausethe transition times are too close together (see Figure 7.10). So if transitiontimes are too close together, we see fewer visible distinct state than the actualnumber of states predicted by the scenario.What happens in other intervals of r ? From (7.11), we know that t becomes greater than t for values of r just beyond the boundaries. From thediagram in Figure 7.6, this happens if the graph of ln T becomes too low, asshown in Figure 7.11. Now, the diagram in Figure 7.11 shows the scenario T −→ T −→ T , (7.13)with transition times t = (cid:18) | k Σ +0 r + ω | k (cid:19) p t = (cid:18) | k | (2 − Σ +0 )3 (cid:19) p . (7.14)9Figure 7.8: f against ln t and r for Σ +0 = − . k = 10 and ω = 5 for theinterval 0 . < r < . r (cid:54) = 1.Figure 7.9: f against ln t and r for Σ +0 = − . k = 10 and ω = 5 for theinterval 100 < r < t (green), t (blue) and t (red)against r for Σ +0 = − . k = 10 and ω = 5 for the interval 100 < r < t < t < t . (7.15)The condition t < t is equivalent to t < t , so it gives (7.9) with theopposite inequality direction: | k Σ +0 r + ω | > (cid:18) | k | p r p (cid:19) . (7.16) t < t implies | k Σ +0 r + ω | < | k | p (cid:18) − Σ +0 (cid:19) p p , (7.17)which gives rise to one interval of r . Together, the condition restricts r toone or more intervals. Continuing with the same example, (7.16) gives theintervals r < . . < r < . , (7.18)1Figure 7.11: Qualitative plot of the log of each term squared against ln t ,showing 3 dominant equilibrium states, for any value of Σ +0 satisfying − < Σ +0 < r < . ≤ r < . . < r < . . (7.19)See Figure 7.12. We plot f against ln t and r on these intervals showing 3distinct states in Figures 7.13 and 7.14. Figure 7.13 shows f against ln t and r for the interval 0 ≤ r < . r greaterthan 0 . f against ln t and r for the interval 1 < r < r which fade away to two visible states as the transition times becomecloser together as r increases. Figure 7.15 shows the log of transition times t (blue), t (green) and t (red) against r for the interval 1 < r < t becomes closer to t as r increases.To complete the example, we now look at what happens beyond r =2 (a) For 0 ≤ r ≤
350 (b) For 0 ≤ r ≤ Figure 7.12: Blue line is the plot of t − t and red line is t − t , whenΣ +0 = − . k = 10, ω = 5. The blue line is positive for a small interval0 ≤ r < . . < r < . r < . f against ln t and r for Σ +0 = − . k = 10 and ω = 5 for theinterval 0 ≤ r < . r greater than 0 . f against ln t and r for Σ +0 = − . k = 10 and ω = 5 for theinterval 1 < r < r which fade away totwo visible states as the transition times become closer together as r increases.4Figure 7.15: The log of transition times t (blue), t (green) and t (red)against r for Σ +0 = − . k = 10 and ω = 5 for the interval 1 < r < t becomes closer to t as r increases.240 . t becomes larger than t . From the diagram in Figure7.6, this happens if the graph of ln T becomes too low, as shown in Figure7.16. Now, the diagram in Figure 7.16 shows the scenario T −→ T −→ T , (7.20)with transition times t = (cid:18) | k Σ +0 r + ω | r (cid:19) p , t = (cid:18) r (2 − Σ +0 )3 | k | (cid:19) p . (7.21)They must satisfy the conditions t < t < t . (7.22)The condition t < t is equivalent to t < t , so it gives (7.10) with theopposite inequality direction, a simple lower bound r > | k | (cid:18) | k | (2 − Σ +0 ) (cid:19) Σ+02 p . (7.23)5Figure 7.16: Qualitative plot of the log of each term squared against ln t ,showing 3 dominant equilibrium states, for any value of Σ +0 satisfying − < Σ +0 < t < t implies | k Σ +0 r + ω | < r p p (cid:18) − Σ +0 | k | (cid:19) p p , (7.24)which restricts r to one or more intervals. Continuing with the same example,(7.23) gives the interval r > . . < r < . r > . . (7.25)Together, the scenario occurs for r > . f against ln t and r showing 3 distinct states with a lower bound on r in Figure7.18.This completes the example. We can summarise the different scenarios thatoccur in this example in another useful diagram, where we plot the transitiontimes of each scenario, and label the dominant term in each cell. See Figure7.19.The fourth scenario is the 2-state sequence T −→ T (7.26)6 (a) For 0 . ≤ r < . ≤ r ≤ Figure 7.17: Blue line is the plot of t − t and red line is t − t whenΣ +0 = − . k = 10, ω = 5. The blue line is positive for a small interval0 . ≤ r < . r > . r > . f against ln t and r for Σ +0 = − . k = 10 and ω = 5 for theinterval 0 ≤ r < r > . r < . +0 = − . k = 10 and ω = 5 showing the different scenarios along each fixed r . Eachcell is labelled with the index of the dominant term.with transition time t = (cid:18) | k Σ +0 r + ω | (2 − Σ +0 )3 k (cid:19) p , (7.27)which is required to satisfy the conditions t < t < t . (7.28)Figure 7.20 shows a qualitative plot of the log of each term squared againstln t , showing 2 dominant equilibrium states for value of Σ +0 satisfying − < Σ +0 <
0. It is clear from the figure that, we have 2 distinct equilibrium statesif and only if the transition times t , t and t , satisfy the condition (7.28). t < t is equivalent to t < t , so it gives (7.24) with opposite inequalitydirection: | k Σ +0 r + ω | > r p p (cid:18) (2 − Σ +0 )3 | k | (cid:19) p p . (7.29) t < t is equivalent to t < t , so it gives (7.17) with the opposite inequal-ity: | k Σ +0 r + ω | > | k | p (cid:18) − Σ +0 (cid:19) p p . (7.30)In the previous example, the condition (7.28) is not satisfied anywhere.Consider a second example. Take Σ +0 = − . k = 0 . ω = 500. (7.29)8Figure 7.20: Qualitative plot of the log of each term squared against ln t ,showing 2 dominant equilibrium states, for any value of Σ +0 satisfying − < Σ +0 < ≤ r < . ≤ r < . r > . . (7.31)Together, they give the interval0 ≤ r < . . (7.32)See Figure 7.21. We plot f against ln t and r for the values Σ +0 = − . k = 0 . ω = 500 on these intervals showing 2 distinct states in Figure7.22. Beyond r = 10 . T −→ T −→ T .9 (a) For 70 . < r < . ≤ r ≤ Figure 7.21: Blue line is the plot of t − t and red line is t − t , whenΣ +0 = − . k = 0 . ω = 500. The blue line is positive for a small interval0 ≤ r < . r except 70 . < r < . f against ln t and r for Σ +0 = − . k = 0 . ω = 500,showing 2 distinct states for the interval 0 ≤ r < Σ +0 = 0 For Σ +0 = 0, we have T = r t , T = k t , T = 3 k t , T = ω . There are only three distinct powers of t , with T dominating at early times, T dominating at late times, and T and T possibly at intermediate times.The first scenario is the 3-state sequence. T −→ T & T −→ T (7.33)with transition times t = (cid:18) ω r + k (cid:19) , t (1&2)3 = (cid:18) r + k )9 k (cid:19) , (7.34)which are required to satisfy the condition t < t (1&2)3 . (7.35)The condition gives a lower bound on rr > (cid:18) | kω | − k (cid:19) . (7.36)If the lower bound is positive, then for r less than this we have the secondscenario, the 2-state sequence T −→ T (7.37)with transition time t = (cid:12)(cid:12)(cid:12)(cid:12) ω k (cid:12)(cid:12)(cid:12)(cid:12) . (7.38)For example, given k = 0 . ω = 2, for r > . r < . f against ln t and r on the interval 0 ≤ r ≤
10 showing both scenarios inFigure 7.24.1Figure 7.23: Blue line is the plot of t (1&2)3 and red line is t , when k = 0 . ω = 2. Figure shows that for r > . t < t (1&2)3 and for r < . t > t (1&2)3 .Figure 7.24: f against ln t and r for k = 0 . ω = 2 for the interval0 ≤ r <
10, showing scenario (7.33) for r > . r < . < Σ +0 < . The case 0 < Σ +0 < . − < Σ +0 < T , T , T , T , in increasing powerof t . The possible scenarios along general worldlines are1. T −→ T −→ T −→ T T −→ T −→ T T −→ T −→ T T −→ T .There are two special worldlines. The first one is r = 0, where T vanishes. Thepossible scenarios along this worldline are the scenarios 2 and 4 above. Thesecond special worldline is r = (cid:113) − ω k Σ +0 , where T vanishes, giving an early-timepermanent spike. The possible scenarios along this worldline are:1. T −→ T −→ T T −→ T .The two special worldlines coincide if ω = 0. In this case the only possiblescenario along this worldline is T −→ T . Figure 7.25 shows a qualitative plot of the log of each term squared againstln t , for the scenario T −→ T −→ T −→ T . (7.39)The transition times t = (cid:18) | k Σ +0 r + ω | k (cid:19) p , t = (cid:18) k r (cid:19) , t = (cid:18) r (2 − Σ +0 )3 | k | (cid:19) p (7.40)3must satisfy the condition t < t < t . (7.41) t < t implies (7.9): | k Σ +0 r + ω | < (cid:18) | k | p r p (cid:19) , (7.42)while t < t implies (7.23): r > | k | (cid:18) | k | (2 − Σ +0 ) (cid:19) Σ+02 p . (7.43)As a concrete example, take Σ +0 = 0 . k = 15 and ω = 6. (7.42) gives r < . r > . . (7.45)Together they give the interval7 . < r < . . (7.46)See Figure 7.26. We plot f against ln t and r for the interval 7 < r <
11 inFigure 7.27, showing 2 visible distinct states because the transition times aretoo close together (see Figure 7.28).From (7.45), we know that t becomes greater than t for values of r justbelow 7 . .
25, this happens if the graph ofln T becomes too low, as shown in Figure 7.29. This gives the scenario T −→ T −→ T , (7.47)with transition times t = (cid:18) | k Σ +0 r + ω | k (cid:19) p , t = (cid:18) | k | (2 − Σ +0 )3 (cid:19) p . (7.48)They must satisfy the condition t < t < t . (7.49)4Figure 7.25: Qualitative plot of the log of each term squared against ln t ,showing 4 dominant equilibrium states, for any value of Σ +0 satisfying 0 < Σ +0 < . (a) For 0 ≤ r ≤
50 (b) For 6 ≤ r ≤ Figure 7.26: Blue line is the plot of t − t and red line is t − t , whenΣ +0 = 0 . k = 15, ω = 6. The blue line is positive for a interval r < . r > . . 11, showing 2 visible distinct states instead of 4, because thetransition times are too close together.Figure 7.28: The log of transition times t (green), t (red) and t (blue)against r for Σ +0 = 0 . k = 15, ω = 6 for the interval 5 < r < 15, showingthat the transition times are close together.6 t < t implies (7.17): | k Σ +0 r + ω | < | k | p (cid:18) − Σ +0 (cid:19) p p , (7.50)while t < t implies (7.10): r < | k | (cid:18) | k | (2 − Σ +0 ) (cid:19) Σ+02 p . (7.51)Continuing with the same example, (7.50) gives r < . r < . . (7.53)Together they give the interval0 ≤ r < . . (7.54)See Figures 7.30. We plot f against ln t and r on the interval 0 ≤ r < t becomes greaterthan t for values of r just above 10 . . T becomes too low, as shown in Figure 7 . T −→ T −→ T , (7.55)with transition times t = (cid:18) | k Σ +0 r + ω | r (cid:19) p , t = (cid:18) r (2 − Σ +0 )3 | k | (cid:19) p . (7.56)They must satisfy the condition t < t < t . (7.57) t < t implies (7.16): | k Σ +0 r + ω | > (cid:18) | k | p r p (cid:19) (7.58)7Figure 7.29: Qualitative plot of the log of each term squared against ln t ,showing 3 dominant equilibrium states, for any value of Σ +0 satisfying − < Σ +0 < t − t and red line is t − t , whenΣ +0 = 0 . k = 15 and ω = 6. The blue line is positive for a small interval r < . r < . ≤ r < . f against ln t and r for Σ +0 = 0 . k = 15 and ω = 6 for theinterval 0 ≤ r < . r < t < t implies: | k Σ +0 r + ω | < r p p (cid:18) (2 − Σ +0 )3 | k | (cid:19) p p . (7.59)Continuing with the same example, (7.58) gives r > . , (7.60)while (7.59) gives r > . . (7.61)Together they give the interval r > . . (7.62)See Figures 7.33. We plot f against ln t and r showing 3 distinct states with alower bound on r in Figure 7.34. We summarise the scenarios in Figure 7.35.9Figure 7.32: Qualitative plot of the log of each term squared against ln t ,showing 3 dominant equilibrium states, for any value of Σ +0 satisfying 0 < Σ +0 < . t < t and red line is t < t whenΣ +0 = 0 . k = 15 and ω = 6. The blue line for a small interval r > . r > . r > . f against ln t and r for Σ +0 = 0 . k = 15 and ω = 6 , showing 3distinct states for r > . r < . +0 = 0 . k = 15 and ω = 6 showing the different scenarios along each fixed r . Eachcell is labelled with the index of the dominant term.1The fourth scenario is the 2-state sequence T −→ T (7.63)with transition times t = (cid:18) | k Σ +0 r + ω | (2 − Σ +0 )3 | k | (cid:19) p (7.64)which are required to satisfy the condition t < t < t . (7.65)Figure 7.36 shows a qualitative plot of the log of each term squared againstln t , showing 2 dominant equilibrium states for value of Σ +0 satisfying 0 < Σ +0 < . 5. It is clear from the figure that, we have 2 distinct equilibriumstates if and only if the transition times t , t and t , satisfy the condition(7.65). t < t implies (7.30): | k Σ +0 r + ω | > | k | p (cid:18) − Σ +0 (cid:19) p p (7.66)while t < t implies (7.29): | k Σ +0 r + ω | > r p p (cid:18) − Σ +0 | k | (cid:19) p p . (7.67)Consider a second example. Take Σ +0 = 0 . k = 10 and ω = 200, (7.66)implies r > . r < . . < r < . f against ln t and r showing 2 distinct states with an upper bound on r in Figure 7.38. Beyond r = 6 . T −→ T −→ T . For 0 ≤ r < . T −→ T −→ T .2Figure 7.36: Qualitative plot of the log of each term squared against ln t ,showing 3 dominant equilibrium states, for any value of Σ +0 satisfying 0 < Σ +0 < . t − t and red line is t − t , whenΣ +0 = 0 . k = 10 and ω = 200. The blue line is positive for a interval r > . r < . . < r < . f against ln t and r for Σ +0 = 0 . k = 10 and ω = 200 for theinterval 0 ≤ r < 10, showing 2 distinct states. Σ +0 = 0 . For Σ +0 = , we have T = r t, T = k , T = 2 kt, T = kr ω . There are only two distinct powers of t , with T and T dominating at earlytimes, and T and T dominating at late times. The scenario is T & T −→ T & T . (7.68)Solving the equation T + T = T + T for t yields the transition time t (2&4)(1&3) = (cid:18) k r + 4 k + 4 kω r + 4 ω r + 4 k ) (cid:19) . (7.69)We now analyse the behaviour of t (2&4)(1&3) as a function of r . Observethat lim r →∞ t (2&4)(1&3) = | k | . t (2&4)(1&3) has one non-trivial critical point at r = (cid:113) k ( − ω + √ k + ω ) k , if k > − (cid:113) − k ( ω + √ k + ω ) k , if k < . (7.70)4Figure 7.39: Transition time t (2&4)(1&3) as a function of r for Σ +0 = 0 . k = 2and ω = 15.Figure 7.40: Transition time t (2&4)(1&3) as a function of r for Σ +0 = 0 . k = − ω = 15.5Figure 7.41: f against ln t and r for Σ +0 = 0 . k = 2 and ω = 15. Thetransition time has a local maximum at r = 2.Figure 7.42: f against ln t and r for Σ +0 = 0 . k = − ω = 15. Thetransition time has a local minimum at r = 4.6If ω k > 0, then t (2&4)(1&3) has a local maximum at the critical point; and if ω k < 0, then t (2&4)(1&3) has a local minimum at the critical point. For exam-ple, if we have Σ +0 = 0 . k = 2 and ω = 15, then at r = (cid:113) k ( − ω + √ k + ω ) k = 1,we have a local maximum (see Figure 7.39). But if we have Σ +0 = 0 . k = − ω = 15 . 1, then at r = (cid:113) − k ( ω + √ k + ω ) k = 4 we have a local minimum (seeFigure 7.40). We plot f against ln t and r for these examples in Figures 7.41and 7.42. Figure 7.42 shows an overshoot transition, which we will discuss inSection ??Note that, we do not have spikes in this case. . < Σ +0 ≤ The case 0 . < Σ +0 ≤ − < Σ +0 < < Σ +0 < . T , T , T , T , in increasing power of t . The possible scenarios along general worldlinesare1. T −→ T −→ T −→ T T −→ T −→ T T −→ T −→ T T −→ T .There are two special worldlines. The first one is r = 0, where T vanishes,giving a late-time permanent spike. This is different from the other cases, as T has the highest power only in this case. The possible scenarios along thisworldline are:1. T −→ T −→ T T −→ T r = (cid:113) − ω k Σ +0 , where T vanishes. The possiblescenario along this worldline are the scenarios 3 and 4 above. The two specialworldlines coincide if ω = 0. In this case the only possible scenario along thisworldline is T −→ T . Figure 7.43 shows a qualitative plot of the log of each term squared againstln t , for the scenario T −→ T −→ T −→ T . (7.71)Note the negative slope for ln T . The transition times t = (cid:18) | k Σ +0 r + ω | k (cid:19) p , t = (cid:18) | k Σ +0 r + ω | (2 − Σ +0 )3 | k | (cid:19) p , t = (cid:18) r (2 − Σ +0 )3 | k | (cid:19) p (7.72)must satisfy the condition t < t < t . (7.73) t < t implies (7.17): | k Σ +0 r + ω | > k p (cid:18) − Σ +0 (cid:19) p p . (7.74)while t < t implies (7.59): | k Σ +0 r + ω | < r p p (cid:18) − Σ +0 | k | (cid:19) p p . (7.75)As a concrete example, take Σ +0 = 0 . k = 0 . ω = 10 (7.74) gives r > while (7.75) gives r < . < r < . . (7.76)See Figures 7.44. We plot f against ln t and r for the interval 0 < r < . r , and3 distincts states for larger values of r . Along r = 0, we have a late-timepermanent spike.8Figure 7.43: Qualitative plot of the log of each term squared against ln t ,showing 4 dominant equilibrium states, for any value of Σ +0 satisfying 0 . < Σ +0 ≤ t − t and red line is t − t , forΣ +0 = 0 . k = 0 . ω = 10. The blue line is positive for all the valuesof r . The red line is positive for r < . (a) For 0 ≤ r ≤ . ≤ r ≤ . Figure 7.45: f against ln t and r for Σ +0 = 0 . k = 0 . ω = 10 for theinterval 0 < r < . 3, showing 4 distinct states. At r = 0, we have a permanentspike at late times.For values of r just above 0 . t becomes greater than t . From Figure7.43, this happens if the graph of ln T becomes too low, as shown in Figure7.46. This gives the scenario T −→ T −→ T , (7.77)with transition times t = (cid:18) | k Σ +0 r + ω | k (cid:19) p , t = (cid:18) | k Σ +0 r + ω | r (cid:19) / p . (7.78)They must satisfy the condition t < t < t . (7.79) t < t implies (7.9): | k Σ +0 r + ω | < (cid:18) | k | p r p (cid:19) , (7.80)while t < t implies (7.24): | k Σ +0 r + ω | < r p p (cid:18) − Σ +0 | k | (cid:19) p p , (7.81) The special worldline r = 0 is excluded. t ,showing 3 dominant equilibrium states, for any value of Σ +0 satisfying 0 . < Σ +0 ≤ r > r > . r > . . (7.82)See Figure 7.47. We plot f against ln t and r on the interval 0 ≤ r < t − t and blue line is t − t , forΣ +0 = 0 . k = 0 . ω = 10. The blue line is negative for a small interval r < . r . Together they give theinterval r > . f against ln t and r for Σ +0 = 0 . k = 0 . ω = 10 for theinterval 0 ≤ r < r > . T −→ T −→ T (7.83)with the transition times t = (cid:18) | k | (2 − Σ +0 )3 (cid:19) p , t = (cid:18) r (2 − Σ +0 )3 | k | (cid:19) p , (7.84)which are required to satisfy the condition t < t < t . (7.85)Figure 7.49 shows a qualitative plot of the log of each term squared againstln t , showing 3 dominant equilibrium states. t < t implies (7.17): | k Σ +0 r + ω | < | k | p (cid:18) − Σ +0 (cid:19) p p (7.86)while t < t implies (7.10): r < | k | (cid:18) | k | (2 − Σ +0 ) (cid:19) Σ+02 p . (7.87)For example, given Σ +0 = 0 . k = 250 and ω = 0 . 1, the first condition t < t implies r < . t < t implies r < . < r < . f against ln t and r showing 3 distinct states in Figure 7.51.03Figure 7.49: Qualitative plot of the log of each term squared against ln t ,showing 3 dominant equilibrium states, for any value of Σ +0 satisfying 0 < Σ +0 ≤ t − t and red line is t − t , forΣ +0 = 0 . k = 250 and ω = 0 . 1. The red line is positive for a interval0 ≤ r < . ≤ r < . ≤ r < . f against ln t and r for Σ +0 = 0 . k = 250 and ω = 0 . ≤ r < 10, showing 3 distinct states. From r greater than 9 . T −→ T (7.88)with the transition time t = (cid:18) k r (cid:19) , (7.89)which is required to satisfy the condition t < t < t . (7.90)Figure 7.52 shows a qualitative plot of the log of each term squared againstln t , showing 2 distinct equilibrium states. t < t implies (7.16): | k Σ +0 r + ω | > (cid:18) | k | p r p (cid:19) , (7.91)while t < t implies (7.23): r > | k | (cid:18) | k | (2 − Σ +0 ) (cid:19) Σ+02 p , (7.92)05Figure 7.52: Qualitative plot of the log of each term squared against ln t ,showing 2 dominant equilibrium states, for any value of Σ +0 satisfying 0 . < Σ +0 ≤ +0 = 0 . k = 2 and ω = − 19, the first condition t < t implies 3 . < r < . t < t implies r > . . < r < . f against ln t and r showing 2 distinct states in Figure7.54.06 (a) For 0 < r < 25 (b) For 2 . < r < . Figure 7.53: Red line is the plot of t − t and blue line is t − t , forΣ +0 = 0 . k = 2 and ω = − 19. The blue line is positive for small interval3 . < r < . r > . . < r < . f against ln t and r for Σ +0 = 0 . k = 2 and ω = − 19 for theinterval 0 ≤ r < 10, showing 2 distinct states for 3 . < r < . We now take a closer look at transient spikes. In section 7.1, we saw a numberof examples with transient and narrow inhomogeneity. See Figures 7.4, 7.24,7.31, 7.42 and 7.54. Do we want to call all these features transient spikes?What is the definition for transient spikes?To explore how to define transient spikes, we begin by looking at the def-inition of permanent spikes. A permanent spike is a feature in a region ofspacetime characterized by a discontinuous limit of the type f → L if r = r L if r (cid:54) = r (7.93)as t → ∞ (late-time) or as t → r = r , which we call the spike worldline. The examples inSection 7.1.6 all have a late-time permanent spike along r = 0. Moreover, thefinal transition time ( t or t ) tends to infinity as r tends to zero. In otherwords, the cell that contains r = 0 is increasingly narrow as t → ∞ . At whattime should a permanent spike begin to be called as such? It is not clear. Inthe early stage of its formation, the inhomogeneous structure is still ratherwide. As it becomes narrower at later times, we become more likely to call theinhomogeneous structure a permanent spike. At late enough time, everyonewould agree to call the structure a permanent spike. Therefore, while there isvagueness about when it starts, a permanent spike is easily identified by theasymptotic narrowing of the inhomogeneous structure.By analogy, we can attempt to characterise transient spikes as a featurein a region of spacetime where an inhomogeneous structure becomes narrowtemporarily. There is the vagueness about how narrow is considered narrow.The second issue is that this definition is too broad. It would include thefeatures in Figures 7.4, 7.42 and 7.54 as transient spikes.In the original context where transient spikes were first named, the world-lines in a small neighbourhood undergo a scenario that is different from those08undergone by worldlines further away. Adding this criterion rules out thefeature in Figure 7.4.The features in Figures 7.42 and 7.54 are overshoot transitions, whichshould be distinguished from transient spikes. We can add a criterion thattransient spike is not a single transition, but something that lasts longer. Inthe next section, we will discuss overshoot transition in details. An overshoottransition can occur inside a transient spike. We noted earlier in Section 7.1 that f has a cascading appearance. Despitethis, Figures 7.42 and 7.54 show that f can fluctuate wildly when it makes atransition between equilibrium states.Under what condition does this happen? If we examine f from (7.2): f = 2( T + T )(2 p T + 2 p T ) + 2( T + T )(1 + p ) T ( T + T ) + ( T + T ) , (7.94)we see that the magnitude of f becomes large if the denominator becomessmall due to cancellation. Among T , T , T , T , only T can become negative,so cancellation is only possible if T is negative. Recall from (7.1) that T = k Σ +0 r + ω , (7.95)So T is negative if and only if r < (cid:114) − ω k Σ +0 , ω k Σ +0 < . (7.96)Cancellation happens when T + T ≈ . (7.97)Its effect is most prominent when cancellation occurs during the T → T (7.98)09transition in a scenario. Heuristically, when T + T = (cid:15) , where | (cid:15) | is small,and suppose T and T are o ( (cid:15) ), then (7.94) implies f ≈ p ) (cid:15) T . (7.99)Then f becomes negative in the first stage of the transition (when (cid:15) < (cid:15) > T and T are much smaller than T and T whenthis happens. We therefore call such a transition an overshoot transition.The overshoot transition occurs on the interval0 ≤ r < (cid:114) − ω k Σ +0 . (7.100)In the exceptional case Σ +0 = 0, T is negative if and only if ω is negative.Additional restriction on the interval is provided by the condition that thetransition T → T occurs. This condition is broken if T and T becomeslarge enough that the scenario changes to T → T → T , T → T → T , or T → T & T → T .An overshoot transition can occur during a transient spike. For example,take Σ +0 = 0, k = 0 . ω = − 2. See Figures 7.55, 7.57 and 7.58. Anovershoot transition occurs on r (cid:46) 1, around ln t ≈ . r (cid:46) − (cid:46) ln t (cid:46) f against ln t and r for Σ +0 = 0, k = 0 . ω = − r (cid:46) 1, around ln t ≈ . r (cid:46) − (cid:46) ln t (cid:46) +0 = 0, k = 0 . ω = − r . Eachcell is labelled with the index of the dominant term. An overshoot transitionoccurs on r (cid:46) 1, around ln t ≈ . r (cid:46) − (cid:46) ln t (cid:46) f against r for Σ +0 = 0, k = 0 . ω = − 2, showing theovershoots occurring on r (cid:46) f against ln t along r = 0 for Σ +0 = 0, k = 0 . ω = − The main aim of this chapter is to explore and describe the transient dynamicsof f . For this, we introduced a new technique. We grouped F into fourterms T , T , T , T based on the power of t . We plotted their power againstthe parameter Σ +0 in Figure 7.1, and we get three specific values and threeintervals of Σ +0 that are analysed separately in subsections of Section 7.1. Theterm with largest and smallest power dominates at late times and early timesrespectively, and a permanent spike occurs where its coefficient vanishes. Theterm with intermediate power may dominate for a finite time, depending onthe size of their coefficient, and give rise to transient structures.When a term dominates, f is in an equilibrium state. The transition timebetween two equilibrium states is considered a boundary between the twostates. Viewed along a worldline, the observer undergoes a sequence of 2-4equilibrium states, which we call a scenario. Viewed at a fixed time, space isdivided into cells of equilibrium states, separated by walls of (spatially depen-dent) transition times. Viewed as a whole, the spacetime is divided into cellsseparated by transition times (see Figures 7.19, 7.35, 7.56). Such a picturereveals the extent of all transient and permanent structures.In Section 7.2, we revised the description of transient spikes. Our newanalysis shows that transient spikes occur on a spatial region rather than alonga single worldline. For an inhomogeneous structure to be called a transientspike, it must meet the following criteria: • Its cell is narrow, • Its worldlines undergo a transition different from neighbouring world-lines, • It lasts longer than a single transition.The last criterion distinguishes a transient spike from a new phenomenon calledovershoot transition, during which f overshoots when it transitions between13equlibrium states. The overshoots occur because the dominant terms involvedhave opposite signs. An overshoot transition can occur inside a transient spike(see Figure 7.55). hapter 8Conclusion The thesis is about the spiky solution with a major focus on finding late-timepermanent spike and analysing transient spike. The journey of the exact spikesolution started in 2008, when the OT G spike solution was discovered. Itsnon-OT G generalised solution was discovered in 2015 by using the Gerochtransformation and its stiff fluid generalised solution was found in 2016 byusing Stephani transformation, which is also used in this thesis.The above solutions produce spikes at early times and our first aim is tofind a solution that produces a late-time permanent spike. To achieve this, weapplied the Stephani transformation using the rotational KVF from the LRSJacobs solution. It generated a new cylindrically symmetric OT G solutionthat produces the desired late-time permanent spike along the axis of rotation.This is the first non-silent solution with a late-time permanent spike. It is alsothe first instance of a spike along a line (previous spikes found occur along aplane). Matter density is higher at the spike. The physical radius of the spiketurns out to be constant.Our second aim was to explore and analyse the k (cid:54) = 0 solutions, whichfeature a rich variety of structures that include a second spike along the cylin-drical shell r = (cid:113) − ω k Σ +0 , transient spikes and the newly discovered overshoottransitions. To achieve this, we introduced a new technique to analyse thedynamics of a key function, f . The analysis helped us revise the description15of transient spikes and describe the overshoot transition.To summarise, in this thesis we have • found the first non-silent solution with a late-time permanent spike. • found the first spike along a line. • introduced a new technique to analyse a key function, f . • revised the description of transient spikes. • discovered and described overshoot transitions.We conclude this thesis by commenting on future research. Firstly, the familyof exact solutions we found in this thesis make up only a set of measure zero inthe class of cylindrically symmetric solutions. How does a typical cylindiricallysymmetric solution evolve? To answer this question, it is necessary to conducta numerical study of the class of cylindrically symmetric solutions, like thenumerical study done for the class of non-OT G vacuum solutions [26].Secondly, we have used the rotational KVF of the LRS Jacobs solution.Exact solutions that admit a rotational KVF include the LRS Taub solution,the NUT (LRS Bianchi type VIII) solution, and the Taub-NUT (LRS Bianchitype IX) solution [45, page 198]. It would be interesting to see what spikysolutions are generated from these solutions.Thirdly, our exact solutions are OT G solutions. In principle, non-OT G solutions and G solutions can be generated from a rotational KVF. Arethere simple enough seed solutions that generate spiky solutions with suchisometries? ppendicesppendix AKinematic variables Let u be unit timelike vector field. The covariant derivative u a ; b can be de-compose into irreducible parts according to [41, Section 1.1.3] u a ; b = σ ab + ω ab + 13 Θ h ab − ˙ u a u b , (A.1)where σ ab is the rate of shear tensor and is symmetric and trace-free, ω ab is therate of vorticity vector and is antisymmetric. Also u a σ ab = 0 = u a ω ab . ˙ u a theacceleration vector, and the scalar Θ is the rate of expansion scalar. It followsthat σ ab = u ( a ; b ) − 13 Θ h ab + ˙ u ( a u b ) , (A.2) ω ab = u [ a ; b ] + ˙ u [ a u b ] , (A.3)˙ u a = u a ; b u b , (A.4)Θ = u a ; a . (A.5)The expansion tensor Θ ab = σ ab + Θ h ab . In a cosmological context we shallreplace Θ by the Hubble Sclar H defined as H = Θ h ab . Below are the18kinematics variables in Iwasawa frame. For more details see [43, Appendix A]. H = − 13 1 N ∂ ( b + b + b ) (A.6)Θ = − N ∂ b (A.7)Θ = − N ∂ b (A.8)Θ = − N ∂ b (A.9) σ = 12 1 N e b − b ∂ n (A.10) σ = 12 1 N e b − b ∂ n (A.11) σ = 12 1 N e b − b ( − n ∂ n + ∂ n ) (A.12)˙ u = − e b ∂ ln | N | (A.13)˙ u = − e b [ − n ∂ ln | N | + ∂ ln | N | ] (A.14)˙ u = − e b [( n n − n ) ∂ ln | N | − n ∂ ln | N | + ∂ ln | N | ] (A.15) n = e b + b − b [ n ∂ n − n ∂ n − ∂ n + ∂ n ] (A.16) n = e b + b − b ∂ n (A.17) n = 0 (A.18) n = e b [( n n − n ) ∂ ( b − b ) + n ∂ n − n ∂ n + ∂ n − ∂ n − n ∂ ( b − b ) + ∂ ( b − b )] (A.19) n = e b ∂ ( b − b ) (A.20) n = e b (cid:2) − ∂ n − n ∂ ( b − b ) + ∂ ( b − b ) (cid:3) (A.21) a = e b ∂ ( b + b ) (A.22) a = e b (cid:2) ∂ n − n ∂ ( b + b ) + ∂ ( b + b ) (cid:3) (A.23) a = e b [( n n − n ) ∂ ( b + b ) − ∂ ( n n − n )+ ∂ n − n ∂ ( b + b ) + ∂ ( b + b )] (A.24) ppendix BKilling vector fields and theirgroup actions A vector field ξ a is a KVF of a given metric g ab if it satisfies the Killingequations ξ a ; b + ξ a ; b = 0 , (B.1)where semicolon denotes covariant derivative. Two vectors ξ a and η a commuteif they satisfy ξ b η a ; b − η b ξ a ; b = 0 . (B.2)It is standard result that the set of all isometries of a given manifold ( M , g )forms a Lie group G r of dimension r called isometry group of ( M , g ). Eachone-dimensional subgroup of G r defines a family of curves whose tangent fieldsis a KVF. In this way the Lie Group G r generates the Lie algebra of KVFs. Iftwo KVFs commute, then they form an Abelian G group. Two KVFs ξ a and η a in an Abelian G group act orthogonally transitively if they satisfy ξ [ a ; b ξ c η d ] = 0 , η [ a ; b η c ξ d ] = 0 . (B.3)A locally rotationally symmetric (LRS) model admits at least 3 KVFs thatform a G group whose group orbits are two dimensional. 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