Cylindrical spikes
aa r X i v : . [ g r- q c ] F e b Cylindrical spikes
M Z A Moughal, W C Lim
Department of Mathematics, University of Waikato, Private Bag 3105,Hamilton 3240, New [email protected], [email protected]
Abstract
The Geroch/Stephani transformation is a solution-generating trans-formation, and may generate spiky solutions. The spikes in solutionsgenerated so far are either early-time permanent spikes or transientspikes. We want to generate a solution with a late-time permanentspike. We achieve this by applying Stephani’s transformation withthe rotational Killing vector field of the locally rotationally symmet-ric Jacobs solution. The late-time permanent spike occurs along thecylindrical axis. Using a mixed Killing vector field, the generated solu-tion also features a rich variety of transient structures. We introducea new technique to analyse these structures. Our findings lead usto discover a transient behaviour, which we call the overshoot transi-tion. These discoveries compel us to revise the description of transientspikes.
Keywords: Geroch’s transformation, Stephani’s transformation, spike,stiff fluid, cylindrical
Spikes are small-scale spatial structures that form and then either remainthere (permanent spikes) or disappear (transient spikes). Spikes were dis-covered incidentally by Berger and Moncrief [1], whose original goal was tounderstand the nature of generic singularities. The well-known BKL conjec-ture by Lifshitz, Khalatnikov and Belinskii [2, 3, 4] are heuristic argumentsthat the approach to generic spacelike singularities is vacuum dominated, lo-cal and oscillatory. The non-local nature of spikes brings the the local natureof the conjecture into doubt. See [5] for a comprehensive introduction. Since2012 the focus has shifted to the possible role of spikes in the formation oflarge-scale structures as the Universe expands [6, 7, 8].In a series of four papers [9, 10, 11, 12], spiky solutions were generated us-ing Geroch’s and Stephani’s transformations [13, 14, 15] in order to study the1ynamics of spikes. One of the goals is to undestand the role of spikes in theformation of filamentary structures. The spikes in solutions generated so farare either early-time permanent spikes or transient spikes. In the conclusionsection of [12], the authors hoped to generate solutions with a late-time per-manent spike, which is more suitable for formation of permanent structures.This became the initial goal for the PhD thesis of the first author [16]. Thegoal was achieved by applying Stephani’s transformation with the rotationalKilling vector field (KVF) of the locally rotationally symmetric (LRS) Jacobssolution. With a mixed KVF, the generated solution unexpectedly featuresa rich variety of transient structures. A new technique was introduced toanalyse these structures, leading to the discovery of a transient behaviour,which we call the overshoot transition, and also leading to the re-examinationof the definition of transient spikes. This paper is an abridged version of thethesis.
Assume zero vorticity (zero shift). The spatial metric components are givenby the formula g ij = e αi e β j δ αβ , where Roman indices i , j = 1 .. α , β = 1 .. e αi (and equivalently e αi ) upper triangular, as follows. The framecoefficients e αi simplify from 9 components to 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 (1) 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)2he frame derivative operators e = N − ∂ , e α = e αi ∂ i in the Iwasawa frameare e = 1 N ∂ (3) e = e b ∂ (4) e = e b [ − n ∂ + ∂ ] (5) e = e b [( n n − n ) ∂ − n ∂ + ∂ ] . (6)In the Iwasawa frame, the metric components in terms of the b ’s and n ’sare given by g = − N (7) g = e − b , g = e − b n , g = e − b n (8) g = e − b + e − b n , g = e − b n n + e − b n (9) g = e − b + e − b n + e − b n . (10)If the metric is given, we can compute the b ’s and n ’s as follows. b = − ln g (11) n = g g (12) n = g g (13) b = − ln( g − g n ) (14) n = ( g − g n ) e b (15) b = − ln( g − g n − e − b n ) . (16) Consider a solution g ab of the vacuum Einstein’s field equations with a KVF ξ a . Geroch’s transformation [13, 14] (see also [18, Section 10.3]) is an algo-rithm for generating new solutions, by exploiting the KVF ξ a . The algorithminvolves solving the following partial differential equations ∇ a ω = ε abcd ξ b ∇ c ξ d , (17) ∇ [ a α b ] = 12 ε abcd ∇ c ξ d , ξ a α a = ω, (18) ∇ [ 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 . Next, define ˜ λ and η a as˜ λ = λ h (cos θ − ω sin θ ) + λ sin θ i − , (20) η a = ˜ λ − ξ a + 2 α a cos θ sin θ − β a sin θ, (21)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 . (22)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 (21) 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 = F g ab + λF β a β b − ξ a β b − β a ξ b , (23)where F = λ + ω . (24)Stephani [15] generalised Geroch’s 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 = ρF . (25)Before applying Geroch’s or Stephani’s transformation, we set up thecoordinates such that the KVF to be used has the form ξ a = (0 , , , , (26)to adapt to the Iwasawa frame for simplicity.In simpler cases, if the seed metric has the form g ab = − N g g g g
00 0 0 g , (27)i.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 . (28)4xpressing the metric ˜ g ab in (28) in b ’s and n ’s gives˜ N = N √ F (29)˜ b = b + 12 ln F (30)˜ b = b −
12 ln F (31)˜ b = b −
12 ln F (32)˜ n = n F − β (33)˜ n = 0 (34)˜ n = 0 . (35) Spacelike KVFs can be classified into two kinds – translational and rotational.The four papers [9, 10, 11, 12] used a linear combination of translationalKVFs. In this paper, we will use a linear combination of KVFs that includesa rotational KVF. Stephani’s transformation requires the matter to be a stifffluid, so we start by looking at locally rotationally symmetric (LRS) solutionswith a stiff fluid. The simplest such solution is the flat FLRW solution, butit does not generate as much structure as the next simplest solution, the LRSJacobs (Bianchi type I) solution, which we shall use as the seed solution.The Jacobs solution [19][20, page 189] is given by the line elementd s = − d t + t p d x + t p d y + t p d z , (36)where the coordinates are ( t, x, y, z ), and p = 13 (1 + Σ +0 + √ − ) , (37) p = 13 (1 + Σ +0 − √ − ) , (38) p = 13 (1 − +0 ) . (39)The non-zero Hubble-normalised shear components [20, Sections 1.1.3, 6.1.1]are Σ +0 and Σ − , and they are constant, with Σ + Σ − ≤
1. The comovingstiff fluid has pressure p and density ρ given by p = ρ = 1 − Σ − Σ − t . (40)5o impose the LRS condition, it is simplest to set Σ − = 0, so the pa-rameter Σ +0 takes values from − +0 = − p , p , p ) = (0 , ,
1) (also known as theTaub form of flat 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 , (41)where the fourth one is rotational. We intend to apply Stephani’s transfor-mation with the general linear combination of the KVFs: c ∂ x + c ∂ y + c ∂ z + c ( − y∂ x + x∂ y ) = ( c − c y ) ∂ x + ( c + c x ) ∂ y + c ∂ z (42)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 KVFreads − y∂ x + x∂ y + k∂ z . (43)This KVF forms an Abelian orthogonally transitive (OT) G group withexactly one other KVF (namely a linear combination of ∂ z and − y∂ x + x∂ y ).By Geroch’s theorem [14, Appendix B], the generated metric will admit anAbelian 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 . (44)We want to simplify the KVF to just ∂ ψ for the application of Stephani’stransformation, so we make a further change of coordinates, by introducing Z = z − kψ. (45)Then, in the coordinates ( t, ψ, Z, r ), the KVF is simply ∂ ψ , but the lineelement now readsd s = − d t + ( k t p + r t p )d ψ + 2 kt p d ψ d Z + t p d Z + t p d r . (46)This shall be the seed solution to which we apply Stephani’s transformation.It has the simple form (27).The state space orbits of a solution, projected onto the (Σ + , Σ − ) plane,can provide some insight into the dynamics of the solution. Recall that6 + -1-0.8-0.6-0.4-0.200.20.40.60.81 - Figure 1: State space orbits of the rotated LRS Jacobs solution projected onthe (Σ + , Σ − ) plane for various values of Σ +0 , assuming k = 0. The r = 0orbits are fixed points. As t increases, r = 0 orbits move away from thesefixed points for Σ +0 >
0, and towards these fixed points for Σ +0 < + , Σ − ) are defined in terms of the diagonal components of the Hubble-normalised expansion shear asΣ + = −
12 Σ , Σ − = Σ − Σ √ . (47)which givesΣ + = −
12 Σ +0 , Σ − = √ (cid:18) l − − Σ +0 (cid:19) , l = t (ln λ ) t . (48)The solution is undefined at r = 0 if k = 0 (coordinate singularity). It isstraightforward to analyse l . If k = 0 then l = 2 p . If k = 0, then l = 2 p at r = 0. For r = 0 write l = p (1 − τ ) + p (1 + τ ) , τ = tanh(Σ +0 (ln t ) + ln | k | − ln r ) . (49)For Σ +0 > l goes from 2 p to 2 p as t goes from 0 (early times) to ∞ (late times). 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 limitalong r = 0 (at late times for Σ +0 >
0, at early times for Σ +0 < r = 0 at late times7or Σ +0 >
0, and at early times for Σ +0 <
0. The false spike is entirely acoordinate effect, due to the rotating Iwasawa frame. Figure 1 shows thatstate space orbits projected on the (Σ + , Σ − ) plane for various values of Σ +0 ,assuming k = 0. The r = 0 orbits are fixed points. As t increases, r = 0orbits move away from these fixed points for Σ +0 >
0, and towards thesefixed points for Σ +0 < We now carry out Stephani’s transformation with the general KVF ∂ ψ . Weobtain λ = k t p + r t p , ω = 2 k p t p + k Σ +0 r + ω , (50) 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 . (51)The generated metric is then given through b ’s and n ’s by the formulas (29)–(35), dropping tildes for brevity. N = F / (52) b = −
12 ln λF (53) b = −
12 ln
F r t p +2 p λ (54) b = −
12 ln(
F t p ) (55) n = F kt p λ − β (56) n = 0 (57) n = 0 . (58)Its ψ - Z area element A = rt p + p (59)is the same as the seed solution’s, and is always expanding. Its volumeelement V = rt √ F (60)is different from the seed solution’s and is not always expanding. Thismeans the Hubble scalar H can become negative for some parameter val-ues, and Hubble-normalised variables would blow up. In this case we use β -normalisation, which is based on the ever-expanding area element [21].8 The initial goal – late-time permanent spikes
For the special case k = 0 (that is, the KVF is purely rotational), (Σ + , Σ − )reduces to Σ + = − Σ +0 + f f , Σ − = √ +0 − f )2 + f , (61)where f = t (ln F ) t = 4 p r t p r t p + ω , p = 13 (1 + Σ +0 ) . (62)Along r = 0, we have f = 0 and(Σ + , Σ − ) = −
12 Σ +0 , √
32 Σ +0 ! . (63)Along r = 0, provided that p = 0 ⇔ Σ +0 > −
1, we have f → ( t → p as t → ∞ , (64)so there is a late-time permanent spike along the cylindrical axis r = 0, forthe case k = 0, − < Σ +0 ≤ ω = 0. Figure 2 shows the state space orbitsprojected on the (Σ + , Σ − ) plane for various values of Σ +0 for k = 0, ω = 0.The r = 0 orbits are fixed points, while the r = 0 orbits move away fromthese fixed points as t increases, mimicking the orbits of Taub (Bianchi typeII) solutions [20, page 136].Following [22], we obtain the coordinate and physical radii of the spike:coordinate radius = | ω | t − p , (65)physical radius = | ω | Z √ u d u. (66)i.e. the physical radius of the spike is time-independent.This is the first generated solution with a late-time permanent spike, andthe first generated solution with a spike along a line. The spike producesan overdensity along the axis at late times, which is conducive to large scalestructure formation. Thus the generated solution can serve as a prototypicalmodel for formation of galactic filaments along web-like strings.What is special about the rotational KVF? Its length vanishes along therotation axis. As the universe expands, the length squared of the KVF λ Such features can also be achieved through silent LTB and Szekeres models [8] withoutusing solution-generating transformations. + -1-0.8-0.6-0.4-0.200.20.40.60.81 - Figure 2: State space orbits projected on the (Σ + , Σ − ) plane for variousvalues of Σ +0 for k = 0, ω = 0. A circle represents the orbit along r = 0,which is a fixed point. r = 0 orbits move away from these fixed points as t increases, mimicking the orbits of Taub solutions.typically increases and dominates the magnitude of its vorticity ω , as seenin [9, 10, 11, 12]. For a rotational KVF, the rotation axis is the exceptionalpoints where this does not happen. This creates a discontinuous limit for f at late times – a late-time permanent spike. While the k = 0 case (rotational KVF) has simple dynamics, the k = 0 case(mixed KVF) has rich transient dynamics which requires the introduction ofa new technique to analyse them. We haveΣ + = − Σ +0 + f f , Σ − = √ l − f − (2 − Σ +0 ))2 + f , (67)where f = t (ln F ) t , l = t (ln λ ) t . If f becomes less than −
2, then it is moreappropriate to use β -normalised (Σ + , Σ − ), which areΣ + = − Σ +0 + f − Σ +0 , Σ − = √ l − f )2 − Σ +0 − √ . (68) Analogous to Hubble-normalisation, β -normalisation is based on the area element A . β = N − ∂ t ln A , and is related to H through β = H + σ + [21]. +0 -1-0.500.511.52 T T T T Figure 3: Power of t of the terms in (69) against Σ +0 . l is a coordinate effect, so we focus on f . f consists of terms involving λ and terms involving ω . λ contains two different power terms, t p and t p ,while ω contains t p and a time-independent term. We group them intofour 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 . (69)Figure 3 plots the power of t of each term in (69) against the parameterΣ +0 . In general, the four powers are distinct, except for 3 special valuesof Σ +0 . For Σ +0 = −
1, there are two distinct powers; for Σ +0 = 0, threedistinct powers; and for Σ +0 = , two distinct powers. The term with thelargest power of t dominates at late times; the term with smallest power of t dominates at early times; and the terms with intermediate power of t mayor may not dominate for a finite time interval, depending on how big theircoefficient 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 ) . (70)Observe that f ≈ p when T dominates4 p when T dominates2(1 + p ) when T dominates0 when T dominates . (71)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 independence11f 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 = . The value of f at successive equilibrium states is strictly increasingin time. Among the four values, 4 p is negative for < Σ +0 ≤
1, with aminimum value of − at Σ +0 = 1, which is still greater than −
2, so theHubble scalar H is positive at each equilibrium state. But we will see laterthat f can become less than − T = T for t yields the transition time t = (cid:18) k r (cid:19) . (72)Comparing the transition times will determine how many transitions an ob-server with fixed r undergoes. The coefficients of T and T have spatialdependence. They can even vanish for certain worldline ( r = 0 for T , and r = q − ω k Σ +0 for T , provided that ω k Σ +0 ≤ t .The transition time between two dominant terms can be regarded asroughly the boundary between the two corresponding equilibrium states.We say “roughly” because the transition is a smooth, continuous process,so there is no sharp boundary. If a transition time has spatial dependence, italso gives the spatial location of the boundary at a fixed time. The spacetimeis partitioned into regions of equlibrium states, separated by transition times.When viewed at a fixed time, we can regard space as being partitioned intocells of equilibrium states, separated by walls (around which spatial gradi-ent is large). If two walls are near each other, we see a narrow cell. Theneighbourhood of the narrow cell shall be called a spike if certain additionalconditions are met. We will discuss these conditions in Section 9.We now give a number of examples to show the various features. For the case − < Σ +0 <
0, Figure 3 gives the ordering T , T , T , T , inincreasing power of t . We have up to 4 distinct equilibrium states, and along12eneral 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 where a term vanishes. 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. Because ofthis, r = 0 is not really special. This suggests that transient spikes do notoccur along a special worldline, but rather require a scenario in which someintermediate term is always sub-dominant. The second special worldline is r = q − ω k Σ +0 , where T vanishes, giving an early-time permanent spike. Thepossible 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 (69), we see that the logarithmof the square of each term is a linear function of ln t . Figure 4 shows aqualitative plot of the log of each term squared against ln t for the scenario T −→ T −→ T −→ T . The plot is useful for determining the order of the transition times. It is clearfrom 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 (73)13 t t ln T ln T ln T ln T ln t Figure 4: Qualitative plot of the log of each term squared against ln t for thescenario T −→ T −→ T −→ T .must satisfy the condition t < t < t (74)in this scenario. The condition then determines the r intervals (the world-lines) where the scenario occurs. t < t implies | k Σ +0 r + ω | < (cid:18) | k | p r p (cid:19) , (75)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 . (76)So the condition (74) restricts r to one or more intervals. As a concreteexample, take the parameter valuesΣ +0 = − . , k = 10 , ω = 5 . (77)(75) can be solved numerically to give the intervals0 . < r < . . < r. (78)Note that r = 1 is the second special worldline, so it must be excluded fromthis scenario. (76) gives r < . . < r < , < r < . . < r < . . (79)14igure 5: f against ln t and r , showing the intervals where the scenario T −→ T −→ T −→ T occurs. The interval 0 . < r < . r = 1. The interval 100 < r <
250 shows only twovisible distinct states because the transition times are too close together.We plot f against ln t and r in Figure 5, showing the intervals where thescenario occurs. The interval 0 . < r < . r = 1. Along the cylindrical shell r = 1, there is a permanentspike at early times. The interval 100 < r <
250 shows only two visibledistinct states because the transition times are too close together. So iftransition times are too close together, we see fewer visible distinct statethan the actual number of states predicted by the scenario.What happens in other intervals of r ? From (78), we know that t becomes greater than t for values of r just beyond the boundaries. Fromthe diagram in Figure 4, this happens if the graph of ln T becomes too low,as shown in Figure 6. Now, the diagram in Figure 6 shows the scenario T −→ T −→ T , (80)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 . (81)They must satisfy the condition t < t < t . (82)The condition t < t is equivalent to t < t , so it gives (75) with theopposite inequality direction: | k Σ +0 r + ω | > (cid:18) | k | p r p (cid:19) . (83)15 t t ln T ln T ln T ln T ln t Figure 6: Qualitative plot of the log of each term squared against ln t for thescenario T −→ T −→ T . t < t implies | k Σ +0 r + ω | < | k | p (cid:18) − Σ +0 (cid:19) p p , (84)which gives rise to one interval of r . Together, the condition restricts r toone or more intervals. (83) gives the intervals r < . . < r < . , (85)while (84) gives the interval r < . ≤ r < . . < r < . . (86)We plot f against ln t and r on these intervals showing the scenario T −→ T −→ T in Figure 7. The interval 0 ≤ r < . < r <
120 shows three 3 distinct states for small r which fade away to two visible states as the transition times become closertogether as r increases.To complete the example, we now look at what happens beyond r =240 . t becomes larger than t . From the diagram in Figure 4,this happens if the graph of ln T becomes too low, as shown in Figure 8.Now, the diagram in Figure 8 shows the scenario T −→ T −→ T , (87)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 . (88)16 f ln t r f
10 1003 ln t r Figure 7: f against ln t and r , showing the intervals where the scenario T −→ T −→ T occurs. The interval 0 ≤ r < . < r <
120 shows three 3 distinct states for small r which fade away to two visible states as the transition times become closertogether as r increases. t t t ln T ln T ln T ln T ln t Figure 8: Qualitative plot of the log of each term squared against ln t for thescenario T −→ T −→ T .They must satisfy the conditions t < t < t . (89)The condition t < t is equivalent to t < t , so it gives (76) with theopposite inequality direction, a simple lower bound r > | k | (cid:18) | k | (2 − Σ +0 ) (cid:19) Σ+02 p . (90)17igure 9: f against ln t and r , showing the scenario T −→ T −→ T for r > . t < t implies | k Σ +0 r + ω | < r p p (cid:18) − Σ +0 | k | (cid:19) p p , (91)which restricts r to one or more intervals. (90) gives the interval r > . . < r < . r > . . (92)Together, the scenario occurs for r > . f against ln t and r showing 3 distinct states with a lower bound on r in Figure 9.This completes the scenarios in this example. We summarise them inanother useful diagram, where we plot the transition times of each scenario,and label the dominant term in each cell. See Figure 10. Are there transientspikes? Not at first sight. Equation (86) gives the two r intervals wherethe worldlines undergo the scenario T −→ T −→ T . The first one couldbe called a transient spike, even though it looks wide in comparison to thenarrow permanent spike around r = 1. The second one however is too wideto be considered a spike. Perhaps we should shift our focus from spikes tocells of various length scales. Very narrow cells are obvious candidates forspikes, but the visual distinction fades for wider cells.Plotting the state space orbits reveals that the solution is future asymp-totic to a state with the following values:Σ + = − , Σ − = − √ , Σ = 0 , Ω = 27256 , Ω k = − , (93)18 .9794 1 1.0226 r -10010 ln t 4 4123 r -10010 ln t 42 13 Figure 10: Plot of the cells and transition times, showing the different sce-narios along each fixed r . Each cell is labelled with the index of the dominantterm. Dashed line indicates the transition time t for l .a presently unidentified non-vacuum state with negative spatial curvatureparameter Ω k . See Figure 11. Figure 10 shows that the transition time of l ,which is t , happens to be close to a transition time of f in this example, sothe state space orbits in Figure 11 do not have a distinctive vertical segmentslike in Figure 1. The second example showcases a transient spike and an overshoot transition.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 (94)with transition times t = (cid:18) ω r + k (cid:19) , t (1&2)3 = (cid:18) r + k )9 k (cid:19) , (95)which are required to satisfy the condition t < t (1&2)3 . (96)19 + -1-0.500.51 - All orbits -1 0 1 + -1-0.500.51 - r=1 -1 0 1 + -1-0.500.51 - r=1.001 -1 0 1 + -1-0.500.51 - r=10 -1 0 1 + -1-0.500.51 - r=200 -1 0 1 + -1-0.500.51 - r=10000 Figure 11: State space orbits along representative worldlines.The condition gives a lower bound on rr > (cid:18) | kω | − k (cid:19) . (97)If the lower bound is positive, then for r less than this we have the secondscenario, the 2-state sequence T −→ T (98)with transition time t = (cid:12)(cid:12)(cid:12)(cid:12) ω k (cid:12)(cid:12)(cid:12)(cid:12) . (99)For example, given k = 0 . ω = ±
2, for r > . r < . f against ln t and r , showing both scenarios in Figure 12. The T −→ T transition is sigmoidfor ω = 2, but has overshoots for ω = − f =2 ln t r
10 4-5 2-10 0 -5100 5 10 f =-2 ln t r
10 4-5 2-10 0
Figure 12: f against ln t and r for Example 2, showing scenario (94) for r > . r < . T −→ T transition issigmoid for ω = 2, but has overshoots for ω = − f has a cascading appearance. Despite this, f canfluctuate wildly with overshoots. Under what condition does this happen?If we examine f from (70): f = 2( T + T )(2 p T + 2 p T ) + 2( T + T )(1 + p ) T ( T + T ) + ( T + T ) , (100)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. Cancellation happens when ω = T + T ≈ . (101)Its effect is most prominent when cancellation occurs during the T −→ T (102)transition in a scenario. Heuristically, when T + T = ǫ , where | ǫ | is small,and suppose T and T are o ( ǫ ) at that instant, then (100) implies f ≈ p ) ǫ T . (103)Then f becomes negative in the first stage of the transition (when ǫ < ǫ > T and T are much smaller than T and T when this happens. We therefore call such a transition an overshoottransition. 21n Example 2, both ω = ± r . − . ln t .
3. The transient spike transition occuringaround ln t ≈ . ω = 2, and is an overshoot tran-sition in the case ω = −
2. This example illustrates that a transient spikeand an overshoot transition are two different phenomena, and an overshoottransition can occur inside a transient spike.
The third example shows a narrow cell near the local extremum of the tran-sition time. Take Σ +0 = − , k = 10 , ω = 10 . . (104)The only scenario for the case Σ +0 = − T & T −→ T & T . (105)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) . (106)which has a global minimum at r = 1 for our example. We plot f againstln t and r in Figure 13, showing the cell becoming narrow near the globalminimum of the transition time. Does this count as a transient spike? No.The wordlines in this example all undergo the same scenario. In the orig-inal context where transient spikes were first named, the worldlines in asmall neighbourhood undergo a different scenario than what worldlines fur-ther away undergo. Adding this criterion rules out the feature in this exampleas a transient spike.The examples have led us to re-examine the definition for transient spikes.Through Example 1, we realise that unlike permanent spikes, transient spikesdo not occur along a special worldline, but only require a scenario where someintermediate term is always sub-dominant. Such a scenario may occur onspatial intervals or cells with various length scales, and only the very narrowones are visually distintive enough to be called a spike. Transient spikes aretherefore visually less distinctive than permanent spikes. Through Example2, we discover a new phenomenon, an overshoot transition, which should not22igure 13: f against ln t and r for Example 3. The cell is narrow near theglobal minimum of the transition time, but is not a transient spike.be confused with transient spike. Through Example 3, we rule out certainnarrow cells as transient spike, because the worldlines all undergo the samescenario.For more examples, see [16]. [16] shows that a late-time permanent spikeforms along the cylindrical axis in the case < Σ +0 ≤ k = 0. In otherwords, in order to generate a solution with a late-time permanent spike, it isnot necessary for the KVF to be purely rotational.
10 Conclusion
To summarise, we have found the first non-silent solution with a late-timepermanent spike; found the first spike along a line; introduced a new tech-nique to analyse a key function, f ; revised the description of transient spikes;discovered and described overshoot transitions. Late-time permanent spikesare more suitable than transient spikes and early-time permanent spikes inmodelling structure formation. Spikes along a line can be used to modelformation of galactic filaments along web-like strings. The new technique toanalyse f reveals the cell-like structure of inhomogeneous spacetimes, andthe possible transition dynamics (regular sigmoid transition, overshoot tran-sition) between cells.We conclude by commenting on future research. Firstly, the family ofexact solutions we found make up only a set of measure zero in the classof cylindrically symmetric solutions. How does a typical cylindirically sym-metric 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 [23]. Sec-23ndly, we have used the rotational KVF of the LRS Jacobs solution. Exactsolutions that admit a rotational KVF include the LRS Taub solution, theNUT (LRS Bianchi type VIII) solution, and the Taub-NUT (LRS Bianchitype IX) solution [20, page 198]. It would be interesting to see what spikysolutions are generated from these solutions. Thirdly, our exact solutions areOT G solutions. In principle, non-OT G solutions and G solutions canbe generated from a rotational KVF. Are there simple enough seed solutionsthat generate spiky solutions with such isometries? Lastly, we are working onapplying the new technique to the exact vacuum non-OT G spike solutionfrom [9] and the stiff spike solution from [10], to help improve the numericalsimulation and matching in an upcoming paper that is an extension of [24]and [5]. Acknowledgements
MZAM is supported by Pakistan’s Higher Education Commission scholar-ship.
References [1] B. K. Berger and V. Moncrief, Phys. Rev. D , 4676 (1993).[2] E. M. Lifshitz and I. M. Khalatnikov, Adv. Phys. , 185 (1963).[3] V. A. Belinskii, I. M. Khalatnikov, and E. M. Lifschitz, Adv. Phys. ,525 (1970).[4] V. A. Belinskii, I. M. Khalatnikov, and E. M. Lifschitz, Adv. Phys. ,639 (1982).[5] J. M. Heinzle, C. Uggla, and W. C. Lim, Phys. Rev. D , 104049(2012), arXiv:1206.0932.[6] A. A. Coley and W. C. Lim, Phys. Rev. Lett. , 191101 (2012),arXiv:1205.2142.[7] W. C. Lim and A. A. Coley, Class. Quant. Grav. , 015020 (2014),arXiv:1311.1857.[8] A. A. Coley and W. C. Lim, Class. Quant. Grav. , 115012 (2014),arXiv:1405.5252. 249] W. C. Lim, Class. Quant. Grav. , 162001 (2015), arXiv:1507.02754.[10] A. A. Coley and W. C. Lim, Class. Quant. Grav. , 015009 (2016),arXiv:1511.07095.[11] A. A. Coley, D. Gregoris, and W. C. Lim, Class. Quant. Grav. ,215010 (2016), arXiv:1606.07177.[12] D. Gregoris, W. C. Lim, and A. A. Coley, Class. Quant. Grav. ,235013 (2017), arXiv:1705.02747.[13] R. Geroch, J. Math. Phys. , 918 (1971).[14] R. Geroch, J. Math. Phys. , 394 (1972).[15] H. Stephani, J. Math. Phys. , 1650 (1988).[16] M. Z. A. Moughal, Generating spiky solutions of Einstein field equationswith the Stephani transformation , PhD thesis, University of Waikato,New Zealand, 2021, arXiv:2102.09776.[17] J. M. Heinzle, C. Uggla, and N. R¨ohr, Adv. Theor. Math. Phys. , 293(2009), arXiv:gr-qc/0702141.[18] D. Kramer, H. Stephani, E. Herlt, M. A. H. MacCallum, andE. Schmutzer, Exact solutions of Einstein’s field equations (CambridgeUniversity Press: Cambridge, Cambridge, 2002).[19] K. C. Jacobs, Astrophys. J. , 661 (1968).[20] J. Wainwright and G. F. R. Ellis,
Dynamical systems in cosmology (Cambridge University Press, Cambridge, 1997).[21] H. van Elst, C. Uggla, and J. Wainwright, Class. Quant. Grav. , 51(2002), arXiv:gr-qc/0107041.[22] W. C. Lim, Class. Quant. Grav. , 045014 (2008), arXiv:0710.0628.[23] L. Andersson, H. van Elst, W. C. Lim, and C. Uggla, Phys. Rev. Lett. , 051101 (2005), arXiv:gr-qc/0402051.[24] W. C. Lim, L. Andersson, D. Garfinkle, and F. Pretorius, Phys. Rev. D79