Symmetry and Equivalence in Szekeres Models
aa r X i v : . [ g r- q c ] F e b Symmetry and Equivalence in Szekeres Models
Ira Georg ∗ Charles Hellaby † Dept. of Maths. and Applied Maths, University of Cape Town, Rondebosch, 7701, South Africa
Abstract
We solve for all Szekeres metrics that have a single Killing vector. For quasi hyperboloidal( ǫ = − ) metrics, we find that translational symmetries are possible, but only in metrics thathave shell crossings somewhere, while metrics that can be made free of shell crossings only permitrotations. The quasi planar metrics ( ǫ = 0 ) either have no Killing vectors or they admit full planarsymmetry. Single symmetries in quasi spherical metrics ( ǫ = +1 ) are all rotations. The rotationscorrespond to a known family of axially symmetric metrics, which for each ǫ value, are equivalentto each other. We consider Szekeres metrics in which the line of dipole extrema is required tobe geodesic in the 3-space, and show the same set of families emerges. We investigate when twoSzekeres metrics are physically equivalent, and complete a previous list of transformations of thearbitrary functions. In 1975 Peter Szekeres [62, 63] discovered a very interesting family of exact inhomogeneous solutions ofthe Einstein field equations, for which the matter source is a comoving, zero pressure fluid (dust). Thereare 6 arbitrary functions that depend on the “radial” coordinate . Although one of these functions canbe used to remove the re-scaling freedom in the “radial” coordinate, there is no canonical choice thatdoes not restrict the physical possibilities [32].There are actually two classes of Szekeres model, the more commonly used one is a generalisationof the Lemaˆıtre-Tolman [49, 64] and Ellis [27] metrics, and the other is a generalisation of the Datt-Kantowski-Sachs [23, 40] metrics. Since the latter can be viewed as a limit of the former [30], wefocus on the Lemaˆıtre-Tolman-Ellis-like metric. Three of the Szekeres arbitrary functions are identicalto those of the “underlying” Lemaˆıtre-Tolmanor Ellis metric, and the other three control the deviationfrom spherical, planar, or hyperboloidal (pseudo-spherical) symmetry.A key result for this metric was the proof by Bonnor, Suleiman and Tomimura (BST) that it hasno Killing vectors [19]. This is despite the constant time 3-surfaces being conformally flat [5], and thelack of gravitational radiation [16]. In fact the BST paper only considered the quasi-spherical case,though one would expect it to generalise; a task which we complete here along the way. ∗ [email protected] † [email protected] Here the “radial” coordinate is the one that becomes a true coordinate radius in the spherically symmetricspecial case. olan and Debnath [54], in investigating shell focussing singularities, have shown that if a quasi-spherical Szekeres spacetime has a “radial” null geodesic, then the spacetime is axially symmetric andthe ray lies along the axis; if there is more than one radial null geodesic, it is spherically symmetric.Further, they showed that all axi-symmetric Szekeres models are equivalent.Krasinski & Bolejko [46] considered light paths and redshifts in Szekeres models. In general, lightrays emitted from the same matter point at different times, and received by the same observer, do notfollow the same comoving spatial path. The authors asked under what conditions the spatial pathsof two such light rays might be repeated. They showed that if all light paths between every emitter-observer pair are repeated the spacetime must be a (dust) FLRW model, and if there exists a repeatedray along a single direction then the model is axi-symmetric about that direction. They also generalisedthe Nolan and Debnath result to quasi-hyperbolic and quasi-planar models. For the Lemaˆıtre-Tolmanand Ellis models, only the “radial” rays have repeatable paths.Neither of the above papers claimed they had found all the axi-symmetric Szekeres models, and thepossibility, in the quasi-hyperbolic and quasi-planar models, of other, non-rotational single symmetrieswas not considered.Sussman & Gaspar [60] studied the location of extrema of density, expansion and spatial curvature,and they produced some very nice numerical examples and plots of Szekeres models with complexstructure. They also mentioned cases where one or two of the non-spherical arbitrary functions areconstant, and suggest some are axially symmetric. More complex Szekeres matter distributions —networks of matter structure — were investigated by Sussman, Gaspar & Hidalgo [61].Though there is a significant body of work on the Szekeres models, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 42, 44, 45,46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68], and there has been a renewalof interest recently, the lack of Killing vectors makes this metric relatively hard to work with. Thequasi-planar and quasi-hyperboloidal cases have been especially neglected. Therefore the symmetricspecial cases with a single Killing vector would be useful as stepping stones between full spherical,planar or hyperboloidal symmetry and the general Szekeres case.A fuller description of the Szekeres metric and its properties can be found in [43, 32, 58]. The line element is d s = − d t + ( R ′ − RE ′ /E ) ( ǫ + f ) d r + R E (cid:16) d p + d q (cid:17) , (1)where ǫ = − , , +1 , f = f ( r ) , E = E ( r, p, q ) , R = R ( t, r ) and we write W = p ǫ + f . (2)The evolution function R obeys a Friedmann equation ˙ R = 2 MR + f + Λ R , (3)where M = M ( r ) , and solving this DE introduces the bang-time function t B = t B ( r ) . The matter iscomoving, u a = δ at , and has a dust equation of state ( p = 0 ), the density being given by κρ = 2( M ′ − M E ′ /E ) R ( R ′ − RE ′ /E ) . (4) ach 2-surface of constant r, t , dl = (d p + d q ) E , (5)is a unit 2-sphere if ǫ = +1 , a unit 2-pseudo-sphere (or hyperboloid) if ǫ = − , and a 2-plane if ǫ = 0 .The shape function E can be written as E = S (cid:26) ( p − P ) S + ( q − Q ) S + ǫ (cid:27) , (6)where S = S ( r ) , P = P ( r ) , Q = Q ( r ) , and ( p, q ) are stereographic coordinates on each 2-surface. Thetransformation between stereographic coordinates ( p, q ) and the more regular “polar” type coordinatescan be found in [34] for each of the three ǫ values. R ( t, r ) gives the evolving scale area of the constant r shell, since its square multiplies the unit-scalesurface, (5). We will refer to r as a “coordinate radius” and R as an “areal radius”, for all ǫ values, itbeing understood that the pseudo-spherical or planar equivalents are intended if ǫ = +1 .The function f ( r ) determines the curvature of the constant t E = f / , of the particles at “radius” r . For ǫ = +1 , M ( r ) is thetotal gravitational mass interior to the comoving shell r ; for other ǫ values, it is a mass-like factorin the gravitational potential energy. The functions S ( r ) , P ( r ) & Q ( r ) determine the strength andorientation of the dipole on each comoving shell.The function S cannot be zero [33, 34], so it is convenient to keep it positive, S > . If ǫ = 0 globally, then S may be absorbed into the other arbitrary functions, so one is free to set S = 1 . Inorder to avoid shell crossing singularities [33], the arbitrary functions must obey a set of conditionsthat limit their ranges relative to each other; these are not very restrictive.Both g rr in (1) and ρ in (4) depend on E ′ /E , which varies over each 2-surface of constant ( t, r ) ,and has extreme values E ′ E (cid:12)(cid:12)(cid:12)(cid:12) e = ± p S ′ + ǫ ( P ′ + Q ′ ) S , (7)located at p e = P + P ′ ( S ′ /S ) + ( E ′ /E ) e , (8) q e = Q + Q ′ ( S ′ /S ) + ( E ′ /E ) e . (9)The quasi-spherical Szekeres metric, with ǫ = +1 , is commonly described as an assembly of evolvingspheres which are non-concentric, and which display a dipole distribution in the density variation aroundeach sphere. The dipole is due to the factor E ′ /E ; on each ( p, q ) E ′ /E | min = − E ′ /E | max . Thestrength and orientation of the dipole depend on coordinate radius r through (7)-(9).The quasi-hyperboloidal Szekeres metric, with ǫ = − , has been much less studied. As shownin [34], it may be thought of as an assembly of evolving right-hyperboloids that are stacked non-symmetrically or “non-concentrically”. The two sheets of the hyperboloid map to separate regions ofthe ( p, q ) plane, one inside the boundary circle ( p − P ) + ( q − Q ) = S , (10) nd one outside it. The boundary circle is the locus of infinity for each sheet. Only one of the twosheets can be free of shell crossings and only if S ′ > P ′ + Q ′ . (11)In this latter case too there is a kind of hyperboloidal (or pseudo-spherical) dipole with strength (7)and orientation defined by (8) and (9). If (11) is not satisfied, (7) is not real, and extrema with respectto p & q do not exist.The quasi-planar Szekeres metric, given by ǫ = 0 , does not have an extremum of E ′ /E , so onecannot talk about a “dipole”.It should be noted that a single spacetime can have both ǫ = +1 and ǫ = − regions, joined by an ǫ = 0 region which may be thin (a 3-surface) or may have finite width. The Szekeres metric — in its fully general form — has no Killing vectors [19]. However, it containsthe spherically-symmetric special case, the Lemaˆıtre-Tolman model [49, 64]. For example, if we choosethe functions S , P & Q to be constant ǫ = +1 , S ′ = P ′ = Q ′ , (12)we will find a Lemaˆıtre-Tolman model. Therefore we expect that axially symmetric special cases exist;and indeed examples such as ǫ = +1 , P ′ = 0 , Q ′ = 0 are known.The spherical symmetry of LT models results from the fact that every const. r, t submanifold isspherically symmetric with respect to a common center. Although a more general quasi-spherical( ǫ = 1 ) Szekeres model also has spherically symmetric r, t constant submanifolds , their centers do notcoincide. When we consider an axially-symmetric arrangement of non-concentric spheres, then there isonly one possibility: the centers of the spheres ought to be in a “straight line”, a geodesic, that formsthe symmetry axis.In quasi-spherical models, the dipole function E ′ /E encodes the distance between two neighbouringspheres with respect to the mean distance at the equator, see figure 1a. If the dipole function isdependent on the coordinates on the sphere p, q , then the distance to the next sphere (with label r + dr ) is different for different p, q . The two neighbouring spheres are non-concentric and because ofthe spherical symmetry of the two shells, E ′ /E must show a dipole structure (see figure 1b), i.e. E ′ /E is axially symmetric (for constant r ), the points of extrema are antipodal, and the extremal valuesare equal with opposite signs. If E ′ /E is constant on all spheres, the model is necessarily sphericallysymmetric. E ′ /E can be considered as the deviation function from global spherical symmetry.We expect an axial symmetry to occur if the centers of the r, t = const. submanifolds are displacedonly along a “straight line”, i.e. if the extrema of E ′ /E form a geodesic. Similarly, for the quasi-planarand quasi-hyperboloidal cases, we expect sub-models that are symmetric, if the dipole functions ofdifferent shells align.In the following we will represent shells of constant r, t as p, q -planes in a Cartesian-like plot andthus unify the representation of the three different types of models ( ǫ = 1 , , − ). Below we solvefor the general case, rigorously deriving the conditions for an axial or other single symmetry from theKilling equations. for short we say “spheres” if ǫ = 1, “hyperboloids” if ǫ = −
1, simply “planes” if ǫ = 0, or “shells” if ǫ is notspecified, instead of “ r, t = const. submanifolds” in max (a) Two const. r, t submanifolds in a general quasi-spherical Szekeres model. The indicator min,max markthe points of extremal distance between the spheres.The dashed curves on the inner sphere are curves oflatitude with constant E ′ /E . (b) Contour lines of E ′ /E after a stereographicprojection relative to fig 1a onto the p, q -planefor an arbitrary choice of parameters. Figure 1: Form of E ′ /E on the sphere (a) and on the p, q plane (b). We now want to formalize the above and assume that there is one Killing vector field ξ µ , i.e. thatthe metric is invariant under the infinitesimal transformation x µ → x µ + κξ µ ( κ is an infinitesimalquantity). Then ξ µ satisfies ξ ν ; µ + ξ µ ; ν . (13)Demanding the existence of one Killing vector field will lead to conditions on three of the free functions.We are only interested in the three free functions S, P, Q , because the other two are already studied inthe LT literature. We assume that f, M and fM / are not constant , (14)and also S, P, Q do not depend on
M, f, t B . (15)We follow the calculation given in Bonnor, Suleiman & Tomimura (BST) [19] with two exceptions.First, we want to keep ǫ general, whereas they specified ǫ = +1 . Second, we insist that exactly oneKilling vector field exists, whereas [19] show that for general S, P, Q there is no Killing vector and thusno symmetry. Adopting the notation of BST, let us write the Szekeres metric (1) in the general form d s = − d t + e λ d r + e ω (d p + d q ) (16)where λ = λ ( t, r, p, q ) , ω = ω ( t, r, p, q ) . More explicit forms of the metric functions will be used afterthe equations have been simplified.Let us now assume a non-vanishing Killing vector field of the form ξ a = ( δ, α, β, γ ) , (17)where each component is an unspecified function of all four coordinates. Appendix A collects therelevant parts of the argumentation in [19], generalising it to all ǫ , and here we just note the key BST assumed W and M are linearly independent, which suffices for ǫ = +1, but not in general. The relation f ∝ M / plus t B = constant is the FLRW case. esults. If we assume the LT functions M and f are arbitrary, we are left with a Killing vector of theform ξ a = (cid:0) , , β ( p, q ) , γ ( p, q ) (cid:1) . (18)It is remarkable that a Killing vector field in a Szekeres model can have no r -component, and its twonon-vanishing components must be independent of r . In these coordinates the Killing vector field looksthe same for all r, t . Now, let us find the Killing vector field and thus the restriction on the functions S, P, Q .In BST’s derivation of (18), four of the Killing equations were not fully used and still have infor-mation that we can further exploit. Let us rewrite the remaining Killing equations rr -component β λ , + γ λ , = 0 , (19) pp -component β , + 12 (cid:0) β ω , + γ ω , (cid:1) = 0 , (20) qq -component γ , + 12 (cid:0) β ω , + γ ω , (cid:1) = 0 , (21) pq -component β , + γ , = 0 , (22)where λ , = ∂λ/∂p , ω , = ∂ω/∂q , etc. It can be seen from (22) and the result of subtracting (21)from (20), β , − γ , = 0 , (23)that β and γ are conjugate harmonic functions.We know that if λ , = 0 and λ , = 0 we have a global spherical ( ǫ = 1 ), planar ( ǫ = 0 ) orpseudo-spherical ( ǫ = − ) symmetry, since the rr component of the metric loses its dependency onthe coordinates p, q . The geometry of the 2 dimensional submanifolds is passed on to the 3 dimensionalspace. We cannot have only one of the two λ , λ vanishing everywhere, because it would result in ǫS ′ = P ′ = Q ′ , hence E ′ /E = 0 , and so the other of the two would also vanish.We now assume that both λ and λ are non-zero. Then equation (19) gives us a way to obtain γ from β , and convert (20) into an exponential differential equation in β alone, β , + 12 (cid:18) ω , + λ , λ , ω , (cid:19) β . (24)In order to derive the Killing vector explicitly, we use the abbreviations Y = p − P , Z = q − Q , A = 2 SE = Y + Z + ǫS , β = e H , (25)and write the solution as H = Z ∂ ββ d p = − Z (cid:18) ω , + λ , λ , ω , (cid:19) d p (26) = − Z (cid:18)
Y Z ′ − Y ′ ZZ A ′ − Z ′ A (cid:19) d p = Z ∂ (cid:0) Z A ′ − Z ′ A (cid:1)(cid:0) Z A ′ − Z ′ A (cid:1) d p (27) ⇒ β = h (cid:0) Z A ′ − Z ′ A (cid:1) , (28) When we say that a function is non-zero, we mean that it does not vanish everywhere, although there mightbe a submanifold where this function vanishes. here h ( r, q ) is a function of integration independent of p . Similarly (19) in (21) gives γ , + 12 (cid:18) λ , λ , ω , + ω , (cid:19) γ (29) ⇒ γ = g (cid:0) Y A ′ − Y ′ A (cid:1) , (30)where g ( r, p ) is a function of integration independent of q . Putting (28) & (30) in (19) we find − βγ λ , λ , = h ( r, q ) g ( r, p ) (31)so clearly h = h ( r ) = − g ( r ) . (32)We find the Killing vector components to be β = h h Z (cid:0) − Y P ′ − Z Q ′ + ǫ S S ′ (cid:1) + Q ′ (cid:0) Y + Z + ǫ S (cid:1)i (33) = h h Q ′ ( p − q ) − P ′ p q + 2( Q P ′ − P Q ′ ) p + 2( P P ′ + Q Q ′ + ǫ S S ′ ) q − P Q P ′ + ( P − Q + ǫ S ) Q ′ − Q ǫ S S ′ i (34) γ = − h h Y (cid:0) − Z Q ′ − Y P ′ + ǫ S S ′ (cid:1) + P ′ (cid:0) Y + Z + ǫ S (cid:1)i (35) = − h h P ′ ( q − p ) − Q ′ p q + 2( P Q ′ − Q P ′ ) q + 2( P P ′ + Q Q ′ + ǫ S S ′ ) p − P Q Q ′ + ( Q − P + ǫ S ) P ′ − P ǫ S S ′ i . (36)The function h is constrained by the fact that the Killing vector components are independent of r asrequired by (18): β ′ = 0 = γ ′ . Therfore we must have the coefficients of the different powers of p & q in β ′ & γ ′ vanish. A short calculation gives us the following three conditions: h ′ P ′ + hP ′′ = 0 , (37) h ′ Q ′ + hQ ′′ = 0 , (38) h ′ ǫSS ′ + 2 hǫSS ′′ + 2 h ( P ′ + Q ′ + ǫS ′ ) = 0 . (39)After integrating, we find hP ′ = c p , (40) hQ ′ = c q , (41) h ( ǫSS ′ + P P ′ + QQ ′ ) = c s , (42)where c p , c q , c s are constants of integration. Also note that h = 0 or we would have vanishing Killingvector components (33) and (35).From these we get P = c p Z d rh + c p , Q = c q Z d rh + c q , ǫS = − P − Q + 2 c s Z d rh + c s , (43)which shows there is only one free function between S , P and Q .So far we have found the Killing vector field for a Szekeres model, equations (28) and (30), andthree conditions that constrain h , S , P , and Q , (40)-(42). Let us be more explicit and consider 3different cases: ase 1: P ′ = 0 = Q ′ . The first two conditions, (40) and (41), are trivial. The third, (42), canbe inserted in β, γ for ǫ = ± . There is no further constraint on S . If ǫ = 0 , P ′ = Q ′ = 0 leads to λ , = λ , = 0 , which we excluded as the full symmetric model. Case 2:
Let one of P ′ , Q ′ be zero and the other one nonzero, say P ′ = 0 , Q ′ = 0 . Then we canuse the first condition and find h = c p /P ′ . The third condition then gives ǫSS ′ = − P P ′ + ( c s /c p ) P ′ and after integrating ǫS = − P + 2 c P + c . (44)If ǫ = 0 we find P ′ ( c − P ) and we must have P ′ = 0 contrary to the assumption of case 2. Thereis no case 2 Killing vector for ǫ = 0 .Equivalently if Q ′ = 0 , P ′ = 0 we find ǫS = − Q + 2 c Q + c and no Killing vector for ǫ = 0 . Case 3:
Let both P ′ , Q ′ = 0 . Then the first two conditions give us h = c q Q ′ = c p P ′ and therefore Q ′ = cP ′ with c = c q c p . The second and third conditions then become, after renaming the constants, Q = cP + c Q , ǫS = − (1 + c ) P + 2 c P + c . (45)If ǫ = 0 we find from the first two conditons Q ′ = c q c p P ′ and the third leads to Q ′ = − c p c q P ′ and thus c p = c q = 0 . There is no case 3 Killing vector for ǫ = 0 .If ǫ = ± we can choose S , P , & Q as described in cases 1, 2, or 3, and find a Szekeres model withone Killing vector field. Examples of such a Killing vector field can be found in figure 2. If ǫ = 0 wecan only have full symmetry or no symmetry. In particular there is no globally axial symmetric modelfor ǫ = 0 apart from full symmetry. (a) Case 1 (b) Case 2, Q ′ = 0 (c) Case 2, P ′ = 0 (d) Case 3 Figure 2: Examples of Killing vector fields for ǫ = ± . The plots show the stream lines in the ( p, q ) plane. The Killing vector field is the same for every constant ( t, r ) surface in a given symmetric model.It is clear that cases 2 and 3 are equivalent; they can be transformed to one another by a simple rotationof the p, q planeRefs [54, 46] showed these cases can be transformed into each other. Conditions (45) are the sameas eqs (B8) & (B11) of [46]. We now show that the fixed points coincide with the dipole extrema. The extrema of the dipolefunction E ′ /E are given by (7)-(9) which hold generally, even if no symmetry exists. The fixed pointsof a Killing vector field are the loci ( p f , q f ) where ξ a = 0 ⇒ β = 0 = γ . (46) he existence of a Killing vector field demands that conditions (40)-(42) hold, so we consider the abovelist of cases. Case 1: P ′ = 0 , Q ′ = 0 . We find ( p e , q e ) = ( P, Q ) = ( p f , q f ) . (47) Case 2: P ′ = 0 , Q ′ = 0 . Expressions for ( p f , q f ) result from case 3 by setting c = 0 . Then P ′ = 0 , Q ′ = 0 results from interchanging P with Q , i.e. a rotation of the p, q plane by ◦ . Case 3:
Both P ′ , Q ′ = 0 . In this case Q = cP + c Q and S = 1 ǫ ( − (1 + c ) P + 2 c P + c ) . (48)Then we find the curves q ( p ) that have β = 0 and γ = 0 as follows, q β = 1 c (cid:18) ± q (1 + c ) p − c p + c c + c − p + cc Q + c (cid:19) (49) q γ = ± p (1 + c ) p − c p − c + cp + c Q . (50)There are two pairs of intersection points for these two curves: p f = c ± p (1 + c ) c + c c + 1 and q f = c Q + c (cid:0) c ± p (1 + c ) c + c (cid:1) c + 1 , (51) p f = c ± c p − (1 + c ) c − c c + 1 and q f = c Q + cc ∓ p − (1 + c ) c − c c + 1 . (52)Depending on the choice of constants, one of the two pairs is complex and the other is the pair of fixedpoints. Using the case 3 conditions (48) we find with E ′ /E | e = ± P ′ ǫS √ d , d := c + (1 + c ) c ≥ (53) ⇒ ( p e , q e ) = (cid:18) P − ǫS − (1 + c ) P + c ± √ d , cP + c Q − c ǫS − (1 + c ) P + c ± √ d (cid:19) = ( c ∓ √ d )(1 + c ) , c ( c ∓ √ d )(1 + c ) + c Q ! = ( p f , q f ) . (54)The points of extrema coincide with the fixed points if d ≥ ( E ′ /E | e does not exist for d ≤ ). Itfollows that the contours of E ′ /E give the congruence of an axial Killing vector field, and we can saythat E ′ /E is the symmetry of an axi-symmetric Szekeres model. The aim of this section is firstly to determine whether a model with a single symmetry is axiallysymmetric or otherwise and secondly to derive the restrictions on the ranges of the functions S , P & Q . The three cases that we found in section 3.1 depend on the choice of the sole free function in E fora model with a single symmetry. Case 1 uses S as the free function, the other cases use either P or Q as the free funciton. The latter two cases are equivalent in the sense that a rotation of the p, q plane a) Example for Case 1 (b) Example for d > d = 0 (d) Example for d < Figure 3: Examples of four possible types of Killing vector fields. The plots show the flow in the ( p, q ) plane. The d refers to the distance between the rotational fixed points and is defined in (53). The fourpictures show a Killing vector field for different choices of constants c, c , c , c Q .will bring us from one to another. The really interesting difference between Killing vector fields arisesfrom the choice of the constants c, c , c , c Q . c is a rotation of the coordinate system and choosing c = 0 results in the Killing vector field aligning with the grid ( p, q ) . c Q is a simple tranlation of thecoordinate system. We can set c = c Q = 0 without loss of generality. However, d as defined in (53)plays a crucial role. As shown in figure 3, d determines the type of the Killing vector field. Case 1 canbe interpreted as having d = ∞ .Now we go back and consider the different ǫ models separately. The Killing vector fields are blindto ǫ , as can be seen in the explicit equations for the Killing vector components in Appendix B. But forevery ǫ we have to read the Killing vector field differently. As we will see now, the four types of Killingvector fields are not all realised for each ǫ , with the exception of Case 1, d = ∞ , which represents thesimplest kind of rotation.In order to keep the equations simple we will assume case 2 ( Q ′ = 0 ). Then the fixed points of theKilling vector field become ( p f , q f ) = (cid:16) c ± √ d , c Q (cid:17) for d ≥ , (55) ( p f , q f ) = (cid:16) c , c Q ± √− d (cid:17) for d ≤ . (56)The line p = c corresponds to a circle with infinite radius. If d ≥ the two fixed points are symmetricwith respect to this line (see figure 3b). If d ≤ the two fixed points lie on this line. Equations (55)and (56) show that the distance between the fixed points is √ d . The three different types of Killingvector fields in figure (3b),(3c),(3d) are due to d > , d = 0 and d < respectively. The condition(44) can be written as ǫS = − ( P − c ) + c + c . (57)Since S > , this leads to range restrictions on P and S . ǫ = 1 ; We can only have Killing vector fields like figure (3a) and (3b).The reason is simply that wewould find a S < , if d = 0 or if d < . The two types of Killing vector fields are equivalent in thesense that a conformal coordinate transformation of the p, q plane can transform one into the other.They are axially symmetric models, because any Killing vector field on a sphere with two fixed points isaxial symmetric. (in figure 3a the second fixed point is at infinity, i.e. at the north pole of the sphere.) he ranges of the functions P, S are bounded by c − √ d < P < c + √ d, S ≤ d . (58)If we think of p, q as coordinates on the sphere, the point ( P, Q ) corresponds to the south pole of thesphere. It can only be between the fixed points on the line joining them. Intuitively we would haveguessed that, since the fixed points must be antipodal on the sphere. ǫ = − ; The quasi-hyperboloidal models can have all four types of Killing vector fields. Figures3a and 3b are hyperbolic rotations, figure 3d is a hyperbolic translation, and 3c is a limit-rotationor horolation . A hyperbolic 2-surface can be represented by the Poincar´e disk on the p, q plane (orequivalently by the upper of a two-sheeted hyperboloid). The center of the Poincar´e disk (or the“south pole” of the hyperboloid sheet) is ( P, Q ) , the radius of the Poincar´e disk is proportional to S (the projection height). If d > we find ( P, Q ) on the line of the two fixed points but outside them,not in between. There is one fixed point on each sheet of the hyperboloid. If d ≤ the fixed pointsare at infinity, which is represented by the edge of the Poincar´e disk. d > P < c − √ d or P > c + √ d, S > , (59) d = 0 P = c , S > , (60) d < P unbounded , S ≥ − d . (61)Thus we can only speak of axial symmetry if the conditions (59) hold. Conditions (60) & (61) violate(11), and only apply in models with shell crossings somewhere. ǫ = 0 ; For purely ǫ = 0 models, there is either no symmetry or full planar symmetry. We cannotconstruct such a model with a single symmetry.We have seen that the Killing vector types of figures 3d & 3c can only occur for ǫ = − , whilethose of 3a & 3b can occur for either ǫ = +1 or − . It therefore seems possible that the latter twotypes can occur for a single ǫ = 0 worldsheet with ǫ flipping across it. Nolan and Debnath [54] found that the existence of a radial null geodesic in an ǫ = +1 model demandsthe existance of a Killing vector field. Their work was generalised to all ǫ by Krasi´nsky and Bolejko[46], who also found that only this family of symmetric Szekeres models has repeatable light rays.The equation following (31) in [54] shows that along a radial geodesic λ , (cid:12)(cid:12)(cid:12) g = λ , (cid:12)(cid:12)(cid:12) g = 0 , (62)which is true for all ǫ as they use the most general form of the geodesic equations. From our perspectivethis result is not surprising because λ , ∝ ( ZA ′ − Z ′ A ) = βh and λ , ∝ ( Y A ′ − Y ′ A ) = γh . (63)Thus λ , , λ , must vanish at points ( p, q ) where β, γ vanish. Furthermore, [54] show that their Q = 0 and Q = 0 is equivalent to (62). Comparing with (33) and (35) shows that Q = γ and Q = β . Since the orbits are open, this is technically a translation. = 0 is special in this regard. We can find a radial geodesic but not a single Killing vectorfield. The reason is, that if a radial geodesic exists then, according to [54], we can find a coordinatetransformation so that P, Q = 0 . And that, for ǫ = 0 , leeds to λ , λ = 0 everywhere. We find the fullplanar symmetry and not just a single symmetry.To sum up, a radial null geodesic exists if and only if β and γ are not globally vanishing, i.e. thereexists a Killing vector. The locus of fixed points coincides with the path of the radial geodesic.In fact, it is a general theorem that the locus of fixed points of a symmetric space(time) is a totallygeodesic sub-space; see p224 of [31], or theorem 1.3 of [50], or p 21 of [41]. In our case, the geodesicsin the subspace are, the symmetry axis on each time slice, the timelike worldlines, and the null geodesicsalong the axis. In the case of axial symmetry, we expect that the symetry axis should be “straight” in the 3-d space;that is, its tangent vector should obey the 3-d geodesic equation. Obviously it can’t obey the 4-dgeodesic equation, since it is well known that even for the RW metric the spacelike geodesics do notlie in constant t ( p e ( r ) , q e ( r )) that is aspatial geodesic.Along the locus of extrema, p e & q e as defined in (8) and (9) ensure that (cid:0) EE ′ p − E ′ E p (cid:1) | e = 0 = (cid:0) EE ′ q − E ′ E q (cid:1) | e (64)Writing the geodesic tangent vector as V a = [ k ( v ) , ℓ ( v ) , m ( v ) , n ( v )] , where v is a parameter, the4-d geodesic equations in the Szekeres metric, and the spacelike condition are: k d v + ℓ W (cid:18) R ′ − RE ′ E (cid:19) ˙ R ′ − ˙ RE ′ E ! + ( m + n ) R ˙ RE , (65) ℓ d v + 2 ℓk ( ˙ R ′ − ˙ RE ′ /E )( R ′ − RE ′ /E ) + ℓ (cid:18) ( R ′′ − RE ′′ /E )( R ′ − RE ′ /E ) − E ′ E − W ′ W (cid:19) + 2 ℓR (cid:18) m ( E ′ E p − EE ′ p ) + n ( E ′ E q − EE ′ q ) E ( R ′ − RE ′ /E ) (cid:19) − ( m + n ) RW E ( R ′ − RE ′ /E ) , (66) m d v + 2 km ˙ RR + ℓ RW (cid:18) R ′ − RE ′ E (cid:19) (cid:0) EE ′ p − E ′ E p (cid:1) + 2 ℓmR (cid:18) R ′ − RE ′ E (cid:19) − ( m − n ) E p + 2 mnE q E , (67) n d v + 2 kn ˙ RR + ℓ RW (cid:18) R ′ − RE ′ E (cid:19) (cid:0) EE ′ q − E ′ E q (cid:1) + 2 ℓnR (cid:18) R ′ − RE ′ E (cid:19) − ( n − m ) E q + 2 mnE p E , (68) +1 = − k + ℓ W (cid:18) R ′ − RE ′ E (cid:19) + ( m + n ) R E . (69)The 3-d geodesics equations are obtained by setting k = 0 = ˙ R = ˙ R ′ . (70) e will now show that, provided the non-spherical arbitrary functions S , P & Q have no dependenceon the spherical arbitrary functions f , M & t B , then the locus of dipole extrema is only geodesic if itis along constant ( p, q ) .Since p e & q e are functions of r only, we write the tangent vector to this locus as V b = [0 , , p ′ e ( r ) , q ′ e ( r )] ℓ ( r ) . (71)Along the pole locus, (64) & (71) reduce the 3-d geodesic equations, (66)-(69) with (70), to ℓ (cid:26) ℓ ′ ℓ + ( R ′′ /R − E ′′ /E )( R ′ /R − E ′ /E ) − E ′ E − W ′ W − ( p ′ e + q ′ e ) W E ( R ′ /R − E ′ /E ) (cid:27) , (72) ℓ (cid:26) p ′′ e + p ′ e (cid:20) ℓ ′ ℓ + 2 (cid:18) R ′ R − E ′ E (cid:19)(cid:21) − ( p ′ e − q ′ e ) E p + 2 p ′ e q ′ e E q E (cid:27) , (73) ℓ (cid:26) q ′′ e + q ′ e (cid:20) ℓ ′ ℓ + 2 (cid:18) R ′ R − E ′ E (cid:19)(cid:21) − ( q ′ e − p ′ e ) E q + 2 p ′ e q ′ e E p E (cid:27) , (74) +1 = ℓ R (cid:26) ( R ′ /R − E ′ /E ) W + ( p ′ e + q ′ e ) E (cid:27) . (75)Now E and its derivatives depend on r through S , P & Q , as do p e & q e , while R ′ /R depends on r through f , M & t B . But, both (72) and (75) show ℓ ′ /ℓ = − R ′ /R − E ′ /E ) , even if p e = 0 = q e .Consequently, equations (73) & (74) show that for ( p e , q e ) to be geodesic, either E ′ /E depends on R ′ /R , which we have excluded, or p ′ e | geodesic = 0 = q ′ e | geodesic . (76)Therefore the only way that the locus of dipole extrema can be made geodesic without putting re-strictions on the spherical arbitrary functions, is if p e and q e are constant. Setting (8) & (9) constantresults in the same set of cases and restrictions on S , P & Q as in section 3.1. The set of axi-symmetric Szekeres models found in section 3.1 are known to be equivalent [54, 46].Do there exist equivalences between general Szekeres models?The non-sphericity or non-pseudo-sphericity of a Szekeres model could be characterised by thevariation of its dipole strength E ′ /E | e , and the path of its dipole extrema, p e & q e , all of whichare functions of r only. We conjecture that two Szekeres models, with distinct sets of non-sphericalfunctions ( S, P, Q ) , are equivalent if one can be “rotated” (in the spherical or hyperboloidal sense) intothe other. Any coordinate transformation of a given metric is necessarily the same physical spacetime.Therefore we are looking for the most general coordinate transformation that preserves the Szekeresform. We already know there is a rescaling freedom in the r coordinate. Since we must retain theforms of (5), (6) & E ′ /E , we are looking for transformations of the ( p, q ) coordinates only which are r -independent. Indirectly, this will result in a transformation of S , P , Q .The symmetries of constant curvature surfaces have been well studied. The transformations thatpreserve the 2-d metric form (5) and (6) are composed of equatorial rotations (Haantjes transforma- The general transformation is often written in the form of a Mobius tranformation,(˜ p + i ˜ q ) = k ( p + iq ) + lm ( p + iq ) + n , (77) where k , l , m & n are complex constants obeying kn − lm = 1. ions) T = 1 + 2 D p + 2 D q + ( D + D )( p + q ) , ˜ p = p + D ( p + q ) T , ˜ q = q + D ( p + q ) T , (78)polar rotations ˜ p = F p − F q p F + F , ˜ q = F p + F q p F + F , (79)inversions ˜ p = − p , ˜ q = q , (80)magnifications ˜ p = µp , ˜ q = µq , (81)and displacements ˜ p = p + p , ˜ q = q + q . (82)in any combination.In order for any of these transformations to preserve the form of the full Szekeres metric (1), werequire that each of (d p + d q ) /E and E ′ /E are invariant under p → ˜ p , q → ˜ q , S → ˜ S , P → ˜ P , Q → ˜ Q . (83)This will result in an associated transformation of S , P , & Q .As an example, we consider an equatorial rotation, and apply (83) plus (78) to (d p + d q ) /E with general E . Simplifying, we get (d˜ p + d˜ q )˜ E → S (d p + d q ) F = (d p + d q ) E , (84)where F = h ( p + q ) − (cid:2) p ˜ P + q ˜ Q + ( p + q )( D ˜ P + D ˜ Q ) (cid:3) + (cid:2) D p + 2 D q + ( p + q )( D + D ) (cid:3) ( ˜ P + ˜ Q + ǫ ˜ S ) i . (85)Since we must have E = F/ (4 ˜ S ) , we compare coefficients of powers of p & q to obtain − D ˜ P + D ˜ Q ] + ( D + D )[ ˜ P + ˜ Q + ǫ ˜ S ]2 ˜ S = 12 S − ˜ P + D [ ˜ P + ˜ Q + ǫ ˜ S ]˜ S = − PS − ˜ Q + D [ ˜ P + ˜ Q + ǫ ˜ S ]˜ S = − QS [ ˜ P + ˜ Q + ǫ ˜ S ]2 ˜ S = S (cid:18) P S + Q S + ǫ (cid:19) , (86) hich gives us the transformation of S , P & Q under an equatorial rotation: U = 1 + 2 D P + 2 D Q + ( D + D )( P + Q + ǫS ) , ˜ S ER = SU , ˜ P ER = P + D ( P + Q + ǫS ) U , ˜ Q ER = Q + D ( P + Q + ǫS ) U . (87)It may be confirmed that the form and value E ′ /E is similarly preserved by the above.Following a similar process for the other transformations, we find ˜ S P R = S , ˜ P P R = ( F P − F Q ) p F + F , ˜ Q P R = ( F Q + F P ) p F + F , (88) ˜ S I = S , ˜ P I = − P , ˜ Q I = Q , (89) ˜ S M = µS , ˜ P M = µP , ˜ Q M = µQ , (90) ˜ S D = S , ˜ P D = p + P , ˜ Q D = q + Q . (91)One may also write down some discrete equivalences, such as (˜ p, ˜ q, ˜ P , ˜ Q ) = ( q, p, Q, P ) , (92) (˜ p, ˜ q, ˜ P , ˜ Q ) = ( p, − q, P, − Q ) , (93)but these are combinations of inversions (80)+(89) and specific polar rotations (79)+(88). Since thesign of S does not affect the metric (provided S is never zero), there is also the trivial equivalence S → − S . (94)Therefore equivalent Szekeres metrics, whether or not they have symmetries, are related by (83),with (78) & (87) or (79) & (88) or (80) & (89) or (81) & (90) or (82) & (91), and these may becombined in the obvious way. A similar list of ( p, q ) transformations with their effects on S , P , & Q , isgiven in [46] equations (B12)-(B20), though magnifications appear to be missing. The transformations(77) & (78) with (82) were used in [54, 12, 46] to show that certain axially symmetric models areequivalent. However, (77) does not provide the relationships between the two sets of S , P , & Q functions in equivalent models. We have considered the conditions under which symmetries do and don’t exist in Szekeres metrics, byattempting to solve the Killing equations and finding all circumstances that make this possible. he theorem of Bonnor, Suleiman and Tomimura, that the Szekeres metric has no Killing vectorsif the free functions f , M , t B , S , P , Q are linearly independent, is easily extended from ǫ = +1 to all ǫ values. Consequently, it is immediately obvious that restrictions on the free functions are needed forKilling vectors to exist.By insisting a solution to the Killing equations exists, the general solution forms (33)-(36) werefound, which are valid for all ǫ , if they can be made independent of the r coordinate. However, theresulting restrictions (40)-(42) on the functions S , P and Q , divide into 3 cases, which restrict theirpossible ranges. In each non-trivial case, S , P and Q depend on a single free function, as shown by(43), and in the simplest case this is obvious. The exact nature of the restrictions depends on the valueof ǫ .The ǫ = +1 Szekeres models are foliated by ( p, q ) t, r shell.Szekeres models with ǫ = − [34] are foliated by ( p, q ) E ′ /E on each ( p, q ) ǫ = 0 Szekeres models, the foliating 2-surfaces are planes. It turns out that the functionalrestrictions force full planar symmetry, so these models cannot have just one symmetry. Interestingly,they cannot be globally free of shell crossings unless there’s full planar symmetry.While the set of axially symmetric solutions was already known [54, 46], as the answer to two otherquestions, the full set of single-symmetry Szekeres models had not been looked for, and it wasn’t knownif the set was complete, till now. The same references also showed the 3 cases for each ǫ = 0 areequivalent since they can be transformed into each other.There is a theorem that in any spacetime with a symmetry, the locus of fixed points is a totallygeodesic submanifold. In particular, with axial symmetry, the locus of fixed points, that is the axis,must be “straight”, i.e. geodesic in the constant t f , M and a , and their S , P & Q are related by one of theseequivalences, or a combination of them, then they are physically equivalent, and can be transformedinto each other. The list in [46] was not quite complete. A BST Revisited
We here review the salient part of [19] and comment on generalising to ǫ = 1 . Assuming a non-vanishing Killing vector field of the form (17), we find the Killing vector equations (13) for metric (16)to be α , + 12 (cid:0) λ , α + λ , β + λ , γ + λ , δ (cid:1) = 0 (95) e λ α , + e ω β , = 0 (96) e λ α , + e ω γ , = 0 (97) δ , − e λ α , = 0 (98) β , + 12 (cid:0) ω , α + ω , β + ω , γ + ω , δ (cid:1) = 0 (99) β , + γ , = 0 (100) δ , − e ω β , = 0 (101) γ , + 12 (cid:0) ω , α + ω , β + ω , γ + ω , δ (cid:1) = 0 (102) δ , − e ω γ , = 0 (103) δ , = 0 . (104)The comoving matter flow lines are geodesic, so their Lie derivative with respect to ξ a must vanish ξ i ; k u k − ξ k u i ; k = 0 = ξ i, , (105)showing ξ a is not dependent on the time coordinate. From (98), (101), (103) and (104) we find that δ , = δ , = δ , → δ is constant . (106)For each of (96) and (97), multiplying by e − ω , taking the derivative with respect to t , and applying(96) gives α , ( ω , − λ , ) e λ − ω = 0 = α , ( ω , − λ , ) e λ − ω , (107)but e λ − ω = 0 corresponds to a singularity in the metric and the mass density (4). Furthermore, ω , − λ , = RR ′ − RE ′ /E ω , = 2 RR ′ − RE ′ /E ˙ RR ! ′ (108)and ω , = 0 corresponds to a FLRW model, because it results in R ( t, r ) = a ( t ) · Φ( r ) , which (14)disallows. Thus, by (107), (96) and (97) we have α , = 0 = α , , and β , = 0 = γ , . (109)We next take the r derivative of (99), apply (106) & (109), (cid:0) ω , α , + ω , α + ω , β + ω , γ + ω , δ (cid:1) , (110)and substitute for α , from (95) α , = − (cid:0) λ , α + λ , β + λ , γ + λ , δ (cid:1) , (111)to get α (cid:18) ω , − λ , ω , (cid:19) + β (cid:18) ω , − λ , ω , (cid:19) + γ (cid:18) ω , − λ , ω , (cid:19) + δ (cid:18) ω , − λ , ω , (cid:19) . (112) oting that (cid:18) ω , − λ , ω , (cid:19) = − (cid:18) R ′ R − E ′ E (cid:19) (cid:18) R ′ R − W ′ W (cid:19) (113) (cid:18) ω , − λ , ω , (cid:19) = 0 = (cid:18) ω , − λ , ω , (cid:19) (114) (cid:18) ω , − λ , ω , (cid:19) = − RR (cid:18) R ′ R − E ′ E (cid:19) , (115)this reduces to − (cid:18) R ′ R − E ′ E (cid:19) R (cid:26) α (cid:18) R ′ − R W ′ W (cid:19) + δ ˙ R (cid:27) . (116)Again we don’t allow e λ − ω = ( R ′ R − E ′ E ) ( EW ) to vanish, in order to avoid singularities, and therefore α (cid:18) R ′ − R W ′ W (cid:19) + δ ˙ R . (117)After a derivation with respect to t and multiplying with ˙ R , we can use the Friedman equation (3) andits t and r derivatives to eliminate ˙ R , ¨ R, ˙ R ′ . In addition we use Equation (117) to eliminate δ . Then, α (cid:18) − ǫ W ′ W − M ′ R + 3 W ′ W MR (cid:19) . (118)Since R is time dependent and M, W are not, the bracket term can only vanish if the following twoequations are satisfied simultaneously: ǫ W ′ W = 0 , M ′ − M W ′ W = 0 . (119)At this point BST assumed ǫ = +1 in order to show that α = 0 . However the result follows for all ǫ , if we alter the assumption regarding M, f slightly. For ǫ = 0 we must have W ′ = 0 and thus also M ′ = 0 . Clearly this is not an interesting model , and by assumption (14), we exclude this choice ofspecial LT-functions.For ǫ = 0 we only have M ′ M = 3 W ′ W ⇒ M = const · f / , (120)which we again exclude by (14). Therefore we must have α = 0 , and from equation (117) it followsthat δ = 0 . The Killing vector now has the form (18), which is used in section 3.1.BST go on to show that either there must be a linear relation between the three functions / (2 S ) , − P/ (2 S ) , − Q/ (2 S ) , or the Killing vector must vanish everywhere. This means a necessary conditionfor a non-vanishing Killing vector field is S ) (cid:0) σ − σ P − σ Q (cid:1) ⇒ P = cQ + c Q , (121) If M ′ = 0 we have a vacuum model, which forces spherical symmetry to avoid shell crossings [33]. Note that fM / =const. allows for a constant t -component δ and an r -component α ( r ) of the Killing vector. here σ i , c , c Q are constants, not all zero. This condition emerged in (43) and (45). Remark:
Strictly speaking, BST only proved their theorem for ǫ = 1 , but they used generalexpressions for most of their proof, so it is not difficult to generalise it. They used the restriction ǫ = 1 on only two occasions. First, it is used to show that in their notation a ( r ) = 0 , which isneeded in deriving their eq (4.22). But this is true for all ǫ models, because there exists a coordinatetransformation in the p, q surface so that a = 0 even if a = 0 initiallly. The proof can be found inPlebanski & Krasinski [58]. Because of this, we are able to write a = S and our notation is welldefined. Second, as mentioned above, they used ǫ = 1 to show δ = 0 = α using (119). A slightcorrection in the assumption regarding M, f will result in the theorem being true for all ǫ . The theoremstatement for all ǫ then is: Generalised BST Theorem
Consider a Szekeres space-time (of class I) satisfying the followingconditions e λ − ω = 0 , (122) R = 0 , E = 0 , W = ǫ + f = 0 , (123) ω , = 0 , (124)for ǫ = ± : M ′ = 0 , W ′ = f ′ = 0 , (125)for ǫ = 0 : fM / = constant . (126) S , − P S , − Q S are linearly independent , (127)then ξ µ = 0 except possibly on isolated submanifolds of the space-time. In the first 3 lines above, zerosmay occur at restricted loci, but not in general. B Details of the Cases
We here collect the results for the cases listed in section 3.
Case 1: P ′ = 0 , Q ′ = 0 .Function conditions P, Q constant (128)Killing field β = 2 c s ( q − Q ) (129) γ = − c s ( p − P ) (130)Fixed points p f = P , q f = Q (131)There is only one fixed point in the ( p, q ) plane, the other is the circle at infinity. The Killing vectorfield consists of concentric circular orbits, as in figure 3a. Case 2a: P ′ = 0 , Q ′ = 0 .Function conditions ǫS = − Q + 2 c Q + c (132)Killing field β = c q (( p − P ) − q + 2 c q + c ) (133) γ = 2 c q ( p − P )( q − c ) (134) ixed points p f = P , q f = c ± q c + c , c ≥ − c (135)or p f = P ± q − c − c , q f = c , c ≤ − c , ǫ = − (136)The first fixed point pair lies on the vertical line p = P , and the second pair lies on the circle Y + Z = Q − c Q − c = S . Case 2b: P ′ = 0 , Q ′ = 0 .Function conditions ǫS = − P + 2 c P + c (137)Killing field β = 2 c p ( q − Q )( c − p ) (138) γ = c p ( p − ( q − Q ) − c p − c ) (139)Fixed points p f = c ± q c + c , q f = Q , c ≥ − c (140)or p f = c , q f = Q ± q − c − c , c ≤ − c , ǫ = − (141)The Killing vector field is as above, but rotated by ◦ in the ( p, q ) plot, as in fig 3d. Case 3: P ′ = 0 , Q ′ = 0 .Function conditions Q = cP + c Q , ǫS = − (1 + c ) P + 2 c P + c (142)Killing field β = c p h ( p − q ) c − pq + 2 q ( cc Q + c ) + 2 pc Q + ( cc − cc Q − c c Q ) i (143) γ = − c p h − p + q − cpq + 2 p ( c Q + c ) + 2 qc Q + c Q + c i (144)Fixed points β = 0 ⇒ q = cc Q + c − p ± p (1 + c ) p − c p + c c + c cγ = 0 ⇒ q = cp + c Q ± p (1 + c ) p − c p − c There are two pairs of intersection points for these two curves, but one pair is complex, depending onthe choice of constants: p f = c ± p (1 + c ) c + c c + 1 and q f = c Q + c (cid:0) c ± p (1 + c ) c + c (cid:1) c + 1 , (145) p f = c ± c p − (1 + c ) c − c c + 1 and q f = c Q + cc ∓ p − (1 + c ) c − c c + 1 . (146)If ǫ = − and c < c / (1 + c ) , the second pair is real, otherwise the first pair. The first pairof fixed points are on a line of slope c that passes through ( p, q ) = (0 , , and the Killing vectorfield is like that of case 2, but rotated through angle tan − c . The second pair are on the circle Y + Z = (1 + c ) P − c P − c = S . References [1] P.S. Apostolopoulos, arXiv:1611.04569 , “Szekeres Models: A Covariant Approach”.[2] P.S. Apostolopoulos, arXiv:1611.09781 , “Intrinsic Conformal Symmetries in Szekeres Models”.
3] P.S. Apostolopoulos & J. Carot,
Internat. J. Mod. Phys. A , 1983-2006 (2007), “Uniquenessof Petrov Type D Spatially Inhomogeneous Irrotational Silent Models”. arXiv:gr-qc/0605130 .[4] J.D. Barrow & J. Stein-Schabes, Phys. Lett.
A103 , 315 (1984), “Inhomogeneous Cosmologieswith Cosmological Constant”.[5] B.K. Berger, D.M. Eardley and D.W. Olson,
Phys. Rev. D , 3086-9 (1977), “Note on theSpacetimes of Szekeres”.[6] K. Bolejko, Phys. Rev. D , 123508 (2006), “Structure Formation in the Quasispherical SzekeresModel”.[7] K. Bolejko, Phys. Rev. D , 043508 (2007), “Evolution of Cosmic Structures in Different Envi-ronments in the Quasispherical Szekeres Model”.[8] K. Bolejko, Gen. Rel. Grav. , 1737-55 (2009), “The Szekeres Swiss Cheese Model and the CMBObservations”.[9] K. Bolejko, Gen. Rel. Grav. , 1585-93 (2009), “Volume Averaging in the Quasispherical SzekeresModel”.[10] K. Bolejko, Astron. Astrophys. , A49 (2011), “Conceptual Problems in Detecting the Evolutionof Dark Energy When Using Distance Measurements”.[11] K. Bolejko & M.-N. C´el´erier,
Phys. Rev. D , 103510 (2010), “Szekeres Swiss-Cheese Modeland Supernova Observations”.[12] K. Bolejko, A. Krasi´nski, C. Hellaby & M-N. C´el´erier, Structures in the Universe by Exact Methods— Formation, Evolution, Interactions , Cambridge U P, 2010, ISBN 978-0-521-76914-3.[13] K. Bolejko, M.A. Nazer, D.L. Wiltshire,
J. Cosm. Astropart. Phys. , 06:035 (2016), “Differ-ential Cosmic Expansion and the Hubble Flow Anisotropy”.[14] K. Bolejko & R. Sussman,
Phys. Lett. B , 265-70 (2011), “Cosmic Spherical Void Via Coarse-Graining and Averaging Non-Spherical Structures”.[15] K. Bolejko & J.S.B. Wyithe,
J. Cosm. Astropart. Phys. , 02:020 (2009), “Testing the Coper-nican Principle Via Cosmological Observations”.[16] W.B. Bonnor,
Comm. Math. Phys. , 191-9 (1976), “Non-Radiative Solutions of Einstein’sEquations for Dust”.[17] W.B. Bonnor, Class. Quantum Grav. , 495-501 (1986), “The gravitational arrow of time and theSzekeres cosmological models”.[18] W.B. Bonnor & D.J.R. Pugh, South African J. Phys. , 169-72 (1987), “Szekeres’s CosmologicalModels and the Postulate of Uniform Thermal Histories”.[19] W.B. Bonnor, A.H. Sulaiman & N. Tomimura, Gen. Rel. Grav. , 549-59 (1977), “Szekeres’sSpace-Times Have No Killing Vectors”.[20] W.B. Bonnor & N. Tomimura, Mon. Not. Roy. Astron. Soc. , 85-93 (1976), “Evolution ofSzekeres’s Cosmological Models”. Errata in:
Mon. Not. Roy. Astron. Soc. , 463 (1976).
21] S. Chakraborty & U. Debnath,
Gravit. Cosmol. , 184-9 (2008), “Shell Crossing Singularities inQuasi-Spherical Szekeres Models”.[22] G.M. Covarrubias, J. Phys. A , 3023-8 (1980), “Gravitational Radiation in Szekeres’s Quasi-Spherical Space-Times”.[23] B. Datt, Zeit. Physik , 314 (1938), “ ¨Uber eine Klasse von L¨osungen der Gravitationsgle-ichungen der Relativit¨at”. Reprinted with historical introduction in:
Gen. Rel. Grav. , 1619-27(1997).[24] U. Debnath, Europhys. Lett. , 29001 (2011), “Thermodynamics in Quasi-Spherical SzekeresSpace-Time”.[25] M.M. de Souza, Rev. Bras. Fiz. , 379-87 (1985), “Hidden Symmetries of Szekeres Quasi-spherical Solutions”.[26] R.J. Gleiser, Gen. Rel. Grav. , 1039-43 (1984), “A Relation Between the Szekeres QuasisphericalGravitational Collapse Solution and the Robinson-Trautman Metrics”.[27] G.F.R. Ellis, J. Math. Phys. , 1171-94 (1967), “Dynamics of Pressure-Free Matter in GeneralRelativity”.[28] S.W. Goode & J. Wainwright, Phys. Rev. D , 3315-26 (1982), “Singularities and Evolution ofthe Szekeres Cosmological Models”.[29] S.W. Goode & J. Wainwright, Mon. Not. Roy. Astron. Soc. , 83 (1982), “Friedmann-likeSingularities in Szekeres’ Cosmological Models”.[30] C. Hellaby,
Class. Quantum Grav. , 2537-46 (1996), “The Null and KS Limits of the SzekeresMetric”. See also: Math. Reviews
97 g 83 , 4592 (July 1997).[31] S. Helgason,
Differential Geometry, Lie Groups, and Symmetric Spaces , Academic Press, 1978,isbn 0-12-338469-5.[32] C. Hellaby,
Proc. Sci.
PoS(ISFTG) , 005 (2009), “Modelling Inhomogeneity in the Universe”. arXiv:0910.0350 [gr-qc] .[33] C. Hellaby & A. Krasi´nski,
Phys. Rev. D , 084011, 1-27 (2002), “You Can’t Get ThroughSzekeres Wormholes: Regularity, Topology and Causality in Quasi-Spherical Szekeres Models”.[34] C. Hellaby & A. Krasi´nski, Phys. Rev. D , 023529, 1-26 (2008), “Physical and GeometricalInterpretation of the ǫ ≤ Szekeres Models”.[35] C. Hellaby & A. Walters,
J. Cosm. Astropart. Phys. , 12:001 (2012), “Constructing RealisticSzekeres Models from Initial and Final Data”.[36] L. Herrera, A. Di Prisco, J. Iba˜nez, J. Carot,
Phys. Rev. D , 044003 (2012), “Vorticity andEntropy Production in Tilted Szekeres Spacetimes”.[37] M. Ishak, A. Peel, Phys. Rev. D , 083502 (2012), “The Growth of Structure in the SzekeresInhomogeneous Cosmological Models and the Matter-Dominated Era”.[38] M. Ishak, J. Richardson, D. Garred, D. Whittington, A. Nwankwo, R. Sussman, Phys. Rev. D ,123531 (2008), “Dark Energy or Apparent Acceleration Due to a Relativistic Cosmological ModelMore Complex than FLRW?”.
39] P.S. Joshi, A. Krolak,
Class. Quantum Grav. , 3069-74 (1996), “Naked strong curvature sin-gularities in Szekeres space-times”.[40] R. Kantowski & R.K. Sachs, J. Math. Phys. , 443-6 (1966), “Some Spatially HomogeneousAnisotropic Relativistic Cosmological Models”.[41] L. Kennard, UCSB, ∼ wziller/math661/LectureNotesLee.pdf , accessed on 2017/01/19.[42] S.M. Koksbang & S. Hannestad, Phys. Rev. D , 023532 (2015), “Studying the Precision ofRay Tracing Techniques with Szekeres Models”.[43] A. Krasi´nski, Inhomogeneous Cosmological Models , Cambridge U P, 1997, ISBN 0 521 48180 5.[44] A. Krasi´nski,
Phys. Rev. D , 064038 (2008), “Geometry and Topology of the QuasiplaneSzekeres Model”.[45] A. Krasi´nski, arXiv:1604.02003 , “Existence of Blueshifts in Quasi-Spherical Szekeres Space-times”.[46] A. Krasinski & K. Bolejko, Phys. Rev. D , 083503 (2011), “Redshift Propagation Equations inthe β = 0 Szekeres Models”.[47] A. Krasi´nski & K. Bolejko,
Phys. Rev. D , 124016 (2012), “Apparent Horizons in the Quasi-spherical Szekeres Models”.[48] A. Krasi´nski & K. Bolejko, Phys. Rev. D , 104036 (2012), “Geometry of the Quasi-HyperbolicSzekeres Models”.[49] G. Lemaˆıtre, Ann. Soc. Sci. Bruxelles
A53 , 51-85 (1933), “L’Universe en Expansion”. Reprintedin English with historical introduction in:
Gen. Rel. Grav. , 641-80 (1997).[50] The Manifold Atlas Project, ,accessed on 2017/01/19.[51] N. Meures & M. Bruni, Phys. Rev. D , 123519 (2011), “Exact Non-Linear Inhomogeneities in Λ CDM Cosmology”.[52] P. Mishra & M.-N. C´el´erier, arXiv:1403.5229 [astro-ph.CO] , “Redshift and Redshift-Drift in
Λ = 0
Quasi-Spherical Szekeres Cosmological Models and the Effect of Averaging”.[53] P. Mishra, M.-N. C´el´erier & T.P. Singh,
Phys. Rev. D , 083520 (2012), “Redshift Drift inAxially Symmetric Quasi-Spherical Szekeres Models”.[54] B.C. Nolan & U. Debnath, Phys. Rev. D , 104046 (2007), “Is the Shell-Focusing Singularityof Szekeres Space-Time Visible?”.[55] A. Nwankwo, M. Ishak, J. Thompson, J. Cosm. Astropart. Phys. , 05:028 (2011), “Lumi-nosity Distance and Redshift in the Szekeres Inhomogeneous Cosmological Models”.[56] A. Peel, M. Ishak & M.A. Troxel,
Phys. Rev. D , 123508 (2012), “Large-Scale Growth Evolutionin the Szekeres Inhomogeneous Cosmological Models with Comparison to Growth Data”.[57] A. Peel, M.A. Troxel, M. Ishak, Phys. Rev. D , 123536 (2014), “Effect of Inhomogeneities onHigh Precision Measurements of Cosmological Distances”.
58] J. Pleba´nski, & A. Krasi´nski,
An introduction to general relativity and cosmology , Cambridge UP,2006, GET ISBN.[59] R.A. Sussman & K. Bolejko,
Class. Quantum Grav. , 065018 (2012), “A Novel Approach tothe Dynamics of Szekeres Dust Models”.[60] R.A. Sussman & I.D. Gaspar, Phys. Rev. D , 083533 (2015), “Multiple Non-Spherical Structuresfrom the Extrema of Szekeres Scalars”.[61] R.A. Sussman, I.D. Gaspar & J.C. Hidalgo, J. Cosm. Astropart. Phys. , 03:012 (2016),“Coarse-Grained Description of Cosmic Structure from Szekeres Models”. Errata in:
J. Cosm.Astropart. Phys. , E03 (2016).[62] P. Szekeres,
Comm. Math. Phys. , 55-64 (1975), “A Class of Inhomogeneous CosmologicalModels”.[63] P. Szekeres, Phys. Rev. D , 2941-8 (1975), “Quasispherical Gravitational Collapse”.[64] R.C. Tolman, Proc. Nat. Acad. Sci. U.S.A. , 169-76 (1934), “Effect of Inhomogeneity onCosmological Models”. Reprinted with historical introduction in: Gen. Rel. Grav. , 935-43(1997).[65] M. Villani, J. Cosm. Astropart. Phys. , 06:015, 1-19 (2014), “Taylor Expansion of LuminosityDistance in Szekeres Cosmological Models: Effects of Local Structures Evolution on CosmographicParameters”.[66] D. Vrba & O. Svitek,
Gen. Rel. Grav. , 1808 (2014), “Modelling Inhomogeneity in SzekeresSpacetime”.[67] J. Wainwright, J. Math. Phys. , 672-5 (1977), “Characterization of the Szekeres InhomogeneousCosmologies as Algebraically Special Spacetimes”.[68] A. Walters & C. Hellaby, J. Cosm. Astropart. Phys. , 12:001 (2012), “Constructing RealisticSzekeres Models from Initial and Final Data”. arXiv:1211.2110 [gr-qc] ..