Redshift propagation equations in the β ′ ≠0 Szekeres models
aa r X i v : . [ g r- q c ] A p r Redshift propagation equations in the β ′ = 0 Szekeres models
Andrzej Krasi´nski
N. Copernicus Astronomical Centre,Polish Academy of Sciences,Bartycka 18, 00 716 Warszawa, Poland ∗ Krzysztof Bolejko
Astrophysics Department, University of Oxford, Oxford OX1 3RH, UK † (Dated: )The set of differential equations obeyed by the redshift in the general β ′ = 0 Szekeres spacetimesis derived. Transversal components of the ray’s momentum have to be taken into account, whichleads to a set of 3 coupled differential equations. It is shown that in a general Szekeres model, and ina general Lemaˆıtre – Tolman (L–T) model, generic light rays do not have repeatable paths (RLPs):two rays sent from the same source at different times to the same observer pass through differentsequences of intermediate matter particles. The only spacetimes in the Szekeres class in which all rays are RLPs are the Friedmann models. Among the proper Szekeres models, RLPs exist only inthe axially symmetric subcases, and in each one the RLPs are the null geodesics that intersect each t = constant space on the symmetry axis. In the special models with a 3-dimensional symmetrygroup (L–T among them), the only RLPs are radial geodesics. This shows that RLPs are veryspecial and in the real Universe should not exist. We present several numerical examples whichsuggest that the rate of change of positions of objects in the sky, for the studied configuration, is10 − − − arc sec per year. With the current accuracy of direction measurement, this drift wouldbecome observable after approx. 10 years of monitoring. More precise future observations will beable, in principle, to detect this effect, but there are basic problems with determining the referencedirection that does not change. I. THE MOTIVATION
The quasi-spherical Szekeres solutions have recentlybegun to be taken seriously as cosmological models [1] –[7]. For this application, one has to know the equationsobeyed by the redshift. The corresponding equation forradial null geodesics in the Lemaˆıtre – Tolman (L–T)model [8, 9] was derived long ago by Bondi [10], see alsoRef. [11]. The generalisation to the Szekeres geometry isnontrivial because in general there are no radial geodesicsin the latter [12, 13]. Consequently, the transversal com-ponents of the ray’s momentum necessarily have to betaken into account, and a set of 3 coupled differentialequations is obtained. These equations can then be ap-plied to nonradial geodesics in the L–T model.The purpose of this paper is to derive the redshiftpropagation equations in a general Szekeres model ofthe β ′ = 0 family [11], so that they can be numericallysolved and applied in various situations.In Sec. II, the Szekeres models are introduced. In Sec.III it is pointed out that the Bondi redshift equation forradial null geodesics in the L–T model is in fact an ap-proximation, the small parameter being the period of theelectromagnetic wave. The same is true for the equations ∗ Electronic address: [email protected] † Electronic address: [email protected] General means not only quasi-spherical. The generalisation tocover the quasi-plane and quasi-hyperbolic cases is immediate,so it would not make sense to leave it out. derived here. In Sec. IV, the general equations of nullgeodesics in Szekeres models are presented. In Sec. V,the set of redshift equations for the Szekeres models is de-rived. In Sec. VI, conditions are discussed under whichlight rays between a given source and a given observerproceed through always the same intermediate matterparticles; such rays are termed “repeatable light paths”,RLPs. In Sec. VII, the equations of Secs. V and VI areapplied to general null geodesics in the L–T model and inthe associated plane- and hyperbolically symmetric mod-els. It is shown there that in these models the only RLPsare the radial null geodesics. Sec. IX is a brief summaryof the results.
II. THE SZEKERES SOLUTIONS
The Szekeres solutions [14, 15] follow when the metricd s = d t − e α ( t,r,x,y ) d r − e β ( t,r,x,y ) (cid:0) d x + d y (cid:1) , (2.1)is substituted in the Einstein equations with a dustsource, assuming that the coordinates of (2.1) are co-moving, so that the velocity field is u µ = δ µ (with( x , x , x , x ) = ( t, r, x, y )).There are two families of Szekeres solutions, depend-ing on whether β, r = 0 or β, r = 0. The first family is asimultaneous generalisation of the Friedmann and Kan-towski – Sachs [16] models. Since so far it has found nouseful application in astrophysical cosmology, we shallnot discuss it here (see Ref. [11]). After the Einsteinequations are solved, the metric functions in the secondfamily becomee β = Φ( t, r )e ν ( r,x,y ) , e α = h ( r )Φ( t, r ) β, r ≡ h ( r ) (Φ , r +Φ ν, r ) , (2.2)e − ν = A ( r ) (cid:0) x + y (cid:1) + 2 B ( r ) x + 2 B ( r ) y + C ( r ) , where the function Φ( t, r ) is a solution of the equationΦ , t = − k ( r ) + 2 M ( r )Φ + 13 ΛΦ ; (2.3)while h ( r ), k ( r ), M ( r ), A ( r ), B ( r ), B ( r ) and C ( r ) arearbitrary functions obeying g ( r ) def = 4 (cid:0) AC − B − B (cid:1) = 1 /h ( r ) + k ( r ) . (2.4)The mass density in energy units is κρ = (cid:0) M e ν (cid:1) , r e β (e β ) , r ; κ = 8 πG/c . (2.5)Whenever (cid:0) e β (cid:1) , r = 0 and (cid:0) M e ν (cid:1) , r = 0, a shell cross-ing singularity occurs. It is similar to the shell crossingsingularity in the L–T models, but with a difference. Ina quasi-spherical model a shell crossing may occur alonga circle, or, in exceptional cases, at a single point, andnot at a whole surface of constant t and r , as was thecase in the L–T models.As in the L–T model, the bang time function followsfrom (2.3): Φ Z d e Φ q − k + 2 M/ e Φ + Λ e Φ = t − t B ( r ) , (2.6)The solutions of the above equation for Λ = 0 involveelliptic functions and were first studied by Barrow andStein-Schabes [17].As seen from (2.1) and (2.2), the Szekeres models arecovariant with the transformations r = f ( r ′ ), where f ( r ′ )is an arbitrary function.The Szekeres metric has in general no symmetry,but acquires a 3-dimensional symmetry group with 2-dimensional orbits when A , B , B and C are all constant(that is, when ν, r = 0).The sign of g ( r ) determines the geometry of the 2-surfaces of constant t and r (and the symmetry of theconstant A , B , B and C limit). The geometry ofthese surfaces is spherical, planar or hyperbolic (pseudo-spherical) when g > g = 0 or g <
0, respectively.With A , B , B and C being functions of r , the surfaces r = const within a single space t = const may have dif-ferent geometries, i.e. they can be spheres in one part ofthe space and the surfaces of constant negative curvatureelsewhere, the curvature being zero at the boundary.The sign of k ( r ) determines the type of evolution; with k > k < k = 0 isthe intermediate case corresponding to the ‘flat’ Fried-mann model ( k = 0 can also occur on a 3-surface asthe boundary between a region with k > k < k ( r ) influences the signof g ( r ). Since 1 /h in (2.4) must be non-negative, wehave the following: With g > g = 0 (planegeometry), k must be non-positive (only parabolic or hy-perbolic evolutions are allowed), and with g < k must be strictly negative, so only thehyperbolic evolution is allowed.The Friedmann limit follows when Φ( t, r ) = Φ ( r ) S ( t ).No further specialization of the Szekeres functions isneeded; the limiting Friedmann model is represented inthe little-known Goode – Wainwright [18] coordinates,see also Ref. [19].The Szekeres models are subdivided according to thesign of g ( r ) into the quasi-spherical (with g > g = 0) and quasi-hyperbolic ones ( g < g ≤ g .The quasi-spherical model may be imagined as a gen-eralisation of the L–T model in which the spheres of con-stant mass are made non-concentric. The functions A ( r ), B ( r ) and B ( r ) determine how the centre of a spherechanges its position in a space t = const when the ra-dius of the sphere is increased or decreased [22]. Still,this is a rather simple geometry because all the arbitraryfunctions depend on one variable, r .It is often convenient to reparametrise the Szekeresmetric as follows [23]. Even if A = 0 initially, a transfor-mation of the ( x, y )-coordinates can restore A = 0, so wemay assume A = 0 with no loss of generality [11]. Thenlet g = 0. Writing A = p | g | / (2 S ), B = − p | g | P/ (2 S ), B = − p | g | Q/ (2 S ), ε def = g/ | g | , k = | g | e k and Φ = p | g | e Φ,we can represent the metric (2.2) as e − ν = p | g |E , E def = ( x − P ) S + ( y − Q ) S + εS , (2.7) /h ( r ) can be zero at isolated points – it is then either a coordi-nate singularity or a neck or belly – but not on open intervals. The tildes were dropped in (2.7) and in all further text. The Φin (2.7) is in fact e Φ and the k ( r ) is e k ( r ). The redefinitions imply,via (2.4), C = p | g | (cid:2)(cid:0) P + Q (cid:1) /S + εS (cid:3) / h = 1 / [ | g | ( ε − e k )]and M = p | g | f M . The M used from now on is in fact f M . d s = d t − (Φ , r − Φ E , r / E ) ε − k ( r ) d r − Φ E (cid:0) d x + d y (cid:1) , where, so far, ε = ± − g = 0, the tran-sition from (2.2) to (2.7) is A = 1 / (2 S ), B = − P/ (2 S ), B = − Q/ (2 S ) and Φ is unchanged. Then (2.7) applieswith ε = 0, and the resulting model is quasi-plane.The parametrisation introduced above makes severalformulae simpler, mainly because the constraint (2.4) isidentically fulfilled in it. However, this parametrisationobscures the fact, evident in (2.1) – (2.4), that the same Szekeres model may be quasi-spherical in one part ofthe spacetime, and quasi-hyperbolic elsewhere, with theboundary between these two regions being quasi-plane;see an explicit simple example in Ref. [20]. In most ofthe literature published so far, these models have beenconsidered separately, but this was either for purposes ofsystematic research, or with a specific application in viewthat fixed the sign of g ( r ).Equation (2.3), is formally identical to the Friedmannequation, but with k and M depending on r , so eachsurface r = const evolves independently of the others.The solutions Φ( t, r ) are the same as the correspondingL–T solutions, and are unaffected by the dependence ofthe Szekeres metric on the ( x, y ) coordinates.As defined by (2.2) – (2.3), the Szekeres models contain8 functions of r , of which only 7 are arbitrary becauseof (2.4). The parametrisation of (2.7) turns g ( r ) to aconstant parameter ε , thus reducing the number to 6.By a choice of r (still arbitrary up to now), we can fixone more function (for example, by defining r ′ = M ( r )).Thus, the number of arbitrary functions that correspondto physical degrees of freedom is 5.In the following, we will represent the Szekeres solu-tions with β, r = 0 in the parametisation introduced in(2.7). The formula for mass density in these variables is κρ = 2 ( M, r − M E , r / E )Φ (Φ , r − Φ E , r / E ) . (2.8)The shear tensor is σ αβ = 13 (cid:18) Φ , tr − Φ , t Φ , r / ΦΦ , r − Φ E , r / E (cid:19) diag(0 , , − , − , (2.9)and the scalar of expansion is θ = u α ; α = 2Φ , t Φ + Φ , tr − Φ , t E , r / E Φ , r − Φ E , r / E . (2.10) III. REMARKS ON THE BONDI REDSHIFTEQUATION IN THE L–T MODEL
The L–T model is a special case of the quasi-sphericalSzekeres models that follows from (2.7) when ε = +1 The implied changes in C and h are then C = ( P + Q ) / (2 S ), h = − /k ; k and M remain unchanged. and the functions P , Q , S are all constant. With a dif-ferent representation of the coordinates on a sphere, theresulting metric is:d s = d t − R, r E ( r ) d r − R ( t, r ) (cid:0) d ϑ + sin ϑ d ϕ (cid:1) , (3.1)and the equation of an incoming radial null geodesic isd t d r = − R, r ( t, r ) √ E . (3.2)Bondi’s derivation [10] of the redshift equation for thisgeodesic is as follows. Take a light signal obeying (3.2),the equation of its trajectory (the solution of (3.2)) is t = T ( r ) (3.3)Take a second light signal, emitted from the same radialcoordinate r , but later (as measured by the time coordi-nate t ) by τ . The equation of its trajectory is: t = T ( r ) + τ ( r ) , (3.4)where ( T + τ ) obeys, from (3.2):d T d r + d τ d r = − R, r ( T ( r ) + τ ( r ) , r ) p E ( r ) . (3.5)From the Taylor formula we have: R, r ( T ( r ) + τ ( r ) , r ) = R, r ( T ( r ) , r ) + τ ( r ) R, tr ( T ( r ) , r )+ O ( τ , r ) , (3.6)where the last term has the property O ( τ , r ) /τ −→ τ → assuming that τ is small , we neglect the last termin (3.6) and obtain from (3.5), taking into account (3.2):d τ d r = − τ ( r ) R, tr ( T ( r ) , r ) p E ( r ) . (3.7)If τ is the period of an electromagnetic wave, then bydefinition: τ ( r obs ) τ ( r em ) = 1 + z ( r em ) , (3.8)where the subscripts ‘obs’ and ‘em’ refer to the pointsof observation and emission, respectively, and z is theredshift. From (3.8), keeping the observer at a fixedposition and letting r em vary, we obtain (d τ / d r ) /τ = − (d z/ d r ) / (1 + z ), and so in (3.7):11 + z d z d r = R, tr ( T ( r ) , r ) p E ( r ) . (3.9)This is Bondi’s radial redshift equation [10]. It does notdescribe the redshift propagation exactly. Neglecting thelast term in (3.6) we have changed the exact equationinto one that only approximates the actual variation of τ along the ray. The approximation is better the smallerthe value of τ . Considering that τ is the period of anelectromagnetic wave, and taking into account the pe-riod range of relevance in observational astronomy (fromgamma rays up to radio waves, the longest observed ofwhich have the wavelength of the order of 15 m, thusthe period of about 5 × − s), we see that, comparedto cosmological time-scales, the periods are short indeedand the approximation is not bad. Moreover, as seenfrom (3.8), by following the rays back from the observa-tion event into the past, we encounter ever smaller valuesof τ , so the approximation gets progressively better withincreasing redshift (or, rather, gets progressively worseas the ray approaches us). Still, it is conceptually impor-tant to remember that (3.9) involves an approximation(this approximation is equivalent to the geometric opticsapproximation [11, 24] that leads to the commonly usedexpression for the redshift 1 + z = ( k α u α ) em / ( k α u α ) obs ).We shall apply the same approach to the redshift equa-tions in the Szekeres models in Sec. V. IV. EQUATIONS OF GENERAL NULLGEODESICS IN A SZEKERES SPACETIME
For reference, the equations of general null geodesics ina Szekeres model are copied from Ref. [12] in AppendixA. They are written there in terms of an affine param-eter s . For our present purpose it is more convenient touse the coordinate r as an independent parameter (whichis non-affine).This is allowed, but with some caution. It is easilyseen from (A1) – (A4) in Appendix A that a geodesicon which d r/ d s = 0 over some open range of s hasd x/ d s = d y/ d s = 0 in that range, and so is timelike.However, (A1) – (A4) do not guarantee that d r/ d s = 0at all points; isolated points at which d r/ d s = 0 canexist. Examples that explain how this can happen arethe non-radial geodesics in an L–T model, considered inSec. VIII. Thus, r can be used as a parameter on nullgeodesics only on such segments where d s/ d r > s/ d r < , r − Φ E , r / E def = Φ , (4.1)Φ , tr − Φ , t E , r / E def = Φ , (4.2)Φ , rr − Φ E , rr / E def = Φ , (4.3) It is shown in Ref. [12] (see also Ref. [13]) that in general thereexists no analogue of a radial null geodesic. Radial geodesicsexist only when the Szekeres model is axially symmetric; thentheir intersections with every space of constant time coordinatelie on the axis of symmetry. E , r E , x −EE , xr def = E , (4.4) E , r E , y −EE , yr def = E . (4.5)In addition, the following replacement will appear useful: (cid:18) d x d r (cid:19) + (cid:18) d y d r (cid:19) = Σ . (4.6)We have, for any coordinate:d x α d s = (cid:18) d r d s (cid:19) d x α d r + d r d s d x α d r . (4.7)Then, from (A2) we have:d r d s = (cid:18) d r d s (cid:19) (cid:26) − Φ d t d r − (cid:18) Φ Φ − E , r E + 12 k, r ε − k (cid:19) − E E Φ d x d r − E E Φ d y d r + Φ E ε − k Φ Σ (cid:27) def = U ( t, r, x, y ) (cid:18) d r d s (cid:19) . (4.8)Consequently, (A1), (A3) and (A4) become, using (4.7):d t d r + Φ Φ ε − k + ΦΦ , t E Σ + U d t d r = 0 , (4.9)d x d r + 2 Φ , t Φ d t d r d x d r −
1Φ Φ ε − k E + 2Φ Φ d x d r − E , x E (cid:18) d x d r (cid:19) − E , y E d x d r d y d r + E , x E (cid:18) d y d r (cid:19) + U d x d r = 0 , (4.10)d y d r + 2 Φ , t Φ d t d r d y d r −
1Φ Φ ε − k E + 2Φ Φ d y d r + E , y E (cid:18) d x d r (cid:19) − E , x E d x d r d y d r − E , y E (cid:18) d y d r (cid:19) + U d y d r = 0 . (4.11) V. THE REDSHIFT EQUATIONS IN THESZEKERES MODELS
Consider, in the Szekeres metric (2.7), two light sig-nals, the second one following the first one after a shorttime-interval τ , both emitted by the same source and ar-riving at the same observer of coordinates ( r, x, y ). Theequation of the trajectory of the first signal is( t, x, y ) = ( T ( r ) , X ( r ) , Y ( r )) , (5.1)the corresponding equation for the second signal is( t, x, y ) = ( T ( r ) + τ ( r ) , X ( r ) + ζ ( r ) , Y ( r ) + ψ ( r )) . (5.2)This means that while the first ray intersects the hy-persurface of a given constant value of the r -coordinateat the point ( t, x, y ) = ( T, X, Y ), the second ray in-tersects the same hypersurface at the point ( t, x, y ) =( T + τ, X + ζ, Y + ψ ). Thus, in general, those two rays willnot intersect the same succession of intermediate mat-ter worldlines on the way. Note that the coordinateswe use throughout the paper are comoving, so both thesource of light and the observer keep their spatial coor-dinates unchanged throughout history. Given this, andgiven that we consider a pair of rays emitted by thesame source and received by the same observer, we have( ζ, ψ ) = (0 ,
0) at the point of emission and at the pointof reception. However, we have to allow that the secondray was emitted in a different direction than the firstone, and is received from a different direction by the ob-server. The directions of the two rays will be determinedby (d x/ d r, d y/ d r ) and (d x/ d r + ξ ( r ) , d y/ d r + η ( r )), re-spectively, where ξ = d ζ/ d r , η = d ψ/ d r . We will assumethat (d τ / d r, ζ, ψ, ξ, η ) are small of the same order as τ ,so we will neglect all terms nonlinear in any of them andterms involving their products.Since ζ = ψ = 0 at the observer, these quantities arenot in fact observable. However, they have to be numer-ically monitored along the ray because, as will be seenbelow, they enter the equation for τ , which is connectedto the redshift by (3.8).In writing out the equations of propagation of red-shift, we will introduce the symbol ∆. It will de-note the difference between the relevant expression takenat ( t + τ, r, x + ζ, y + ψ ) and at ( t, r, x, y ), linearizedin ( τ, ζ, ψ ), for example Φ ( t + τ, r, x + ζ, y + ψ ) − Φ ( t, r, x, y ) def = ∆Φ + O ( τ , τ ζ, τ ψ, ζ , . . . ). We have: ∆Φ = Φ , t τ, ∆ (Φ , t ) = Φ , tt τ, ∆ d t d r = d τ d r , ∆ d x d r = ξ, ∆ d y d r = η, (5.3)∆ E = E , x ζ + E , y ψ, ∆ E , x = ζ/S, ∆ E , y = ψ/S (5.4)∆Φ = Φ τ + Φ E E ζ + Φ E E ψ, (5.5)∆Φ = (Φ , ttr − Φ tt E , r / E ) τ + Φ , t E E ζ + Φ , t E E ψ, (5.6)∆Φ = (Φ , trr − Φ t E , rr / E ) τ + Φ E ( E , rr E , x −EE , rrx ) ζ + Φ E ( E , rr E , y −EE , rry ) ψ. (5.7) This means that in a general inhomogeneous and anisotropicUniverse the observed objects should drift across the sky. See abrief quantitative discussion of this effect in Sec. VIII. A quick way to calculate (5.3) – (5.16) is to take the differ-ential of the corresponding quantity at constant r and replace(d t, d x, d y, d(d x/ d r ) , d(d y/ d r )) by ( τ, ζ, ψ, ξ, η ). In the next two equations account is taken of the factthat E , xy ≡ E = ( E , r E , xx −EE , rxx ) ζ + ( E , ry E , x −E , y E , rx ) ψ, (5.8)∆ E = ( E , rx E , y −E , x E , ry ) ζ + ( E , r E , yy −EE , ryy ) ψ, (5.9)∆Σ = 2 d x d r ξ + 2 d y d r η, (5.10)∆ U = 2 (cid:18) − ∆Φ Φ + Φ ∆Φ Φ (cid:19) d t d r − Φ d τ d r − ∆Φ Φ + Φ ∆Φ Φ + ∆ E , r E − E , r ∆ EE + 2 (cid:18) − Φ , t E τ E Φ + 2 Φ∆ E E E Φ − Φ∆ E E Φ + Φ E ∆Φ E Φ (cid:19) d x d r − E ξ E Φ + 2 (cid:18) − Φ , t E τ E Φ + 2 Φ∆ E E E Φ − Φ∆ E E Φ + Φ E ∆Φ E Φ (cid:19) d y d r − E η E Φ + ( ε − k )ΦΣ E Φ (cid:18) Φ , t τ Φ − EE − ∆Φ Φ + ∆ΣΣ (cid:19) . (5.11)Applying the ∆-operation to (4.9) – (4.11) we obtain:d τ d r + Φ ∆Φ + Φ ∆Φ ε − k + (cid:0) Φ , t + ΦΦ , tt (cid:1) Σ τ E − , t ∆ E Σ E + ΦΦ , t ∆Σ E + ∆ U d t d r + U d τ d r = 0 , (5.12)d ζ d r + 2 (cid:18) Φ , tt Φ − Φ , t Φ (cid:19) d t d r d x d r τ + 2 Φ , t Φ d x d r d τ d r + 2 Φ , t Φ d t d r ξ − ∆Φ E ( ε − k )Φ + Φ , t Φ E τ ( ε − k )Φ − Φ ∆ E ( ε − k )Φ+ 2 (cid:18) ∆Φ Φ − Φ Φ , t τ Φ (cid:19) d x d r + 2 Φ Φ ξ − (cid:18) d x d r (cid:19) (cid:18) ζS E − E , x ∆ EE (cid:19) − E , x ξ E d x d r − x d r d y d r (cid:18) ψS E − E , y ∆ EE (cid:19) − E , y E (cid:18) d y d r ξ + d x d r η (cid:19) + (cid:18) d y d r (cid:19) (cid:18) ζS E − E , x ∆ EE (cid:19) + 2 E , x η E d y d r + ∆ U d x d r + U ξ = 0 , (5.13)d ψ d r + 2 (cid:18) Φ , tt Φ − Φ , t Φ (cid:19) d t d r d y d r τ + 2 Φ , t Φ d y d r d τ d r + 2 Φ , t Φ d t d r η − ∆Φ E ( ε − k )Φ + Φ , t Φ E τ ( ε − k )Φ − Φ ∆ E ( ε − k )Φ+ 2 (cid:18) ∆Φ Φ − Φ Φ , t τ Φ (cid:19) d y d r + 2 Φ Φ η + (cid:18) d x d r (cid:19) (cid:18) ψS E − E , y ∆ EE (cid:19) + 2 E , y ξ E d x d r − x d r d y d r (cid:18) ζS E − E , x ∆ EE (cid:19) − E , x E (cid:18) d y d r ξ + d x d r η (cid:19) − (cid:18) d y d r (cid:19) (cid:18) ψS E − E , y ∆ EE (cid:19) − E , y η E d y d r + ∆ U d y d r + U η = 0 , (5.14)In addition, we have the first integral of the geodesicequations (4.9) – (4.11): (cid:18) d t d r (cid:19) = (Φ ) ε − k + Φ E "(cid:18) d x d r (cid:19) + (cid:18) d y d r (cid:19) , (5.15)Applying the ∆-operation to this we getd τ d r d t d r = Φ ∆Φ ε − k + (cid:18) ΦΦ , t τ E − Φ ∆ EE (cid:19) "(cid:18) d x d r (cid:19) + (cid:18) d y d r (cid:19) + Φ E (cid:18) d x d r ξ + d y d r η (cid:19) . (5.16)Note: d t/ d r < VI. REPEATABLE LIGHT PATHS
As attested by (5.12) – (5.14), in a generic Szekeresmodel two light rays connecting a given source to a givenobserver at different instants of emission do not proceedthrough the same succession of intermediate matter par-ticles. We will now investigate under what conditionsthis intermediate succession is the same. This propertywill be called repeatable light paths (RLP).For a RLP we have ζ = ψ = ξ = η = 0 (6.1)all along the ray. Then (5.12) decouples from (5.13) –(5.14) and just determines τ (and, with it, the redshift),if the null geodesic equations are solved first. Equations(5.13) – (5.14) become then:2 (cid:18) Φ , tt Φ − Φ , t Φ (cid:19) d t d r d x d r τ + 2 Φ , t Φ d x d r d τ d r − ∆Φ E ( ε − k )Φ + Φ , t Φ E τ ( ε − k )Φ +2 (cid:18) ∆Φ Φ − Φ Φ , t τ Φ (cid:19) d x d r + ∆ U d x d r = 0 , (6.2) 2 (cid:18) Φ , tt Φ − Φ , t Φ (cid:19) d t d r d y d r τ + 2 Φ , t Φ d y d r d τ d r − ∆Φ E ( ε − k )Φ + Φ , t Φ E τ ( ε − k )Φ +2 (cid:18) ∆Φ Φ − Φ Φ , t τ Φ (cid:19) d y d r + ∆ U d y d r = 0 . (6.3)These equations can be understood in 2 ways:1. As equations defining special Szekeres spacetimesin which all null geodesics are RLPs.2. As equations defining special null geodesics whichare RLPs in subcases of the Szekeres spacetimes.In the first interpretation, (6.2) – (6.3) should be iden-tities in the components of d x α / d r . They are polynomi-als of degree 3 in these components, and since (d t/ d r ) does not appear in them, the constraint (5.15) plays norole – all powers of d x α / d r that do appear are indepen-dent. Equating to zero the coefficient of (d x/ d r ) in (6.2)(which arises inside ∆ U , within Σ), and taking into ac-count that ∆ E = ∆Σ = 0 when (6.1) holds, we getΨ def = Φ , tr − Φ , t Φ , r / Φ = 0 . (6.4)The integral of this is Φ = S ( t ) f ( r ), where S and f are arbitrary functions. It is seen from (2.9) that thismeans zero shear, i.e. the Friedmann limit. With (6.4)fulfilled, (6.2) and (6.3) become identities, and (4.9) –(4.11) reduce to the equations of general null geodesics ina Friedmann spacetime. With the observer placed at theorigin, the geodesics become radial, d x/ d r = d y/ d r =0, and then (5.16) becomes equivalent to the ordinaryRobertson – Walker redshift formula, 1+ z = S ( t o ) /S ( t e ).To verify this, some calculations are needed, in which Ref.[11] may prove helpful.Thus, we have proven the following: Corollary 1:
The only spacetimes in the Szekeres family in which allnull geodesics have repeatable paths are the Friedmannmodels. In the second interpretation of (6.2) – (6.3), we con-sider 2 cases:
A. The general case: d x/ d r = 0 = d y/ d r every-where. Then we multiply (6.2) by d y/ d r , (6.3) by d x/ d r andsubtract the results. Disregarding the familiar case Ψ = 0we get E d y d r − E d x d r = 0 . (6.5) We recall, however, that the Friedmann limit is represented inthe Goode – Wainwright [18] coordinates (see the remark in para4 after (2.6)). Consequently, all equations representing the Fried-mann model will look unfamiliar. This is one more piece of evidence of how exceptional the Robert-son – Walker class of models is.
This, together with (6.2), (5.12) and (4.9) – (4.11) definesa certain subcase of the Szekeres model and a class ofcurves in it. Since both the subcase and the class willturn out to be empty, but the calculations proving it arerather elaborate, we present them in Appendix B.
B. The special cases: d x/ d r = 0 or d y/ d r = 0 . These two cases are equivalent under the coordinatetransformation ( x, y ) = ( y ′ , x ′ ), so we consider only thefirst one. Again disregarding Ψ = 0, we get from (6.2) E = 0. Then, (4.10) implies two possibilities: Ba) E , x = 0.This is possible only when P is constant, and then thegeodesic lies in the subspace x = P . Equations (4.11) and(6.3) still have to be obeyed, while (4.10) and (6.2) arefulfilled identically. The simple coordinate transforma-tion x = x ′ + P has then the same effect as if P = 0 and x = 0 along the geodesic. We show in Appendix E thatin this case, apart from the axially symmetric subcasementioned below, RLPs may exist only when the Szek-eres metric has a 3-dimensional symmetry group. Suchspacetimes are considered in Sec. VII. Bb) d y/ d r = 0.The case d x/ d r = d y/ d r = 0, ε = +1 was investigatedin detail in Ref. [12]. It turned out that this can happenonly when the Szekeres spacetime is axially symmetric,and then along only one sub-family of null geodesics –those that intersect each t = constant space on the sym-metry axis. We show in Appendix F that this resultapplies also with ε ≤
0, and that other RLPs may existonly with higher symmetries.
VII. RLPS IN THE G /S MODELS
The symbol G /S denotes such models that have 3-dimensional symmetry groups acting on 2-dimensionalorbits [11]. They result from the general β ′ = 0 Szekeresfamily when the functions ( P, Q, S ) are all constant. Thesymmetry of the model is then spherical when ε = +1(this is the L–T model), pseudospherical (also called hy-perbolic) when ε = − ε = 0.Using the G symmetry, the origin of the ( x, y ) coor-dinates at ( x, y ) = ( P, Q ) can be moved to any locationon the S surfaces. So let us consider the S on whichthe first light ray is emitted, and let us choose the originof ( x, y ) at the position of the emitter. Thus, in (4.9)– (4.11) the initial point of the earlier null geodesic willhave the coordinates ( x, y ) = ( P, Q ), and, at this point, E , x = E , y = 0. In addition, the isotropy subgroup of G ,existing in each case at every point of the manifold, allowsus to rotate the ( x, y ) coordinates, with no loss of gen-erality, so that the initial value of d y/ d r for our chosengeodesic is zero, i.e. so that the ray is initially tangentto the y = constant subspace. Equation (4.11) showsthat with such initial conditions (and with E , y = 0 atthe initial point) we have d y/ d r = 0 initially, and sod y/ d r = 0 = d y/ d r all along the geodesic.With coordinates chosen in such a way, equations (4.11) and (6.3) are fulfilled identically. However, (6.2)is not an identity and reduces to:d x d r (cid:20) (cid:18) Φ , tt Φ − Φ , t Φ (cid:19) d t d r τ + 2 Φ , t Φ d τ d r +2 (cid:18) Φ , tr Φ − Φ , t Φ , r Φ (cid:19) τ + 2 (cid:18) − Φ , ttr Φ , r + Φ , tr Φ , r (cid:19) d t d r τ − , tr Φ , r d τ d r − Φ , trr Φ , r τ + Φ , rr Φ , tr Φ , r τ + ( ε − k ) τ E (cid:18) d x d r (cid:19) (cid:18) Φ , t Φ , r − ΦΦ , tr Φ , r (cid:19) def = d x d r χ = 0 . (7.1)One solution of this is d x/ d r = 0, which together withd y/ d r = 0 defines a radial null geodesic. Then, (4.10) –(4.11) are fulfilled identically, while (5.15) – (5.16), to-gether with (4.1) – (4.2) and (5.5) reproduce the Bondiequation (3.7) when ε = +1. So, we found that in the G /S models all radial null geodesics are RLPs. There would exist other RLPs in these models if χ in(7.1) were zero along any null geodesic – possibly in somesubcases of the models. It is shown in Appendix G thatthis does not happen, so the radial null geodesics are theonly RLPs in these models.
VIII. NUMERICAL EXAMPLES OF NON-RLPSIN THE L–T MODEL
For illustration, we first consider a configurationthat is not realistic, but shows the non-RLP effect ina clearly visible way. It is an LT model specifiedby the following functions: t B = 0 and ρ ( t , r ) = ρ (cid:2) δ − δ exp (cid:0) − r /σ (cid:1)(cid:3) , (where t is the current in-stant, r is defined as R ( t , r ), and ρ is the density atthe origin and equals 0 . × (3 H ) / (8 πG ), where H =72 km s − Mpc − and G is the gravitational constant).This model is the so-called giant void model discussed indetail in [25], with the best-fit parameters: δ = 4 .
05 and σ = 2 .
96 Gpc. We use this model to study the configu-ration presented in Fig. 1, where, for the middle curve,the angle between the radial direction and the incominggeodesic is γ = 0 . π . We consider 3 light paths. Thefirst one corresponds to photons received by the observer5 × years ago, the second one corresponds to photonsreceived at the current instant, and the third one corre-sponds to photons which will be received in 5 × yearin the future. Figure 1 shows these 3 geodesics projectedon the space t = now along the flow lines of the matter The null geodesics with d x/ d r = d y/ d r = 0 can properly becalled radial only in the L–T model, where ε = +1. What thiscondition means in the other two cases is not clear, so the term“radial” is used here only as a brief label. r [ G p c ] Θ [rad] ρ / ρ FIG. 1: Three nonradial null geodesics in an L–T model, pro-jected on the space t = now along the flow lines of the L–Tdust. Each geodesic runs between the same observer and thesame emitter, which, at present, lie at the same distance of3.5 Gpc from the center. The solid line represents the ray thatthe observer receives at the current instant, γ = 0 . π , thedashed line represents the ray that was received 5 × yearsago, γ = 0 . π , and the dotted line represents the ray thatwill be received 5 × years in the future, γ = 0 . π . Asseen, nonradial null geodesics in the L–T model do not havethe RLP property. The inset shows the density profile ( ρ is the density at the origin) evaluated at the current instantalong these three different paths. source in the L–T model. Since in each case the lightpaths are different, the profile of matter density alongeach projected light ray is different. This feature is pre-sented in the inset in Fig. 1. Even though the densityvariation along the light path is of small amplitude, theeffect is clearly visible. The average rate of change of theposition of the source in the sky, seen by the observer, is ∼ − arc sec per year.Now we will study a more realistic configuration. Theparameters of the L–T model will be the same as above,but the placement of the observer and of the source willbe different, see Fig. 2. The observer (O) is located at R ( R = R ( t , r ) is the present-day areal distance) andobserves a galaxy (*), the angle between the directiontowards the galaxy and towards the origin is γ . We study3 configurations: (1) R = 3 Gpc, (2) R = 1 Gpc, (3) R = 1 Gpc but with δ = 10. All 3 cases have d = 1 Gyr( ≈ . γ ) we find anull geodesic that joins the observer and the galaxy. Wethen calculate the rate of change γ , which is equivalentto the change of the position of the galaxy in the sky.A detailed description of the algorithm is presented inAppendix H. The results are presented in Fig. 3.As seen, the rate of change of the position of the sourcein the sky depends on the angle γ . The amplitude ofthe change is of the order ∼ − arc sec per year forcase (2) and ∼ − arc sec per year for cases (1) and FIG. 2: A schematic view of the considered configurations.The observer (O) is located at R and observes a galaxy (*),the angle between the direction towards the galaxy and to-wards the origin is γ . Because of the non-RLP effect the angleat which the galaxy is observed at some other time instant isdifferent. (3). Given Gaia accuracy of position measurement,5 − × − arc sec, we would need to wait at least afew years to detect the change of position due to non-RLPeffects. However, this estimate assumes that we have areference direction that does not change. This will bea difficult practical problem, since cosmological observa-tions are done under the assumption that our Universeis precisely represented in large scales by the Robert-son – Walker class of models, in which there is no suchdrift. We would have to identify a direction that doesnot change with time even in an inhomogeneous modelor measure a relative change of position between variousobjects. IX. SUMMARY
By a method analogous to that of Bondi [10] we havederived the equations to be obeyed by the redshift in ageneral Szekeres β ′ = 0 spacetime, (5.12) – (5.14). Thenull geodesic equations parametrised by r , which must besolved together with (5.12) – (5.14), are given by (4.9) –(4.11). Although the physically most interesting quantityis the longitudinal redshift determined by τ , the othertwo components, ζ and ψ , must be numerically moni-tored along the ray because the equations that determine( τ, ζ, ψ ) are coupled. -20-15-10-5 0 5 10 15 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 γ / - [ a r c s e c / y ea r ] γ / π • FIG. 3: The rate of change of position in the sky ( ˙ γ ) dueto the non-RLP effect, expressed as a change of an angle inarc sec per year × . The solid line presents case (1) where R = 3 Gpc, the dashed line presents case (2) where R = 1Gpc, and the dotted line presents case (3) where R = 1 Gpcand δ = 10. We have shown that, in general, two light rays sentfrom the same source at different times to the same ob-server do not proceed through the same succession ofintermediate matter particles; we refer to this propertyby saying that the light paths are not repeatable. In atoy model, with the present spatial distance between thelight source B and the observer being of the order of 1.5Gpc, the estimated rate of the drift of B across the skywould be ≈ × − arc sec per year. In a more realisticconfiguration, this number is ≈ − arc sec per year.The Gaia is expected to have the precision of positiondetermination 5 − × − arc sec.We have derived the equations defining repeatable lightpaths (RLPs), (6.2) – (6.3); they must hold together with(4.9) – (4.11) and (5.12). We have shown that all nullgeodesics are RLPs only in the Friedmann models. Theonly other cases in which RLPs exist are the following:( i ) The axially symmetric Szekeres models, in whichthe RLPs are the null geodesics intersecting every spaceof constant time on the axis of symmetry.( ii ) The radial null geodesics in the G /S subcases(i.e. in the spacetimes that have 3-dimensional symmetrygroups). Appendix A: Equations of null geodesics in aSzekeres spacetime in an affine parametrisation
For convenience of the readers, the equations of nullgeodesics in a Szekeres spacetime are copied here fromRef. [12]. They are given in an affine parametrisation.d t d s + Φ , tr − Φ , t E , r / E ε − k (Φ , r − Φ E , r / E ) (cid:18) d r d s (cid:19) + ΦΦ , t E "(cid:18) d x d s (cid:19) + (cid:18) d y d s (cid:19) = 0 , (A1)d r d s + 2 Φ , tr − Φ , t E , r / E Φ , r − Φ E , r / E d t d s d r d s + (cid:18) Φ , rr − Φ E , rr / E Φ , r − Φ E , r / E − E , r E + 12 k, r ε − k (cid:19) (cid:18) d r d s (cid:19) + 2 Φ E E , r E , x −EE , xr Φ , r − Φ E , r / E d r d s d x d s + 2 Φ E ( E , r E , y −EE , yr )Φ , r − Φ E , r / E d r d s d y d s − Φ E ε − k Φ , r − Φ E , r / E "(cid:18) d x d s (cid:19) + (cid:18) d y d s (cid:19) = 0 , (A2)d x d s + 2 Φ , t Φ d t d s d x d s −
1Φ Φ , r − Φ E , r / E ε − k ( E , r E , x −EE , xr ) (cid:18) d r d s (cid:19) + 2Φ (cid:18) Φ , r − Φ E , r E (cid:19) d r d s d x d s − E , x E (cid:18) d x d s (cid:19) − E , y E d x d s d y d s + E , x E (cid:18) d y d s (cid:19) = 0 , (A3)d y d s + 2 Φ , t Φ d t d s d y d s −
1Φ Φ , r − Φ E , r / E ε − k ( E , r E , y −EE , yr ) (cid:18) d r d s (cid:19) + 2Φ (cid:18) Φ , r − Φ E , r E (cid:19) d r d s d y d s + E , y E (cid:18) d x d s (cid:19) − E , x E d x d s d y d s − E , y E (cid:18) d y d s (cid:19) = 0 . (A4) Appendix B: Solutions of (6.5).
Since (6.5) should hold along certain null geodesics, itsderivative by r along those geodesics must be zero. Thisderivative, denoted by D / d r , of any quantity χ definedalong the geodesic, χ ( t ( r ) , r, x ( r ) , y ( r )), is: D χ d r = ∂χ∂t d t d r + ∂χ∂r + ∂χ∂x d x d r + ∂χ∂y d y d r . (B1)Calculating D / d r of (6.5) we get: (cid:18) E ,r + E ,x d x d r + E ,y d y d r (cid:19) d y d r − (cid:18) E ,r + E ,x d x d r + E ,y d y d r (cid:19) d x d r + E d y d r − E d x d r = 0 . (B2)0The expression in the last line can be calculated from(4.10) – (4.11) using (6.5); it is: E d y d r − E d x d r = ( E , y E , rx −E , x E , ry ) "(cid:18) d x d r (cid:19) + (cid:18) d y d r (cid:19) . (B3)Substituting (B3) and (4.4) – (4.5) in (B2), and takinginto account the identities E , xy = E , xx −E , yy = 0 we get: E ,r d y d r − E ,r d x d r = 0 . (B4)This should hold simultaneously with (6.5). Since weassumed d x/ d r = 0 = d y/ d r , (6.5) and (B4) imply: E E ,r − E E ,r = 0 . (B5)When (4.4) – (4.5) are substituted in (B5), E factors out,and the other factor is: E , r ( E , y E , rrx −E , x E , rry )+ E , rr ( E , x E , ry −E , y E , rx )+ E ( E , rx E , rry −E , ry E , rrx ) = 0 . (B6)This simplifies to a polynomial of second degree in x and y , which should vanish identically. Using the algebraicprogram Ortocartan [26, 27] we find that the coefficientof ( x + y ) is P, rr Q, r − P, r Q, rr = 0 . (B7)One of the solutions of this is P, r = 0; then no limitationfor Q follows. This case we consider separately below.When P, r = 0, (B7) implies Q = C P + D , (B8)where C and D are arbitrary constants. When this issubstituted in (B6), the coefficient of y implies: ε (cid:0) SS, r P, rr − S, r P, r − SS, rr P, r (cid:1) − (cid:0) C (cid:1) P, r = 0 , (B9)and this guarantees that the whole of (B6) is fulfilled.The case ε = 0 is seen to be incompatible with P, r = 0.This means that no RLPs exist in the ε = 0 models with P, r = 0. Further calculations apply only to ε = ± S, r = 0 because S, r = 0 immediately implies P, r = 0, which we have leftfor a separate investigation. Therefore we can introduce Should there exist any point P n in the spacetime at which thepolynomial would be nonzero, this would mean that the deter-minant of the set { (6.5) , (B4) } is nonzero at P n , which in turnwould mean d x/ d r = d y/ d r = 0 at P n – contrary to our initialassumption. S ( r ) as the new independent variable in (B9), which thenbecomes: ε ( SP, SS − P, S ) − (cid:0) C (cid:1) P, S = 0 . (B10)Since the case P = constant was left for later, we assume P, S = 0, and then (B10) is easily integrated with theresult: εS + (cid:0) C (cid:1) P = C P + D , (B11)where C and D are new arbitrary constants.When ε = +1, eqs. (B8) and (B11) are equivalent tothose that were shown in Ref. [12] (sec. 3.3.1) to besufficient conditions for the Szekeres metric to be axiallysymmetric. However, this equivalence is nontrivial, andthe extension of the proof to ε = 0 , − x, y ) = ( x ′ + x , y ′ + y ) , (B12)where ( x , y ) are arbitrary constants. This changes( P, Q ) to ( e P , e Q ) = ( P − x , Q − y ) . (B13)(2) The metric (2.7) does not change in form when( x, y ) are transformed by a general orthogonal transfor-mation: x = ax ′ + by ′ √ a + b , y = − bx ′ + ay ′ √ a + b , (B14)which implies the change of ( P, Q ) to: e P = aP − bQ √ a + b , e Q = bP + aQ √ a + b . (B15)(3) The metric (2.7) does not change in form under thediscrete transformations:( x, y ) = ( y ′ , x ′ ) , ( x, y ) = ( − x ′ , y ′ ) , ( x, y ) = ( x ′ , − y ′ ) , (B16)which induce, respectively( e P , e Q ) = ( Q, P ) , ( e P , e Q ) = ( − P, Q ) , ( e P , e Q ) = ( P, − Q ) . (B17)(4) The metric (2.7) does not change in form when( x, y ) are transformed by a conformal symmetry of a Eu-clidean 2-plane – a 2-dimensional Haantjes transforma-tion by the terminology of Ref. [11]. It has the form: x = x ′ + λ (cid:16) x ′ + y ′ (cid:17) T , y = y ′ + λ (cid:16) x ′ + y ′ (cid:17) T , (B18) T def = 1 + 2 λ x ′ + 2 λ y ′ + (cid:0) λ + λ (cid:1) (cid:16) x ′ + y ′ (cid:17) , where λ and λ are arbitrary constants – the group pa-rameters. This group is Abelian, the inverse transforma-tion to (B18) being of the same form, but with parame-ters ( − λ , − λ ). The characteristic properties of (B18),useful in calculations, are: x + y = x ′ + y ′ T , d x + d y = d x ′ + d y ′ T . (B19)Under (B18) – (B19), ( P, Q, S ) change, respectively, to: e P = 1 U (cid:2) P − λ (cid:0) P + Q + εS (cid:1)(cid:3) , e Q = 1 U (cid:2) Q − λ (cid:0) P + Q + εS (cid:1)(cid:3) , e S = S/U,U def = 1 − λ P − λ Q + (cid:0) λ + λ (cid:1) (cid:0) P + Q + εS (cid:1) . (B20)Let e E denote E with ( x, y, P, Q, S ) replaced by( x ′ , y ′ , e P , e Q, e S ). Then calculation shows that E = e E /T, (B21)and since T does not depend on r , it follows that the g rr component in (2.7) is also covariant with (B18) – (B19).Now we will use the properties listed above to interpretthe consequences of (B8) and (B11) for the metric (2.7).The D in (B8) can be set to zero by (B12) with( x , y ) = (0 , D ). The C in (B8) can be set to zeroby (B15) with b = − aC ; the result of these two trans-formations is Q = 0. Finally, the C in (B11) (with C = 0 taken into account) can be set to zero by(B12) with ( x , y ) = ( − C / , D = C = C = 0 with no loss of generality.We carry out a combination of (B12) with (B18): x = x + x ′ + λ (cid:16) x ′ + y ′ (cid:17) T ,y = y ′ , (B22)and get the following generalisation of (B20) with λ = 0: e P = 1 U n P − x − λ h ( P − x ) + Q + εS io , ( e Q, e S ) = ( Q, S ) / U , (B23) U def = 1 − λ ( P − x ) + λ h ( P − x ) + Q + εS i . Using (B8) and (B11) with D = C = C = 0, the abovebecomes e P = 1 U (cid:2) P − x − λ (cid:0) − x P + D + x (cid:1)(cid:3) , e Q = 0 , (B24) U def = 1 − λ ( P − x ) + λ (cid:0) − x P + D + x (cid:1) . Now it can be seen that if the constants ( x , λ ), so fararbitrary, obey: 1 + 2 λ x = 0 ,x + λ (cid:0) D + x (cid:1) = 0 , (B25)then e P = e Q = 0, and in the ( x ′ , y ′ ) coordinates theSzekeres metric is explicitly axially symmetric. However,two things must be noted:(1) The set (B25) has no solutions when D ≤ P = Q = 0, eq. (B9) is fulfilled identically,and (B11) no longer follows, thus there is no limitationon S .Looking at (B11) with C = C = 0 we see that D < ε = +1 or ε = 0. The case D = 0,although possible with ε = +1 or ε = 0, need not beconsidered with these two values of ε , for the followingreasons: With ε = +1 this would imply S = 0, whichis an impossibility in (2.7), and with ε = 0 = C = C ,(B11) implies P = 0. With Q = 0 now being considered, P = ε = 0 guarantees that S may be set to 1 by a suit-able reparametrisation of the other metric functions [20].Consequently, with P = Q = 0, the ε = 0 Szekeres met-ric is already plane symmetric even with non-constant S ,and the Szekeres metrics with 3-dimensional symmetrygroups are considered in Sec. VII.So, finally, D ≤ ε = − P, r = 0.By a transformation of x this can be reduced to P = 0.Then, the whole of (B6) becomes: x (cid:2) ε (cid:0) S, r Q, r − SS, r Q, rr + SS, rr Q, r (cid:1) + Q, r (cid:3) = 0 . (B26)This is equivalent to the subcase C = 0 of (B9) under thecoordinate transformation ( x, y ) = ( e y, e x ) and the associ-ated renaming ( P, Q ) = ( e Q, e P ). This case was includedin the consideration above.Thus, apart from the special cases D ≤ x/ d r = 0 = d y/ d r maypossibly exist only when the Szekeres metric is reducible,by a coordinate transformation, to one with P = Q = 0.In this case, (6.5) becomes: ε S, r S (cid:18) x d y d r − y d x d r (cid:19) = 0 . (B27)But with ε = 0 and P = Q = 0 now being consid-ered, the quasi-plane Szekeres metric is plane symmetriceven with non-constant S ; see the paragraph following(B25). The Szekeres metrics with 3-dimensional symme-try groups are considered in Sec. VII, so we need notconsider ε = 0 here.2When S, r = 0 = P, r = Q, r , all Szekeres metrics acquirea 3-dimensional symmetry group and are considered inSec. VII. Thus, we need not consider S, r = 0 in (B27).What remains of (B27) is x d y/ d r − y d x/ d r = 0. Onesolution of this is x = 0 along the null geodesic. Theother solution is y = G x along the geodesic, where G isa constant. However, we are now considering the axiallysymmetric Szekeres solutions in which P = Q = 0. Inthis case, a rotation (B14) can be chosen so that y = G x is transformed to x ′ = 0. So, apart from the special case ε = − , D ≤
0, for the other Szekeres solutions thefollowing result applies:
Corollary 2:
The Szekeres spacetimes in which RLPs exist withd x/ d r = 0 = d y/ d r either are the Friedmann models(in which all null geodesics are RLPs) or are inhomo-geneous and axially symmetric or have a 3-dimensionalsymmetry group. In the first two cases, coordinates maybe chosen so that x = 0 along the hypothetic RLP and P = Q = 0 in the metric. The second case is consideredin Appendix F. The third case is considered in Sec. VIIand in Appendix G. Appendix C: The special metric with Q = 0 , ε = − and D ≤ . We consider here the special case D ≤ E = x − P x + y + D S ,E = P, r S (cid:0) y − x + D (cid:1) , E = − P, r S xy. (C1)The solution of (6.5) is then either P, r = 0, which belongsto the axially symmetric case considered in appendix E,or x + y − Cy − D = 0 , (C2)where C is the arbitrary constant that arises while inte-grating (6.5). By writing the above as x + ( y − C/ = D + C / D + C / > . (C3)(With this quantity being negative, (C2) has no solu-tions, i.e. there are no RLPs. When it is zero, the onlysolution of (C2) is ( x, y ) = (0 , C/ C = 0, since D ≤ r and substi-tuting in it the expressions for x, rr and y, rr from (4.10)– (4.11), we obtain an identity. This means that (C2) isconsistent with the geodesic equations (4.9) – (4.11) anddefines a special class of null geodesics. We will verify inthe following that this class does not contain any RLPs. We note the following auxiliary formulae. In Eqs. (C4)– (C11) asterisks mark those equations that hold onlyalong the null geodesics obeying (C2), those without theasterisk are general. After (C11) all further calculationsare done only along these geodesics, so we omit the as-terisks for better readability.( ∗ ) E = − P x + Cy + 2 D S , (C4) E , x = x − PS , E , y = yS , E , r = − xP, r S − S, r S E , (C5) E , rx = − P, r S − S, r S ( x − P ) , E , ry = − yS, r S , (C6)( ∗ ) E = P, r S (cid:0) y − Cy (cid:1) , (C7)( ∗ ) y, r = − x y − C x, r , (C8)( ∗ ) x, r + y, r = 4 D + C (2 y − C ) x, r . (C9)( ∗ ) 2 x, r E + 2 x, r y, r E = P, r S y (2 y − C ) (cid:0) x, r + y, r (cid:1) . (C10)From (5.15) we have x, r + y, r E = t, r Φ − Φ ( ε − k )Φ . (C11)In order to write the equations in a compact way itis convenient to use the symbol Ψ introduced in (6.4).Recall: this is a factor of shear, and when it vanishes, theSzekeres model reduces to Friedmann. Thus, in searchingfor RLPs in nontrivial models we will assume Ψ = 0.Using (C11) in (5.16) adapted to RLPs we obtain τ, r t, r = Φ Φ τε − k + Φ , t t, r τ Φ − Φ , t Φ τ ( ε − k )Φ ≡ Φ Ψ τε − k + Φ , t t, r τ Φ . (C12)We also have:Φ − Φ , t Φ / Φ ≡ Φ , tr − Φ , t Φ , r / Φ = Ψ , (C13)Φ , ttr − Φ , tt Φ , r / Φ = Ψ , t + Φ , t Φ Ψ , (C14)Φ , trr − Φ , t Φ , rr / Φ = Ψ , r + Φ , r Φ Ψ . (C15)Assuming Ψ = 0 we now multiply (6.2) by ΦΦ t, r / [( ε − k )Ψ], use (C7) – (C15), cancel τ that factors out, andwrite the result in the form: x, r (cid:0) t, r + c t, r + c t, r + c (cid:1) = B t, r + B t, r , (C16)where c = 2Φ (cid:0) Φ Ψ , t − Ψ (cid:1) ( ε − k )Ψ , (C17)3 c = − + ΦΦ Ψ , r / Ψ − ΦΦ , rr ε − k + Φ E , rr / E + Φ Φ , r ε − k , (C18) c = 2ΦΦ Ψ( ε − k ) , (C19) B = P, r y (2 y − C )( ε − k ) S , (C20) B = − ε − k ) B . (C21)Then, using (C1), (C7) and (C11) we can rewrite (4.9)in the form: t, rr = c t, r + c t, r + c t, r + c + Ax, r t, r , (C22)where c = − ε − k ΦΦ , (C23) c = 2 ΨΦ + Φ , t Φ , (C24) c = Φ , rr − Φ E , rr / E Φ − E , r E + k, r ε − k ) + Φ Φ , (C25) c = − ΨΦ ε − k , (C26) A def = (cid:0) D + C (cid:1) P, r y Φ(2 y − C ) S E Φ . (C27)Combining (C9) and (C11) we get: x, r = (2 y − C ) E (4 D + C ) Φ (cid:18) t, r − Φ ε − k (cid:19) . (C28)Equations (C16) and (C28) determine d t/ d r along thehypothetic RLP. Formally, a solution for d t/ d r of theseequations always exists, but it must be consistent withthe geodesic equations, and this is what we will inves-tigate next. Namely, every solution of these equationsmust be preserved along the null geodesics. To seewhether it is, we first transform this set into a singlepolynomial equation for d t/ d r .We square (C16) and use (C28) in the result. We thusobtain an 8-th degree polynomial in t, r , whose coefficientat t, r is α = (2 y − C ) E (4 D + C ) Φ . (C29)It is seen that it cannot vanish except when y = C/
2, butthis defines a “radial” geodesic that exists only in the ax-ially symmetric case [12]. Thus we divide the 8-th degreepolynomial by α and obtain the following equation t, r + 2 c t, r + a t, r + a t, r + a t, r + a t, r + a t, r + a t, r + a = 0 , (C30) where: a = 2 c + c − Φ ε − k − (cid:0) D + C (cid:1) Φ P, r y ( ε − k ) S E , (C31) a = 2 c + 2 c c − c Φ ε − k , (C32) a = 2 c c + c − (cid:0) c + c (cid:1) Φ ε − k +3 Φ ( ε − k ) (cid:0) D + C (cid:1) Φ P, r y S E , (C33) a = 2 c c − (2 c + 2 c c ) Φ ε − k , (C34) a = c − (cid:0) c c + c (cid:1) Φ ε − k − ε − k ) (cid:0) D + C (cid:1) Φ P, r y S E , (C35) a = − c c Φ ε − k , (C36) a = − c Φ ε − k . (C37)Now we differentiate (C30) along the null geodesic by therule (B1), and use (C30) to eliminate t, r , t, r and t, r from the result. In this way we obtain: b t, r + b t, r + b t, r + b t, r + b t, r + b t, r + b t, r + b + x, r (cid:0) β t, r + β t, r + β t, r + β t, r + β t, r + β t, r + β t, r + β ) = 0 , (C38)where b = 8 c + 2 c ,r + a ,t − a c − a c − c c − c c ,t + 6 a c c + 4 c c − c c , (C39) b = a ,r + a ,t + 14 c c − a c − a c − a c − a c ,t + 2 a c c + 2 a c c + 2 a c − a c c , (C40) b = a ,r + a ,t + 6 a c − a c − a c − a c − a c ,t + 2 a c c + 2 a c c + 2 a a c − a c c , (C41) b = a ,r + a ,t + 5 a c − a c − a c − a c − a c ,t + 2 a c c + 2 a c c + 2 a a c − a c c , (C42) b = a ,r + a ,t + 4 a c − a c − a c − a c − a c ,t + 2 a c c + 2 a c c + 2 a a c − a c c , (C43) b = a ,r + a ,t + 3 a c − a c − a c − a c − a c ,t + 2 a c c + 2 a c c + 2 a a c − a c c , (C44) b = a ,r + a ,t + 2 a c − a c − a c − a c ,t +2 a c c + 2 a c c + 2 a a c − a c c , (C45) b = a ,r + a c − a c − a c ,t + 2 a c c +2 a a c − a c c , (C46) β = 2 (cid:18) c ,x − x y − C c ,y (cid:19) − c A, (C47) β = a ,x − x y − C a ,y − a A, (C48) β = a ,x − x y − C a ,y − a A, (C49) β = a ,x − x y − C a ,y − a A, (C50) β = a ,x − x y − C a ,y − a A, (C51) β = a ,x − x y − C a ,y − a A, (C52) β = a ,x − x y − C a ,y − a A, (C53) β = a ,x − x y − C a ,y − a A. (C54)We provisionally assume that the coefficient of x, r in(C16) is nonzero. (We will later come back to this pointand investigate what happens when it is zero.) Thenwe determine x, r from (C16) and substitute the resultin (C38). After multiplying out to get a polynomial in t, r , we again use (C30) to eliminate t, r and t, r (butnot t, r ). Then we assume that the coefficient of t, r ,denoted d , is nonzero (we will check the case d = 0later), and divide the equation by d . In this way weobtain: t, r + δ t, r + δ t, r + δ t, r + δ t, r + δ t, r + δ t, r + δ t, r + δ = 0 , (C55)where δ i def = d i /d , i = 2 , . . . ,
9, and d = b − b c + b c + β B + β B − a b − a β B − β B c + 2 b c + 4 β B c , (C56) d = b + b c + b c + b c + β B + β B − a b − a b − a β B − a β B + a b c +2 a β B c , (C57) d = b + b c + b c + b c + β B + β B − a b − a b − a β B − a β B + a b c +2 a β B c , (C58) d = b + b c + b c + b c + β B + β B − a b − a b − a β B − a β B + a b c +2 a β B c , (C59) d = b + b c + b c + b c + β B + β B − a b − a b − a β B − a β B + a b c +2 a β B c , (C60) d = b + b c + b c + b c + β B + β B − a b − a b − a β B − a β B + a b c +2 a β B c , (C61) d = b c + b c + b c + β B − a b − a b − a β B − a β B + a b c + 2 a β B c , (C62) d = b c + b c + β B − a b − a β B + a b c + 2 a β B c , (C63) d = b c . (C64)Every solution of (C30) is a candidate RLP, and everyRLP must obey (C30). Equation (C55) is the conditionthat (C30) is preserved along null geodesics. Thus, everysolution of (C30) must also be a solution of (C55). Since(C30) and (C55) are of the same degree in t, r , it followsthat both must have the same set of zeros. Consequently,their coefficients must be the same. After we make surethat they are the same, we may next investigate whichzeros define RLPs. Thus, the following equations arenecessary conditions for the existence of RLPs:2 c = δ , a i = δ i , i = 3 , . . . , ⇐⇒ c d − d = 0 , d a i − d i = 0 . (C65)By far the simplest condition, as seen from (C64), is theone with i = 9. Even so, further calculations are socomplicated and involve intermediate equations so largethat they could be done only using the computer algebrasystem Ortocartan [26, 27], and we only describe howthey were done.First we observe that the functions (Φ , y, E ) are lin-early independent, and in the resulting final equation canbe used as independent variables. Although the proof isa simple exercise, it requires careful inspection of specialcases that we had earlier excluded for separate investiga-tion, so we give it in the separate Appendix D.The condition (C65) corresponding to i = 9 is d a − d = 0 . (C66)In this, one has to do the whole cascade of substitutions,listed in (C17) – (C64). In the result, we use (C5) – (C6).However, we use the last of (C5) only to eliminate E , rr .For E , r we substitute from (4.1), i.e. E , r = E (Φ , r − Φ ) / Φ , (C67)in order to express E , r through Φ .We then use (C15) to express Φ , trr through Ψ, and(6.4) to express Φ , tr through Ψ. In the result we use(C2) to eliminate x and (C4) to express x through E .From the resulting equation we can factor out Φ , and5we must multiply it by E to get rid of negative powersof E . The final equation thus obtained has on its l.h.s.a polynomial of 4th degree in E , of 4th degree in y andof 6th degree in Φ (recall, we determined that (Φ , y, E )are independent variables). So, if this polynomial is tobe zero, the coefficients of all powers of the independentvariables must vanish separately.Now we take this large polynomial as data for a secondprogram, in which (Φ , y, E ) are treated as independentvariables, no longer as functions. In it, we determine thecoefficient of y and substitute E = 0 to find the termindependent of E . The resulting equation is:16 (cid:0) D + C (cid:1) Φ ΨΨ , t P, r / (cid:2) S ( ε − k ) (cid:3) = 0 . (C68)In this equation we can discard several alternatives: Φ =0 obviously, 4 D + C = 0 because of (C3), Ψ = 0 becauseit defines the Friedmann limit and P, r = 0 because, inview of (B11) and the paragraph above (B22), it leads tothe G /S symmetric cases considered in Appendix G.The only case to consider is thus Ψ , t = 0.In order to investigate it, we substitute Ψ , t = 0 in thelarge main polynomial, and in the resulting somewhatsmaller polynomial we take the coefficient of y . Theequation that results is: − (cid:0) D + C (cid:1) (cid:0) P + C (cid:1) E Φ Ψ × P, r / (cid:2) P S ( ε − k ) (cid:3) = 0 . (C69)The factors E and Φ obviously cannot vanish, and whyzero values of (cid:0) D + C (cid:1) , Ψ and P, r are discarded wasexplained above. Thus, the only case left is 4 P + C = 0,which means P = 0 = C . But this is just a special case of P, r = 0 discarded above. Thus, (C68) does not includeany case that would define any new RLP, apart fromthose considered elsewhere.We go back to (C55) to consider the case d = 0. Thecalculation is almost the same as we did for (C66), withonly minor differences: this time the expression is some-what simpler, and Φ does not factor out. We employthe algebraic program to calculate E d , with the samecascade of substitutions as before, take the coefficient of y at E = 0, and obtain an equation almost identical to(C68):4 (cid:0) D + C (cid:1) Φ Ψ , t P, r / (cid:2) S ( ε − k ) Ψ (cid:3) = 0 . (C70)As explained above, only Ψ , t could possibly be zero, sowe substitute Ψ , t = 0 in the main large polynomial, andin the resulting expression take the coefficient of y . Theresult is almost the same as (C69) − (cid:0) D + C (cid:1) (cid:0) P + C (cid:1) E Φ Ψ × P, r / (cid:2) P S ( ε − k ) (cid:3) = 0 , (C71) The whole equation would take 1830 print lines on paper. and again does not include any case that would define anew RLP.Finally, we go back to (C54), where we assumed thatthe coefficient of x, r in (C16) was nonzero, and investi-gate what happens when it is zero. Then B t, r + B t, r = 0 , (C72)and one of the solutions of this is t, r = 0. This we im-mediately discard because it defines a spacelike curve,while our RLPs must be null geodesics. In consequenceof (C21), another solution of (C72) is B = 0. But thisimplies P, r = 0 or y = 0 or y = C/
2. The first caseleads to the G /S solutions considered in Appendix G,the other two to the axially symmetric solutions of Ap-pendix F. So the only possibility left to fulfil (C72) is t, r = ± s ε − k ) Φ . (C73)Putting this into the coefficient of x, r in (C16) we get: (cid:20) ε − k ) (cid:21) / Φ (" ∓ / ΦΨ , t p ε − k )Ψ Φ + " − / ΦΨ3 p ε − k ) ± (cid:18) Φ Ψ , r Ψ + Φ , r (cid:19) Φ ±
23 Φ (cid:18) Φ E , rr E − Φ , rr (cid:19)(cid:27) = 0 . (C74)In this expression, we do the same series of substitu-tions that we did in the large polynomial that resultedfrom (C66): we express E , rr through E , r using (C5), E , r through Φ , r and Φ using (C67), then x through E and y using (C4), and multiply the whole expression by E . Theresult is: " ∓ / ΦΨ , t p ε − k )Ψ E Φ + " − / ΦΨ3 p ε − k ) ± (cid:18) Φ Ψ , r Ψ + Φ , r +Φ S, r S (cid:19) E Φ ∓
23 ΦΦ , rr E ∓
23 Φ (cid:20)(cid:18) P, r S (cid:19) , r (cid:18) − S E P + Cy P + D P (cid:19) + (cid:18) S, r S (cid:19) , r E + S, r Φ , r S Φ E (cid:21) = 0 . (C75)We then use the fact that (Φ , y, E ) are linearly indepen-dent and require that each coefficient of an independentfunction is zero. There is only one term containing y ,with the coefficient C ( P, r /S ) , r . But C cannot be zero, asexplained below (C3). Thus ( P, r /S ) , r = 0 is the uniqueimplication of this (it will be seen from the following thatwe need not consider whether this condition is consistentwith the other equations that P and S must obey). Thismeans: P, r = α S, (C76)6where α is an arbitrary constant. Taking this into ac-count, and taking the term independent of Φ in (C75)we get: Φ , rr +Φ (cid:18) S, r S (cid:19) , r + S, r Φ , r S = 0 . (C77)Using (C76), this is easily integrated with the result:Φ = χ ( t ) Pα S + χ ( t ) S , (C78)where ( χ ( t ) , χ ( t )) are arbitrary functions of t . Bothappear as integration “constants” of (C77). Using suchΦ in the definition of Ψ, (6.4), we get:ΨΦ = γ ( t ) S , γ ( t ) def = χ χ ,t − χ χ ,t . (C79)But with Ψ = γ ( t ) / ( S Φ) the last three terms in the coef-ficient of Φ in (C75) sum up to zero, and what remainsof that coefficient is the equation ΦΨ = 0. The only so-lution of this can be Ψ = 0, but we know it leads to theFriedmann model.Consequently, the coefficient of x, r in (C16) is alwaysnonzero.Since (C68) and (C70) were, in their respectivebranches of the calculation, among the necessary condi-tions for the existence of RLPs, we conclude that the spe-cial quasi-hyperbolic Szekeres solution defined by (C1)does not contain any RLPs except (possibly) when it be-comes axially symmetric or G /S , but these cases areconsidered in Appendices F and G. Appendix D: Proof that (Φ , y, E ) are linearlyindependent We take the equation α Φ + βy + γ E = 0 (D1)with constant coefficients α, β, γ and prove that it implies α = β = γ = 0.We substitute for Φ from (4.1), then multiply theequation by 2 S E and use (C2) to eliminate x . We thusobtain a polynomial of degree 1 in x and degree 2 in y ,which we denote by P . We take the second derivative of P by xy . The result is:2 P ( βS − γC ) = 0 . (D2)We discard the solution P = 0 because this implies con-stant S (see (B11) and the remarks above (B22)), andthen the metric acquires a G /S symmetry group – theseare discussed in Appendix G. We also discard the case β = 0 because then S , and consequently P , are constant,again leading to the G /S case. So finally, the implica-tion of (D2) is β = 0 = γC. (D3) With this, we go back to P and take its second deriva-tive by y . The result is − γP /S = 0, and the solutionof this is γ = 0. We again go back to P with β = γ = 0,and take its first derivative by y . The result is: αC (Φ , r +Φ S, r /S ) = 0 . (D4)When the expression in parentheses vanishes, Φ becomesa product of the form R ( t ) /S ( r ), where R ( t ) is an ar-bitrary function. Such form of Φ defines the Friedmannlimit, in which we know that all null geodesics are RLPs,so we discard this case. Thus, we follow the case αC = 0.Putting this, together with β = γ = 0, in P and takingthe derivative of the result by x we obtain2 α ( − P Φ , r + P, r Φ − Φ P S, r /S ) = 0 . (D5)If the expression in parentheses should vanish, then thesolution is Φ = R ( t ) P ( r ) /S ( r ), which again leads back tothe Friedmann model. Thus, finally, (D5) implies α = 0,which completes the proof. Appendix E: The RLPs with P = 0 and x = 0 alongthe geodesic. We can leave aside the case when Q, r = 0 because thenthe metric is axially symmetric from the beginning.It is useful to turn this case back to that of AppendixB by the transformation (B14) – (B15) with a = 0 = b . After the transformation we have x ′ = − by ′ /a , i.e.d x ′ / d r = 0 if d y ′ / d r = 0, and e P = − b e Q/a , i.e. e P , r = 0 if e Q, r = 0. Thus the new e P and e Q obey (B8) with D = 0from the beginning, and the RLP condition reduces to(B11) alone, with ( P, Q ) replaced by ( e P , e Q ). The rest ofthe reasoning of Appendix B then applies, unchanged, to( e P , e Q ), with the same result: Corollary 3:
RLPs with P = 0 and x = 0 along the geodesic mayexist only in the special case when the coordinates maybe transformed so that Q = 0 as well, i.e. the metricis axially symmetric, or has a 3-dimensional symmetrygroup. Appendix F: The axially symmetric case P = Q = 0 :only the axial geodesics x, r = y, r = 0 are RLPs We know from Ref. [12] that in the quasi-sphericalcase ε = +1 null geodesics on which x and y are constantexist only when the Szekeres model is axially symmetric.Then coordinates may be chosen so that P = Q = 0, andthe constant-( x, y ) null geodesics have x = y = 0, i.e.intersect each t = constant space on the symmetry axis.In this appendix we show that the statement above ap-plies also with ε = 0 and ε = −
1, that the constant-( x, y )null geodesics are RLPs, and that other RLPs may existonly when the Szekeres spacetime has more symmetries.7
1. Constant- ( x, y ) null geodesics exist only in theaxially symmetric case This thesis was proven in Ref. [12] for ε = +1. Theassumption made there in the proof, that E >
0, doesnot hold for ε = 0 and ε = −
1, so we first verify whathappens when E = 0.It is seen from (4.10) and (4.11) that constant ( x, y )imply E = E = 0 along the geodesic. With E = 0(4.4) and (4.5) then imply that either E , r = 0 at all r ,which means a G /S symmetry (discussed in AppendixG), or E , x = E , y = 0 along the geodesic, which means P and Q being constant, i.e. axial symmetry. Thus, E = 0 along a constant-( x, y ) null geodesic implies axialsymmetry anyway. The equations of Sec. 3.3.1 in Ref. [12] that are im-posed on (
P, Q, S ) by the condition of constant ( x, y )along the geodesic become subcases of our (B8) and(B11) for a general ε . As shown in our Appendix B,they imply axial symmetry for any ε . This is true evenfor the special solution discussed in Appendix C, as wenow show.When x, r = y, r = 0 along a null geodesic, as statedabove, (4.10) and (4.11) imply E = E = 0 along thisgeodesic. Then (C1) implies that either ( i ) P, r = 0, or( ii ) x = 0 and y + D = 0, or ( iii ) y = 0 and x − D = 0.Case ( i ) is axially symmetric. Case ( ii ) implies E = 0along the geodesic, and this was discussed above. Case( iii ) implies D ≥
0. However, the solution of AppendixC has D ≤ D = 0 = ⇒ x = 0 along this geodesic.But then we have again E = 0, which completes the proof.
2. Constant- ( x, y ) null geodesics are RLPs As stated above, along null geodesics of constant ( x, y )we have E = E = 0. Then (6.2) and (6.3) are fulfilledidentically, which means that such geodesics are RLPs.
3. Other RLPs may exist in the axially symmetriccase only with higher symmetries
We will now show that, in the axially symmetric case,(6.2) –(6.3) may have other solutions than constant ( x, y )only when the spacetime has more symmetries than justthe axial.The whole reasoning and calculation is closely analo-gous to the one presented in Appendix C for the specialSzekeres solution. We proved in Appendix B that in the Moreover, as shown in Ref. [20], the location E = 0 is infinitelyfar from any point within the spacetime, i.e. does not in factbelong to the spacetime. axially symmetric case coordinates may be chosen so that P = Q = 0 and the candidate RLP has x = 0. Then: E = x + y S + 12 εS,E = εxS, r /S, E = εyS, r /S. (F1)Note that with ε = 0, this axially symmetric Szekeressolution is in fact plane symmetric. Thus, it will be con-sidered together with other G /S symmetric solutionsin Appendix G. From here on in the present appendixwe assume ε = 0, i.e. ε = ± x = 0. From (5.15) we obtain: y, r E = t, r Φ − Φ ( ε − k )Φ , (F2)and (C12) applies unchanged. We then multiply (6.3)by ΦΦ t, r / [( ε − k )Ψ] and use (F2), (C12) – (C15) and x = 0 in the result. As before, τ factors out and iscancelled, and we obtain an equation almost identical to(C16), with the same definitions of ( B , c , c , c ), butwith y, r in place of x, r , and with the definition of B changed to: B = 2 εyS, r ( ε − k ) S . (F3)We proceed in strict analogy to Appendix B. From(4.9) using (F2) we again obtain (C22), with y, r in placeof x, r , and with the same definitions of ( c , . . . , c ), butwith the definition of A changed to: A def = 2 εy Φ S, r S E Φ . (F4)Then we square the current analogue of (C16) and use(F2) to eliminate y, r from the result. We obtain an 8-thdegree polynomial in t, r , but this time the coefficient of t, r is α = E / Φ (F5)and is sure to be nonzero. Dividing the polynomial by α we obtain (C38) again, but with the definitions of someof the coefficients changed as follows: a = 2 c + c − Φ ε − k − ε Φ S, r y ( ε − k ) S E , (F6) a = 2 c c + c − (cid:0) c + c (cid:1) Φ ε − k + 12 ε Φ Φ S, r y ( ε − k ) S E , (F7) a = c − (cid:0) c c + c (cid:1) Φ ε − k − ε Φ Φ S, r y ( ε − k ) S E , (F8)the remaining ones are the same as given by (C32), (C34)and (C36) – (C37).8Now we differentiate the current analogue of (C30) by r along the null geodesic by the rule (B1). This time,however, x = 0 along our candidate RLP, so no coeffi-cient depends on x . We then use our analogue of (C30)to eliminate t, r , t, r and t, r from the result. Theequation that emerges is an analogue of (C38) with y, r in place of x, r , with the same definitions of ( b , . . . , b ),and with the definitions of ( β , . . . , β ) changed to β = 2 c ,y − c A, (F9) β = a ,y − a A, (F10) β = a ,y − a A, (F11) β = a ,y − a A, (F12) β = a ,y − a A, (F13) β = a ,y − a A, (F14) β = a ,y − a A, (F15) β = a ,y − a A, (F16)where for A the definition (F4) must be used.Here we can assume that the coefficient of y, r in thepresent analogue of (C16) is nonzero – the explanationgiven in the paragraphs containing (C72) – (C75) stillapplies, except that the B given by (F3) cannot vanishfor somewhat different reasons. Then we determine y, r from that equation and substitute the result in the cur-rent analogue of (C38). After multiplying out to get apolynomial in t, r , we again use (C30) to eliminate t, r and t, r (but not t, r ). Then we assume that the coef-ficient of t, r , denoted d , is nonzero (we will check thecase d = 0 later), and divide the equation by d . In thisway we obtain an exact copy of (C55), with the same def-initions (C56) – (C64) of the coefficients; but it is to beremembered that some of the symbols in these formulae(namely B , B , a , a , a and all of ( β , . . . , β )) nowhave different definitions from those in Appendix C.Consequently, eqs. (C65) must still hold, and we againchoose (C66) to investigate, by exactly the same methodas before. By the method of Appendix D we show that(Φ , y, E ) are still linearly independent in the present case(i.e. with E given by (F1), and along the x = 0 geodesic).The explanation given under (C67) still applies, with themodification that now x is nowhere present, so does nothave to be eliminated. In place of (C68) we now obtain:256 ε Φ ΨΨ , t S, r / (cid:2) S ( ε − k ) (cid:3) = 0 . (F17)We recall that we excluded the case ε = 0 (since it istreated in Appendix G), and Ψ = 0 because it reduces the The cases ε = 0 and S, r = 0 define metrics of higher symmetry,treated in Appendix G, while y = 0 (together with x = 0 assumedthroughout this appendix) defines an axial geodesic, which wealready know is an RLP. Szekeres model to Friedmann. We can also exclude S, r =0 because then the metric acquires a G /S symmetryand is also treated in Appendix G. So, as before, theonly case left to investigate is Ψ , t = 0.We substitute Ψ , t = 0 in the large main polynomial,and in the resulting smaller polynomial we take the co-efficient of y . The equation that results is:256 ε E Φ Ψ S, r / (cid:2) S ( ε − k ) (cid:3) = 0 . (F18)The only factors that could vanish here are Ψ and S, r ,but, as explained above, their vanishing leads to simplercases of higher symmetry. Thus, (F18) does not includeany case that would define any new RLP, apart fromthose considered elsewhere.We go back to the paragraph after (F16) to considerthe case d = 0. The explanation given above (C70) stillapplies, but this time, in the expression E d calculatedby the algebraic program, we take the coefficient of y at E = 0, and obtain:64 ε Φ Ψ , t S, r / (cid:2) S ( ε − k ) Ψ (cid:3) = 0 . (F19)As explained above, only Ψ , t could possibly be zero, sowe substitute Ψ , t = 0 in the main large polynomial, andin the resulting expression take the coefficient of y . Theresult is:64 ε E Φ Ψ S, r / (cid:2) S ( ε − k ) (cid:3) = 0 , (F20)and again does not include any case that would define anew RLP.So, the final conclusion is: Corollary 4:
In the axially symmetric Szekeres solutions, apart fromcases of higher symmetry, the only RLPs are the axialnull geodesics that intersect each 3-space of constant t on the symmetry axis. Appendix G: There are no non-radial RLPs in any G /S model. We will investigate the equation χ = 0 (see (7.1))and will show that it has no solutions defining nonradialRLPs, unless the model reduces to Friedmann.Several equations in this Appendix follow from the cor-responding ones in the Appendix C as the special case E , r = 0; they are similar but not identical.We will use all equations adapted to the special casediscussed in Sec. VII, i.e. ζ = ψ = ξ = η = E , r =d y/ d r = 0. From (5.15) we find x, r E = t, r Φ − Φ r ( ε − k )Φ . (G1)Then, using (G1) in (5.16) we obtain τ, r t, r = τ Φ r Ψ ε − k + τ Φ , t Φ t, r , (G2)9where Ψ is defined by (6.4) – as seen from (2.9) this isa coefficient of shear, whose vanishing defines the Fried-mann limit.We now substitute (G1) and (G2) in χ = 0, where χ isgiven by (7.1). We multiply the result by ΦΦ , r t, r / [( ε − k )Ψ], and cancel τ that factors out. The result is: W = t, r + c t, r + c t, r + c = 0 , (G3)where c = 2Φ (cid:0) Φ , r Ψ , t − Ψ (cid:1) ( ε − k )Ψ (G4) c = Φ (Φ , r Ψ , r − Φ , rr Ψ) − , r Ψ( ε − k )Ψ (G5) c = 2ΦΦ , r Ψ( ε − k ) . (G6)Adapting (4.9) to the G /S case, eliminating x, r withuse of (G1) and using (G3) to eliminate t, r we obtain: t rr = c t, r + c t, r + c , (G7)where c = 2Ψ , t Ψ + Φ , t Φ , (G8) c = Ψ , r Ψ − Φ , r Φ + k, r ε − k ) , (G9) c = Φ , r Ψ ε − k . (G10)Now we differentiate (G3) along a null geodesic (sincethe equation must hold all along it), by the rule givenin (B1), and use (G7) to eliminate t, rr . The resultingequation is of 4-th degree in t, r . We eliminate the 4thpower of t, r by using (G3). In the end, we obtain anequation of degree 3 in t, r , which we write symbolicallyas follows: d t, r + d t, r + d t, r + d = 0 . (G11)The expressions for the coefficients in (G11) are: d = c ,t + 3 c − c c , (G12) d = c ,r + c ,t + 3 c + 2 c c − c c , (G13) d = c ,r + c ,t + 2 c c + c c − c c , (G14) d = c ,r + c c . (G15)For the beginning, let us assume that d = 0. Forfurther considerations it will be more convenient to write(G11) as follows: W = t, r + δ t, r + δ t, r + δ = 0 , (G16)where δ i def = d i /d , i = 2 , , χ = 0 in (7.1). Thus(G3), just like (7.1), defines the collection of RLPs to-gether with the conditions of their existence. Every solu-tion of (G3) and (7.1) is a candidate RLP, and every RLP must obey (G3) and (7.1). Then, (G16) is the conditionthat the solutions of (G3) are preserved along the nullgeodesics, thus every solution of (G3) must be a solutionof (G16) and vice versa. But if (G3) and (G16) havethe same set of solutions, then they must be identical,i.e. their respective coefficients must be equal. Thus, thenecessary conditions for the existence of RLPs are: c i = δ i ⇐⇒ c i d − d i = 0 , i = 2 , , . (G17)As with the previous calculations, we employed the al-gebraic program Ortocartan [26, 27]. We consider (G17)with i = 4, the simplest one. In it, we substitute (G4) –(G15) and multiply the result by (Ψ / Φ , r ) to get a poly-nomial in Φ, Ψ and their derivatives (Φ , r factors out inthe original expression). The resulting expression is sim-ple enough to be shown here: W = − ΦΦ , r Ψ k, r +8Φ Φ , r Ψ Ψ , t +4ΦΦ , t Φ , r ΨΨ , t ( ε − k ) + 4Φ Φ , r ΨΨ , tt − Φ , r Ψ , t ( ε − k ) + 3ΦΦ , r ΨΨ , r − , r Ψ − , rr Ψ ( ε − k ) = 0 . (G18)We assume Λ = 0 and take the k > t, r ) = Mk (1 − cos η ) ,η − sin η = k / M [ t − t B ( r )] , (G19)where η is a parameter (dependent on t and r ), and t B ( r )is an arbitrary function, the bang time. We introduce theabbreviations: I M def = 3 k, r k − M, r M , D M def = k t B,r
M . (G20)The derivatives of Φ and Ψ can then be written asΦ , r = (cid:18) Mk (cid:19) , r (1 − cos η ) + M I M k sin η ( η − sin η )1 − cos η − M D M k / sin η − cos η , (G21)Φ , t = √ k sin η − cos η , (G22)Φ rr = (cid:18) Mk (cid:19) , rr (1 − cos η )+ (cid:18) Mk (cid:19) , r sin η (cid:20) I M η − sin η − cos η − D M √ k (1 − cos η ) (cid:21) + M ( I M ) , r k sin η ( η − sin η )1 − cos η + M I M k (2 sin η − sin η cos η − η )( η − sin η )(1 − cos η ) − M I M D M k / η − sin η cos η − η (1 − cos η ) − (cid:16) √ kt B,r (cid:17) , r sin η − cos η − M D M k (1 − cos η ) , (G23)Ψ = √ kI M η − η cos η − η (1 − cos η ) + D M η (1 − cos η ) , (G24)Ψ , r = (cid:16) √ kI M (cid:17) , r η − η cos η − η (1 − cos η ) + I M η + η sin η (5 + cos η ) − − η (1 − cos η ) × h √ kI M ( η − sin η ) − D M i + ( D M ) , r η (1 − cos η ) − D M sin η (5 + cos η )(1 − cos η ) (cid:20) I M ( η − sin η ) − D M √ k (cid:21) , (G25)Ψ , t = k I M M η + η sin η (5 + cos η ) − − η (1 − cos η ) − k / D M M sin η (5 + cos η )(1 − cos η ) , (G26)Ψ , tt = k / I M M (1 − cos η ) ( − η − η cos η +2 η sin η + 33 sin η + 9 sin η cos η (cid:1) − k D M M × −
19 cos η + 2 sin η − − cos η ) . (G27)From here on, the intermediate expressions become solarge that we cannot reproduce them here; we only de-scribe how the calculation is done. We substitute (G21)– (G27) in (G18) and multiply the result by (1 − cos η ) to get a polynomial in (1 − cos η ) (of 6th degree). It isalso a polynomial of 4th degree in η . The function η isthe only one in this polynomial that depends on t , thecoefficients of η , (1 − cos η ) and their powers depend onlyon r . Thus we treat η and (1 − cos η ) as independent vari-ables, and the coefficients of their different powers mustvanish separately.In W transformed in this way we now take the co-efficient of η , the resulting equation is a polynomial ofdegree 4 in (1 − cos η ): M I M ( ε − k ) (cid:2) ε/k − − cos η ) − ε/k − − cos η ) +(216 ε/k − − cos η ) − (189 ε/k − − cos η ) − . (G28)The term independent of (1 − cos η ) clearly shows that I M = 0 is a unique solution of this, independently of thevalue of k . So, we substitute I M = 0 in the main large polynomialand in the resulting smaller polynomial we take the termindependent of (1 − cos η ). The equation that results is: − M D M / [ k ( ε − k ) ] = 0 . (G29)Here the unique solution is D M = 0. But with I M = D M = 0 we get Ψ = 0 from (G24), i.e. the Friedmannlimit. Thus, c d − d = 0, which is one of the necessaryconditions for the existence of RLPs, can in this casebe fulfilled only when the Szekeres model trivializes toFriedmann.The calculation above was done for k >
0. The calcu-lation with k < k, η ) in allequations with ( − e k, i e η ).When k = 0, we have necessarily ε = +1 and thecalculation must be done separately. Then we have:Φ = (cid:18) M (cid:19) / ( t − t B ) / , Ψ = 23 (cid:18) M (cid:19) / t B,r ( t − t B ) − / . (G30)With the r -coordinate chosen so that M = M r , where M is a constant, this simplifies W in (G18) to W = − (64 / M r t B,r ( t − t B ) − − (14 / × / M / r t B,r ( t − t B ) − / +(256 / M r t B,r ( t − t B ) − +14 × / M / r t B,r ( t − t B ) − / − M r t B,r ( t − t B ) − (G31) − × / M / r t B,r ( t − t B ) − / +3 × / M / r t B,rr t B,r ( t − t B ) − / = 0 . Now the independent variables are t and r , and t ap-pears always in the combination ( t − t B ). Thus differentpowers of ( t − t B ) are linearly independent, and their coef-ficients must vanish separately. Whichever term we take,except for the last two, the result is always the same: t B,r = 0 (G32)(because M = 0 is the vacuum, i.e. Schwarzschild, limitof the L–T model). This guarantees that all the termsin (G31) vanish. However, as seen from (G30), t B,r = 0means Ψ = 0, i.e. zero shear (see (6.4) and (2.9)), i.e.the Friedmann limit. Thus, there are no non-radial RLPsalso when k = 0, which completes the proof in the case d = 0.We go back to (G15), where we assumed d = 0 andproceed from there on to consider the case d = 0. In-stead of (G18) we now get: W = − , r Ψ , t / Ψ + 2ΦΦ , r Ψ , tt / Ψ ε − k
1+ 4ΦΨ , t +2Φ , t Φ , r Ψ , t / Ψ + (3 / k, r ε − k − , r Φ + 3 Ψ , r Ψ = 0 . (G33)As before, we begin by considering the case k >
0. Wemultiply W by Ψ Φ(1 − cos η ) and substitute for Φ andΨ from (G19) – (G27). What results is a polynomial ofdegree 6 in (1 − cos η ) and of degree 3 in η . Taking thecoefficient of η we obtain: W = M I M sin ηε − k (cid:2) − ε − k )(1 − cos η ) +45( ε − k )(1 − cos η ) − ε − k )(1 − cos η )+30 k (1 − cos η ) − k ] = 0 . (G34)Looking at the term independent of (1 − cos η ) we seethat the unique solution of this is I M = 0.So we substitute I M = 0 in the main polynomial, andin the resulting expression we take the term independentof (1 − cos η ). The resulting equation is:36 sin ηM D M √ k ( ε − k ) = 0 . (G35)The unique solution of this is D M = 0, which, togetherwith I M = 0, leads back to the Friedmann limit. Thus,no RLPs exist in nontrivial Szekeres spacetimes in thiscase, either.The argument given before, that the result for k < k, η ) → ( − e k, i e η ), is still valid.So we now consider d = 0 with k = 0. We substitute k = 0 (and, as is necessary, ε = +1) in (G33), multiplythe result by Ψ, substitute then for Ψ and Φ from (G30),and obtain: W = (16 / M r t B,r ( t − t B ) − +2 × / M / rt B,r ( t − t B ) − / − (56 / M r t B,r ( t − t B ) − +36 / M / rt B,rr ( t − t B ) − / = 0 . (G36)The coefficients of independent powers of ( t − t B ) have tovanish separately, as explained before. Whichever termwe take, except for the last one, the result is always thesame: t B,r = 0 (G37)and this implies the Friedmann limit in the same wayas explained after (G32). This also guarantees that thewhole of (G36) is fulfilled.Thus, in every case considered, the assumption thatnon-radial RLPs could exist leads to either the Fried-mann limit or the Schwarzschild limit. The final conclu-sion is that the only RLPs in the G /S models are theradial null geodesics. (cid:3) Appendix H: A detailed description of the modelpresented in Sec. VIII.
The algorithm used in the calculations discussed in Sec.VIII consists of following steps:1. First we set the observer at R (the present-dayareal distance) and consider sources which are, atthe present instant, away from the observer by 1Gly ( ≈ . • The radial coordinate is chosen to be the arealradius at the present instant: ¯ r = Φ( t , r ).However, to simplify the notation we will omitthe bar and denote the new radial coordinateby r . • The chosen asymptotic cosmic background isan open Friedman model, i.e. Ω m = 0 . ρ b = Ω m × ρ cr = 0 . × H πG (1 + z ) , (H1)where the Hubble constant is H = 72 km s − Mpc − . • The initial time t is calculated from the fol-lowing formula for the background FriedmanUniverse t ( z ) = ∞ Z H − (1 + e z ) − d e z p Ω m (1 + e z ) + (1 − Ω m )(1 + e z ) , (H2) • The age of the universe is assumed to be ev-erywhere the same: t B = 0. • The function M ( r ) follows from (2.5), wherethe present-day density is ρ ( t , r ) = ρ (cid:2) δ − δ exp (cid:0) − r /σ (cid:1)(cid:3) • Because of the assumed spherical symmetrye ν = 1. • The function k ( r ) can be calculated from(2.6). • Then the evolution of the model can be calcu-lated from eq. (2.3).3. We then find a null geodesic that joins the observerand the source. The angle between the directiontowards the source and the direction towards theorigin, at the present instant, is denoted as γ .24. The null geodesics are found in the following man-ner: • Because of spherical symmetry we may set oneof the angular components of the null vectorto zero. We set k φ = 0. • The second angular component, k θ , followsfrom R k θ = J = const = R sin γ , where R = R ( t , r ), i.e. at the observer’s position.This relation is a consequence of (4.10) and(4.11) and was derived in [12] (see equation(3.26) in [12]). • The radial component is evaluated from (4.8)with E = 0 = E = E , r , and Σ = E (d θ/ d r ) . • The time component of the null vector is foundfrom k α k α = 0. 5. We then find two other null geodesics: one thatwill reach the observer in 1 Gy time, the other onethat arrived at the observer’s position 1 Gy ago.Because of the non-RLP effect these geodesics ap-proach the observer at angles that are different from γ .6. The difference between these angles allows us toevaluate the rate of change of the angle γ . Acknowledgments:
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