Conformal Methods in General Relativity with application to Conformal Cyclic Cosmology: A minicourse given at the IXth IMLG Warsaw 2018
aa r X i v : . [ g r- q c ] F e b Conformal methods in General Relativity withapplication to Conformal Cyclic Cosmology:A minicourse at IX IMLG
Paul TodJune 18th-22nd 2018
Abstract
In these lectures my aim is to review enough of conformal differ-ential geometry in four dimensions to give an account of Penrose’sconformal cyclic cosmology.
My approach will be fairly concrete. For more abstract approaches see [9]or [12].
These largely follow [52]: we have a metric g ab with signature (+ − −− )and will use abstract indices where possible. The Christoffel symbols, whichneed concrete indices, here written as bold, areΓ abc = 12 g ad ( g db , c + g dc , b − g bc , d ) , (1)and the metric or Levi-Civita covariant derivative in concrete indices is ∇ a X b = ∂∂x a X b + Γ bac X c . The Riemann tensor is defined in abstract indices by( ∇ a ∇ b − ∇ b ∇ a ) X c = R cabd X d . (2)1t decomposes into irreducible parts as R abcd = C abcd + 12 ( g ac R bd + g bd R ac − g bc R ad − g ad R bc ) − R ( g ac g bd − g ad g bc ) , (3)with C abcd the Weyl tensor and R ac = R dadc , R = g ab R ab , the Ricci tensor and Ricci scalar respectively. The Einstein tensor is G ab = R ab − Rg ab , and with the conventions used here, the Einstein field equations (or EFEs)with cosmological constant λ are G ab = − πGT ab − λg ab , (4)and for vacuum R ab = λg ab . (5)We recall the definition L ab = −
12 ( R ab − Rg ab ) (6)of a tensor sometimes called the Schouten tensor , which will be useful below.It is called P ab by some authors (where the kernel letter is a capital rho),and allows a slight simplification of (3): R cdab = C cdab − δ [ c [ a L d ] b ] . For the record, recall the Bianchi identity in the form ∇ d C dabc = 2 ∇ [ a L b ] c . (7)The tensor on the right here is sometimes called the Cotton tensor .The Lie derivative of a tensor by a given vector field X a is obtainedrecursively by L X f = X a ∇ a f, L X Y a = X b ∇ b Y a − Y b ∇ b X a = [ X, Y ] a f and a vector field Y , where [ X, Y ] a is the Lie bracket, andthen extended so as to satisfy Leibniz rule. It is independent of torsion-freeconnection, and in particular therefore L X g ab = X c ∇ c g ab + g ac ∇ b X c + g bc ∇ a X c = ∇ a X b + ∇ b X a . A vector field X a is Killing if L X g ab = 0 and conformal Killing if L X g ab = φg ab for some function φ . The importance of such vector fieldsis in defining conserved quantities for the geodesic equation: • Recall a curve Γ with tangent vector V a is a geodesic if V b ∇ b V a = f V a , and affinely-parametrised if this holds with f = 0. • A geodesic is time-like, space-like or null according as g ab V a V b is pos-itive, negative or zero (and of course this holds at all points of Γ if itholds at one). • For an affinely-parametrised geodesic Γ with tangent vector V a and aKilling vector X a , the contraction g ab V a X b is constant along Γ. If thegeodesic is null this is also true with a conformal Killing vector. • One can define Killing tensors and conformal Killing tensors as ten-sors K a ...a n which give rise to higher-order constants of the motion K a ...a n V a . . . V a n for geodesics or null geodesics. • This discussion can be phrased in terms of the phase space or cotangentbundle T ∗ ( M ) = { ( p a , q b ) } of space-time, where the indices can betaken to be abstract, and the Poisson bracket { , } on T ∗ ( M ): thereis a vector field { g ab p a p b , ·} on T ∗ ( M ) called the geodesic spray , whoseintegral curves are lifted geodesics. A Killing vector defines a linearpolynomial in momentum X a p a which is constant along the geodesicspray: { g ab p a p b , X c p c } = 0 . This language is useful in discussing the Einstein-Vlasov system.
We shall be interested in conformally rescaling a space-time metric, whichis the transformation g ab → ˆ g ab = Ω g ab , (8)3or a real-valued, usually smooth function Ω. Necessarily thenˆ g ab = Ω − g ab and it’s easy to see thatˆΓ abc − Γ abc = δ ab Υ c + δ ac Υ b − g bc g ad Υ d , (9)where Υ a = Ω − ∇ a Ω (and we’re allowed abstract indices as the differencebetween two connections is a tensor).It is now straightforward if tedious to calculate the transformation of thecurvature. One findsˆ C dabc = C dabc , (10)ˆ R ab = R ab + 2 ∇ a Υ b − a Υ b + g ab ( ∇ c Υ c + 2Υ c Υ c ) , (11)ˆ R = Ω − ( R + 6 (cid:3) ΩΩ ) . (12)The Schouten tensor (6) transforms asˆ L ab = L ab − ∇ a Υ b + Υ a Υ b − g ab Υ c Υ c . (13)It’s worth noting the transformation of the volume form: ǫ abcd → ˆ ǫ abcd = Ω ǫ abcd , whence also ˆ ǫ cdab = ǫ cdab (14)so that this tensor, which defines duality on 2-forms, is therefore conformallyinvariant.The tensor ǫ abcd ǫ pqrs which defines the volume form on the phase space T ∗ ( M ) is also conformally invariant. A scalar ω is said to have conformal weight k if it transforms as ω → ˆ ω = Ω k ω when g ab → ˆ g ab = Ω g ab . Evidently scalars of conformal weight k can be regarded as sections of abundle E k and one can define a weighted covariant derivative ∇ ( cw ) a ω := ( ∇ a − k Υ a ) ω, g ab has con-formal weight 2, the duality operator ǫ cdab has conformal weight 0, and soon. A few points about the transformation of geodesics under conformal rescal-ing: • A null geodesic for g ab continues to be a null geodesic for ˆ g ab ; affine-parametrisation is preserved if one sets ˆ V a = Ω − V a or equivalentlyˆ V a = V a . • Time-like or space-like geodesics are not preserved. • A Killing vector X a for g ab is a Killing vector for ˆ g ab iff X a Υ a = 0,otherwise it is a conformal Killing vector. Conformal Killing vectorstransform to conformal Killing vectors but Killing tensors in generaldon’t transform nicely. • de Sitter space : This is the space of constant curvature obtained asthe hyperboloid T − X − Y − Z − W = − H − = constantin the 5-dimensional Minkowski space with metric g = dT − dX − dY − dZ − dW . It’s an exercise to introduce coordinates (start with T = H − sinh Ht )and find the metric as g = dt − H − cosh ( Ht )( dr + sin r ( dθ + sin θdφ )) , and then it isn’t difficult to discover that C dabc = 0 , R ab = 3 H g ab . Evidently the underlying manifold is R × S and the R -factor can becompactified by conformal rescaling with Ω = H sech( Ht ):ˆ g = Ω g = dτ − ( dr + sin r ( dθ + sin θdφ )) , (15)5here dτ = Hdt cosh
Ht , and w.l.o.g. e Ht = tan( τ / , so that the range −∞ < t < ∞ corresponds to 0 < τ < π .The metric ˆ g is referred to in this context as the Einstein static cylin-der . We can add boundaries to the de Sitter space to compactify it inthe Einstein cylinder: a past boundary at τ = 0 and a future boundaryat τ = π . • anti-de Sitter space : One can similarly consider the hyperboloid T + W − X − Y − Z = H − = constantin the 5-dimensional space with metric g = dT + dW − dX − dY − dZ . This is not simply-connected as it contains circles with
X, Y, Z con-stant and T + W = constantwhich are time-like (and therefore a causal problem) but the universalcover is free of these pathologies and is the space of constant curvatureknown as anti-de Sitter space. It’s an exercise to introduce coordinates(start with T + iW = H − e iτ cosh( HR )) and find the metric as g = H − cosh ( HR ) dτ − ( dR + H − sinh ( HR )( dθ + sin θdφ )) , and then calculate C dabc = 0 , R ab = − H g ab . This is a warped product metric on R × H . If we rescale with Ω = H sech( HR ) and introduce the new radial coordinate r bysin r = tanh( HR )then ˆ g = dτ − ( dr + sin r ( dθ + sin θdφ )) , which is (15) again, but the range 0 ≤ R < ∞ of the anti-de Sitterradial coordinate corresponds to 0 ≤ r < π/ S of the Einsteinstatic cylinder. Thus anti-de Sitter space is also conformally related toa piece of the Einstein static cylinder consisting of the product of thetime-axis with a ball of finite radius in the S factor. The boundary r = π/ R × S .6 Minkowski space : Taking this in spherical polars for the space part,the metric is g = dT − ( dR + R ( dθ + sin θdφ )) . Introduce null coordinates u = ( T − R ) / , v = ( T + R ) / g = 4 dudv − ( v − u ) ( dθ + sin θdφ ) , and then set u = tan p, v = tan q and Ω = cos p cos q to obtainˆ g = 4 dpdq − sin ( q − p )( dθ + sin θdφ ) . Now put p = ( τ − r ) / , q = ( τ + r ) / p, q )-plane − π < p ≤ q < π − π < τ − r < π, − π < τ + r < π, r ≥ . The boundary of Minkowski space in the Einstein static cylinder con-sists of the past light cone (conventionally called I + and pronounced‘scri-plus’) of the point i + = { τ = π, r = 0 } and the future light cone( I − or ‘scri-minus’) of the point i − = { τ = − π, r = 0 } , which meetat the point i = { τ = 0 , r = π } . These parts of the boundary haveconventional names, motivated by the classes of geodesics which havethem as end-points: i + is future time-like infinity , i − is past time-likeinfinity , i is space-like infinity , I + is future null infinity and I − is past null infinity . In the early days of GR, one understood an isolated system to be onewhich could be expressed in coordinates such that the metric approached flatspace at a suitable rate in a suitable radial coordinate. A geometrical andcoordinate-independent notion capturing this idea was introduced by Pen-rose in the early 1960’s, motivated by the examples above. The definition[52] can be given as follows: 7 efinition : A smooth space-time M with metric g is asymptoticallysimple if there is a smooth manifold ˆ M with boundary I and metric ˆ g anda smooth scalar function Ω such that1. M = int ˆ M ,2. ˆ g ab = Ω g ab in M ,3. Ω > M ; Ω = 0 , ∇ a Ω = 0 on I ,4. every null geodesic in M has a future and a past endpoint on I .One can modify the definition to have lower degrees of differentiability, defin-ing e.g. C k -asymptotic simplicity .Condition 4 is included to ensure that one has all of I and to excludetrivial examples with empty I , but would be too strong for example in space-times with black holes. Thus one introduces weak asymptotic simplicity : aspace-time M is weakly asymptotically simple (or WAS) if there exists anasymptotically-simple M ′ and a neighbourhood O ′ of I in ˆ M ′ such that O ′ ∩ M ′ is isometric to a subset of M .We draw some conclusions from the assumption that M is WAS: • From (12), replacing Ω by its inverse:Ω − R = ˆ R + 6Ω ˆ (cid:3) (Ω − ) = ˆ R − − ˆ (cid:3) Ω + 12Ω − ˆ g ab Ω a Ω b , and so R = Ω ˆ R −
6Ω ˆ (cid:3)
Ω + 12ˆ g ab Ω a Ω b . (16)Now at I , Ω vanishes, Ω a is the tangent to I , nonvanishing by assump-tion, and we can assume ˆ R and ˆ (cid:3) Ω are finite. The Einstein equationswith cosmological constant λ imply R = 4 λ + 8 πGT, where T is the trace of the energy monmentum tensor. We concludeby evaluating (16) at I that, provided T is zero in a neighbourhood of I , the surface I is space-like, time-like or null (i.e. its normal is time-like, space-like or null) according as λ is positive, negative or zero .Note that this is in line with the explicit examples of de Sitter space,anti-de Sitter space and Minkowski space treated earlier.8 From (11) replacing Ω by its inverse:Ω R ab = Ω ˆ R ab − ∇ a Ω b − ˆ g ab ˆ g cd ( ˆ ∇ c Ω d − − Ω c Ω d ) . Take the trace-free part to deduce that if the trace-free part of T ab iszero (or just bounded) in a neighbourhood of I then the trace-free partof ˆ ∇ a Ω b vanishes at I . This means that I is umbilic if time-like orspace-like and shear-free if null. Refining the choice of Ω allows w.l.o.g.the assumption that I is actually extrinsically flat in the time-like orspace-like cases and expansion-free in the null case. • Using (11) and (9) we calculateˆ ∇ d ˆ C dabc = ∇ d C dabc + C dabc Υ d . Multiply by Ω and take the limit at I . From the Bianchi identity (7)and the Einstein field equations, if the space-time is vacuum near I or if the matter content (and therefore the physical Ricci curvature)falls off fast enough then the term ∇ d C dabc goes to zero at I whilethe term on the left is bounded: we conclude that ˆ C dabc Ω d vanishes at I . In the time-like or space-like case this is sufficient to conclude thatˆ C dabc = 0 at I . In the null case the conclusion follows if one can showthat each component of I is topologically R × S (which is possiblebut intricate; see [47],[42]). These are a class of curves with better properties under conformal rescalingthan metric geodesics, ([6],[17],[56]). There are two slightly different ways todescribe them: first, a conformal geodesic is a curve γ with tangent vector v a and a one-form b a given along it and satisfying the system: v c ∇ c v a = − v c b c ) v a + g ac b c ( g ef v e v f ) , (17) v c ∇ c b a = ( v c b c ) b a − g ac v c ( g ef b e b f ) + L ac v c , (18)where L ab as in (6). In this form, the system transforms to itself underconformal rescaling with ˆ v a = v a , (19)ˆ b a = b a − Υ a . (20)9t’s clear from (17) that a null conformal geodesic is in fact a null metricgeodesic and, conversely, given a null metric geodesic one can find ( v a , b a ) tomake it a conformal geodesic. In [56] an interpretation of non-null conformalgeodesics was given as follows: given a segment of a non-null conformalgeodesic on which ( v a , b a ) are finite and nonzero solve for Ω in v a Ω a = Ω v a b a ;this gives Ω along the segment; find it in a neighbourhood of the segmentsuch that Υ a = b a at the segment and ∇ a Υ b = b a b b − g ab b c b c + L ab , again, along the segment. There will be many such Ω but now rescaling byone such reduces b a to zero by (20) and L ab to zero at the segment by (13).Now by (17) the conformal geodesic is a metric geodesic so a conformal geodesic γ is a metric geodesic in a rescaled metric for whichthe Ricci tensor vanishes at γ . A conformal geodesic admits a preferred parameter σ defined up to choiceof origin by v c ∇ c σ = 1 . (21)There is a reparametrisation freedom of M¨obius transformations in σ : σ → ˜ σ = aσ + bcσ + d , (22)with ad − bc = 1, so that σ may be called a projective parameter , providedthis is accompanied by a transformation of v a and b a :˜ v a = ( cσ + d ) v a (23)˜ b a = b a + f g ac v c (24)where f = − g ef v e v f ) − c ( cσ + d ) − . This transformation draws attention to a problem in working with confor-mal geodesics: it is possible for ˜ v a to vanish and ˜ b a to be singular at aregular point both of the manifold and of the curve, by choice of projectiveparameter.One way to avoid this problem is to introduce a third-order form ofthe equations. How this is done depends slightly on the signature so we’ll10estrict to a time-like conformal geodesic in a Lorentzian space-time. (See[33] for a discussion and application of null conformal geodesics.)Introduce the unit tangent u a = χ − v a , where χ = g ef v e v f . Now thecurve is parametrised by proper-time and (17) can be solved for b c in termsof the acceleration a c = u a ∇ a u c as b c = a c + ( u a b a ) u c , (25)where u a b a = − χ − u a ∇ a χ = − χ/χ and the overdot is differentiationw.r.t. proper-time. Use this in (18) to find u c ∇ c a b = u b ( − g ef a e a f − L ef u e u f ) + L bc u c , (26)and ¨ χ = −
14 ( g ef a e a f + 2 L ef u e u f ) χ. (27)In this form of the equations, (26) is a third-order equation for the curvein proper time and then (27) is a second-order equation for χ which withthe acceleration a b determines the one-form b c . To obtain the projectiveparameter σ from proper-time note that1 = v c ∇ c σ = χ u c ∇ c τ = χ dσdt , so that dσ = dt/χ . If we choose two solutions χ , χ of (27) with unitWronskian: χ ˙ χ − χ ˙ χ = 1 , so that ddt (cid:18) χ χ (cid:19) = 1 χ and χ − dt = d (cid:18) χ χ (cid:19) , then an allowed choice for σ is χ /χ . Different choices of the solutions χ i determine projective transformations of σ , and zeroes and poles of theprojective parameter are tied to zeroes and poles of the solutions of (27). Given a conformal geodesic γ , one may define a conformally-invariant prop-agation of vectors along it according to v b D b e a := v b ∇ b e a + ( v c b c ) e a + ( e c b c ) v a − g ac b c ( g ef v e v f ) = 0 . (28)From the way this is written, it is evidently parallel propagation in a con-nection which differs from the metric connection by a change of Christoffelsymbols: Γ → ˜Γ = Γ + δ ab b c + δ ac b b − g ad b d g bc . D a , is symmetric and preserves the metric up to scale: D a g bc = − b a g bc , so it’s a Weyl connection . In particular v a is propagated in this connection,since (17) is just v b D b v a = 0 , and the statement v b D b e a = 0is conformally invariant (i.e. preserved with the rescaling ˆ e a = e a ). We candefine an orthogonal basis of vectors e aα with dual basis θ αa along γ and thenthe components C δαβγ := C dabc e aα e bβ e cγ θ δd of the Weyl tensor are conformally-invariant, as are components of deriva-tives D a · · · D a n C dabc . This will be relevant in section 3.2.4. • In 4 dimensions, there is a 9-parameter family of unparametrised con-formal geodesics and a 5-parameter family of unparametrised metricgeodesics (so there are clearly ‘more’ conformal geodesics). • Metric geodesics are not conformal geodesics in general but will be ina Ricci-flat space, or if the velocity vector is an eigenvector of the Riccitensor (an example of this case is provided by the matter flow lines ina perfect-fluid FRW cosmological model). Null geodesics however arealways conformal geodesics. • In Euclidian space, the conformal geodesics are planar circles (or straightlines). In Minkowski space-time the constant acceleration world-linesare conformal geodesics – they go through I and so components in aconstant basis of the velocity u a become infinite. (See [62] for moreexamples and results). • In general, neither Killing vectors nor conformal Killing vectors giveconserved quantities for conformal geodesics, so the equations havebeen hard to integrate explicitly (again, there are examples when itcan be done in [62]). 12
In the Einstein static cylinder, the vector field ∂/∂τ is tangent toconformal geodesics and to metric geodesics. A choice of projectiveparameter is σ = tan τ so that σ has an infinite sequence of poles andzeroes for an infinite range in τ . If we consider an FRW metric with S space sections in the form: g = dt − t k ( dr + sin r ( dθ + sin θdφ )) , then with conformal factor Ω = t − k this rescales to the Einstein staticcylinder ˆ g = t − k ( dt − t k ( dr + sin r ( dθ + sin θdφ ))= dτ − ( dr + sin r ( dθ + sin θdφ ))with τ = ± t − k / (1 − k ). Now with k > t = 0 is sent to τ = ±∞ and in particular ithasn’t been added as a boundary. This can be seen to have happenedbecause there are infinitely many zeroes and poles of the projectiveparameter along the conformal geodesic which is a matter flow linein the FRW metric. Thus it makes sense to say that for k > The Maxwell field can be regarded as a 2-form F and then the source-freeMaxwell equations require this and its dual to be closed: d F = 0 = d F ∗ . If we assume that under conformal rescaling F → ˆF = F (i.e. conformalweight 0) then also F ∗ → ˆF ∗ = F ∗ by (14) so that both Maxwell equationsare preserved. In index notationˆ F ab = F ab , ˆ F ∗ ab = F ∗ ab = 12 ǫ cdab F cd . For the energy-momentum tensor we have T ab = − π (cid:18) F ac F cb − F cd F cd g ab (cid:19) , T ab = Ω − T ab , (29)and that this preserves the conservation equation.Note • for any trace-free T ab the transformation (29) preserves the conserva-tion equation, but if T ab is not trace-free then no simple transformationpreserves conservation; • while any Killing vector K a generates a conserved current J a := T ab K b from any conserved T ab , a conformal Killing vector X a will producea conserved current ˜ J b := T ab X b from a conserved trace-free T ab , butnot otherwise. We’ll see an application of this in section 3.6.2. The Klein-Gordon or massless scalar field equations do not have a simpletransformation but there is a modification which does. Following the pre-sentation in [34], consider the tensor D ab [ φ, g cd , α ] := 4 φ a φ b − g ab g cd φ c φ d − φ ∇ a φ b + 2 φ L ab + 2 αφ g ab with L ab = − R ab + 112 Rg ab as usual, α a real constant and φ a real scalar field. Define Q ( φ ) := (cid:3) φ + 16 Rφ − αφ . We claim that, if ˆ φ = Ω − φ (so φ has conformal weight -1) then D ab [ ˆ φ, ˆ g cd , α ] = Ω − D ab [ φ, g cd , α ] , (this is just a matter of checking). Note that g ab D ab = − φQ ( φ ) , and that ∇ a D ab = 4 Qφ b − φQ b . Therefore if we require φ to satisfy the field equation Q = 0 i.e. (cid:3) φ + 16 Rφ − αφ = 0 (30)14hich is a non-minimally coupled Klein-Gordon equation , then D ab [ φ, g cd , α ]is trace-free and divergence-free and by the argument around (29) so is D ab [ ˆ φ, ˆ g cd , α ] in the rescaled connection. Therefore ˆ φ satisfies (30) w.r.t.ˆ g (which can be checked directly). In this sense the field equation (30) isconformally invariant.A scalar field satisfying (30) is conveniently called a conformal scalar and D ab [ φ, g cd , α ] can be taken as its energy momentum tensor. We’ll comeback to this. In statistical mechanics a distribution of matter is defined by its distribu-tion function, a non-negative function f ( q a , p a ) on the phase space T ∗ ( M ).For simplicity, the particles are often thought of as belonging to a singlespecies and the support of f is confined to the future mass shell { g ab p a p b = m , p > } . This matter model typically does not have good behaviourunder conformal rescaling (unsurprisingly) but it’s a different story if onerestricts the support to the future null cone N + q at the point labelled q –this is massless Vlasov or massless Boltzmann .Vlasov is the case of collisionless matter and then f is constant alongthe geodesic flow: { g ab p a p b , f ( q, p ) } = 0equivalently L f := g ab p a ∂f∂q b − p a p b ∂g ab ∂q c ∂f∂p c = 0 , (31)which can conveniently be called the Vlasov equation.For the Boltzmann case one needs a collision-term on the right-hand-sideof the Vlasov equation and this is typically quadratic in f .In either case, from f one defines the energy-momentum tensor by T ab = Z N q p a p b f ( q, p ) ω p = 1 √− g Z N q p a p b f ( q, p ) d pp , (32)where ω p = d p/ ( √− gp ) is a Lorentz-invariant volume form on N q whichwe’ll explain below, and then this can be used in the Einstein field equations.It is clear that T ab in (32) is trace-free and it’s an exercise to check that itis divergence-free by virtue of (31) (this is much easier to see in inertalcoordinates). 15o find the behaviour under conformal rescaling we first note that ω p can be written ω p = ǫ abcd T a dp b ∧ dp c ∧ dp d g ef T e p f = d p √− gp where T e is an arbitrary time-like vector. This evidently picks up a factorΩ − under rescaling if we assume ˆ p a = p a , so with ˆ f = f we’ll have ˆ T ab =Ω − T ab which is the correct rescaling to preserve the conservation equation.For Boltzmann we need to know more about C ( f, f ). As is standard,we assume binary collisions so that a pair of particles with null 4-momenta p a , q a collide to produce a pair with null 4-momenta p ′ a , q ′ a (or vice versa)and we assume conservation of 4-momentum so that the total 4-momentumin the collision is P a := p a + q a = p ′ a + q ′ a . The Boltzmann equation is L f = C ( f, f ) , where the R.H.S., called the collision term is an integral C ( f, f ) = Z ( f ( t, p ′ ) f ( t, q ′ ) − f ( t, p ) f ( t, q )) k ( s, θ ) ω q ξ p ′ over allowed q, p ′ , q ′ . Here k ( s, θ ) is a function to be specified and (frequentlybut not always) known as the scattering-cross-section, the scalar s is definedas s = g ab P a P b = 2 g ab p a q b ,θ is the usual scattering angle, determined from g ab p a p ′ b = s − cos θ ) , and ω q is the 3-form introduced above, on the null-cone for q a . The integralis over allowed q , which is the 3-dimensional null cone for q , and allowed p ′ , q ′ constrained by p ′ + q ′ = p + q = P say. There is a 2-dimensional allowedset of p ′ , since g ab p ′ a p ′ b = 0 = g ab P a ( P b − p ′ b ) , and given p ′ and q there is no freedom in q ′ . Therefore ξ p ′ is a 2-form onthe space of allowed p ′ a (called a Leray form in the French literature, [8]),and it can be defined as ξ p ′ = − ǫ abcd p ′ a q ′ b dp ′ c ∧ dp ′ d g kl p k q l ) . ξ p : it is in fact unchangedby conformal rescaling.(The definition of ξ p can be motivated as follows: on the submanifoldΣ pqP of N p × N q , the product of null-cones in p and in q , on which p a + q a = P a for fixed P a we want to define a 2-form ξ so that, on N p × N q , ξ ∧ ( 124 ǫ abcd dP a ∧ dP b ∧ dP c ∧ dP d ) = ω p ∧ ω q . This is because it ensures that Z N p × N q F δ (4) ( P a − p ′ a − q ′ a ) ω p ′ ∧ ω q ′ = Z Σ p ′ q ′ T F ξ p ′ with F suitably restricted, and the δ -function included to enforce conserva-tion of 4-momentum.)If we now have an explicit expression for k ( s, θ ) then we will know thetransformation of the Boltzmann equation. While not actually a matter equation of motion, it is interesting to considerthe equation (studied in [31] and the work of Friedrich e.g. [15]): ∇ a ∇ b σ + σL ab + ρg ab = 0 . (33)This is equivalent to the statement that the trace-free part of ∇ a ∇ b σ + σL ab is zero, since ρ can be eliminated in terms of σ and (cid:3) σ : ρ = −
14 ( (cid:3) σ − Rσ ) . Equation (33) has a significance that we’ll come to but the first observationis that it is conformally-invariant if we accompany (8) with σ → ˆ σ = Ω σ, that is, σ has conformal weight 1. This follows rapidly from (13), and wealso obtain ρ → ˆ ρ = Ω − ( ρ − Υ c σ c − σ Υ c Υ c ) . Now suppose that one has a solution of (33), then in regions in which it hasno zeroes we may use Ω = σ − as a conformal factor to obtain ˆ σ = 1, whenˆ ∇ a ˆ ∇ b ˆ σ = 0 and so the trace-free part of ˆ L ab is zero. Therefore a space-timeadmits nontrivial solutions of (33) if and only if it is locally conformal to anEinstein space, [31]. 17 .3.5 Two, more complicated examples We’ll review two, more complicated examples, both of which have actuallyarisen in cosmological studies. • Massless viscous magnetohydrodynamics (MHD)
Following [60] we note that the equations for this are conformally-invariant. The model is radiation fluid which moves through a mag-netic field which it itself generates. Suppose the fluid has unit velocity u a , density ρ and pressure p = ρ/
3. There is a standard kinematicdecomposition of the covariant derivative of the velocity: ∇ a u b = u a A b + ω ab + σ ab + 13 θh ab with h ab = g ab − u a u b , which is the projection orthogonal to the fluidflow, and A b , ω ab , σ ab , θ are respectively the acceleration, twist, shearand expansion of the fluid flow. These are fixed by requiring u a A a = 0 = ω ab u b = σ ab u b = ω ( ab ) = σ [ ab ] = h ab σ ab . From the discussion in [68], we’ll suppose that there is shear viscosity but no bulk viscosity . This is expressed by adding a term − ησ ab , with η the coefficient of shear viscosity, to the fluid energy-momentum tensorso that this becomes T Fab = 13 ρ (4 u a u b − g ab ) − ησ ab . (34)In what follows, we allow η to be a function of position.There is a Maxwell field F ab which is generated by a current density J a from Ohm’s Law J a − u a ( u b J b ) = σF ab u b (35)where σ is conductivity, again allowed to be a function of position, sothat ∇ [ a F bc ] = 0 , ∇ a F ab = 4 πJ b . The presently undefined component J a u a is the charge density andwe’ll suppose this is zero. The Maxwell contribution to the energy-momentum tensor is T Mab = − π (cid:18) F ac F cb − F cd F cd g ab (cid:19) , g ab → ˆ g ab = Ω g ab , we’ll have u a → ˆ u a = Ω − u a , whence the kinematic quantities musttransform asˆ A b = A b − (Υ b − u b ( u c Υ c )) , ˆ ω ab = Ω ω ab , ˆ σ ab = Ω σ ab , ˆ θ = Ω − θ. From the remarks in Section 1.3.1 we want the energy-momentumtensors to have conformal weight − ρ = Ω − ρ, ˆ η = Ω − η, (36)and Ohm’s Law transforms properly withˆ J a = Ω − J a , ˆ σ = Ω − σ. (37)Note once again that these rescalings require, for consistency, that σ and η are functions of position i.e. are not constant.The conformal invariance of this system, pointed out by [60], was usedby them in an FRW cosmology to solve curved-space MHD from flat-space MHD. • Massless Vlasov with Yang-Mills
Here we follow [8] and [64]. The model has massless Vlasov mattercarrying a Yang-Mills charge. Quantities will typically carry Yang-Mills indices which we’ll take to be Greek. Thus there is Yang-Millspotential A αa which gives rise to a Yang-Mills field F αab according to F αab = ∇ a A αb − ∇ b A αa + c αβγ A βa A γb , (38)where c αβγ are the structure constants of whichever Yang-Mills group G , conveniently assumed to be semi-simple, has been chosen. To de-scribe the matter there is a distribution function f ( q a , p a , Q α ) sup-ported on future null-cones N + q = { g ab ( q ) p a p b = 0 , p > } and de-pendent on a Yang-Mills charge vector Q α . This defines an energy-momentum tensor as in (32) and also a current vector T Vab = Z N q p a p b f ( q, p ) ω p , J αa = Z Z Q α p a f ω p ω Q , (39)19here ω Q is a volume form on the space of Q α .The distribution function satisfies a Vlasov equation ∂f∂q a dq a ds + ∂f∂p a dp a ds + ∂f∂Q α dQ α ds = 0 , (40)and the particles follow a Lorentz force-law adapted for Yang-Millstheory: dq a ds = g ab p b (41) dp a ds = − g bc,a p b p c + Q α η αβ F βab g bc p c (42) dQ α ds = − c αβγ A βb g bc p c Q γ . (43)Here η αβ is the metric on G , which exists by assumption, and themetric g bc has been written inexplicitly to help with the book-keeping.The remaining Yang-Mills equation is ∇ a F abα + c αβγ A βa F abγ = 4 πJ bα , (44)and the Yang-Mills contribution to the energy-momentum tensor is T Y Mab = − η αβ ( F αac F c βb − F αcd F cd β g ab ) . (45)Under conformal rescaling, following the example of Sections 1.3.1 and1.3.3, f, p a , Q α and A αa are unchanged, therefore so is F αab , while T Vab , J αa and T Y Mab all have weight −
2. Proper time in the Lorentz force-law andthe Vlasov equation will change according to d/d ˆ s = Ω − d/ds , whilethe right-hand-sides in the Lorentz force-law all have weight − g cd p c p d = 0) so these equations are invariant. Finally onereadily checks that (44) is invariant ( F abα and J bα both have weight − As a preamble to CCC, we’ll review some elementary mathematical cosmol-ogy with particular interest in the case of positive λ .20 .1 Friedmann-Lemaitre-Robertson-Walker (FLRW) models For these the metric can be written g = dt − ( a ( t )) dσ k where the spatial metric is dσ k = dr + ( f k ( r )) ( dθ + sin θdφ )with the familiar choices f = sin r, f = r, f − = sinh r. The Ricci tensor R ab necessarily takes the form A ( t ) u a u b + B ( t ) g ab with u a = ∇ a t so these are naturally adapted to the perfect fluid matter model withfluid velocity u a , and with a cosmological constant. The energy-momentumtensor for the matter is T ab = ( ρ + p ) u a u b − pg ab and the EFEs including λ are R : 3 a − ¨ a = − πG ( ρ + 3 p ) + λR ij : a − ¨ a + 2 a − ˙ a + 2 ka − = 4 πG ( ρ − p ) + λ Eliminate ¨ a from these to arrive at the Friedmann equation˙ a + k = 83 πGρa + 13 λa . (46)Dot this and eliminate ¨ a again to arrive at the conservation equation˙ ρa + 3 ˙ a ( ρ + p ) = 0 . (47)The complete set of EFEs is equivalent to (46, 47), together with an equationof state p = f ( ρ ).Easy cases to solve are the polytropic equations of state p = ( γ − ρ with 1 ≤ γ ≤
2, and the usual understanding is that γ = 1 , p = 0 is dust, γ = 4 / T aa = 0), and γ = 2 , dp/dρ = 1 is ‘stiff’ matter (forwhich the speed of sound equals the speed of light). In these cases, theconservation equation integrates to give ρa γ = 3 µ/ πG = constant (48)21here µ is a constant of integration characterising the density, and theFriedmann equation becomes˙ a + k = µa − γ + 13 λa . If solutions start with a Big Bang at t = 0 then for small ta ∼ t / γ , ρ ∼ t − , (49)so that the Ricci tensor is indeed singular at the Bang (since ρ is a componentof the Ricci tensor in a parallelly-propagated frame, and diverges at theBang) while if λ = 3 H then for large ta = a e Ht + a − e − Ht + h.o. , ρ ∼ e − γHt , (50)with a − fixed by a and k .An interesting special case is radiation when ρ = 3 a − µ/ πG and˙ a + k = µa − + 13 λa . Introduce conformal time τ by dτ = dt/a and then (cid:18) dadτ (cid:19) = µ − ka + 13 λa , (51)an equation that we shall revisit. This will be helpful below, so suppose then that the universe is filled witha mixture of dust and radiation which don’t interact. From (47) and thediscussion below it the two components have respective densities ρ m = Aa , ρ γ = Ba , (52)for constants of integration A, B . Here the dark matter and what is conven-tionally called the baryon contribution is in ρ m and photons and masslessneutrinos are in ρ γ . The Friedmann equation (46) becomes˙ a = − k + κa ( ρ m + ρ γ ) + H a , κ = 8 πG/ H = λ/
3. The general consensus is that k con-tributes very little to this equation – it is swamped by the densities at earlytimes and by the cosmological constant term at late times – so we shall setit to zero. Then there is a conventional parametrisation of the densities bycomparison with the H -term: introduce the quantities α = κρ m H , β = κρ γ H , where these are calculated at the present time . One can find these tabulatedin the literature and commonly accepted values are ( α, β ) = (0 . , . × − )(e.g.[45] or [7]) while the e -folding time H − is 1 . × years, which issurprisingly close to the accepted age of the universe which is t = 1 . × years (so H − = 1 . t ).Writing a for the scale factor at the present time, one can solve for A, B as κA = αa H , κB = βa H , and write the Friedmann equation as an equation for S = a/a as S ˙ S = H ( β + αS + S ) . (53)Recall dS/dτ = a ˙ S = a S ˙ S , so we can integrate (53) to obtain either τ or t as Z S dS ( β + αS + S ) / = a Hτ, (54)or Z S SdS ( β + αS + S ) / = Ht, (55)where we’ve chosen the origins to coincide. In particular we can find thetotal age of the universe from (52): with ( α, β ) = (0 . , . × − ) calculate Z ∞ dS ( β + αS + S ) / = 3 . , so τ F = 3 . / ( a H ). At the present time S = 1 and correspondingly Z dS ( β + αS + S ) / = 2 . , so that now τ = τ = 2 . / ( a H ) and the universe has used up 74% of itstotal conformal time – just 26% is left!In terms of proper time, (55) will give the present age of the universe:integrate to S = 1 to obtain t = 1 . × years, as expected.Equations (54, 55) are exact but there are two simple approximationsgiving an answer in elementary functions:23 at late times we can neglect the radiation as compared to the dust and(55) is solved by S = α / (sinh(3 Ht/ / , (56)so that as expected S ∼ e Ht at large t . (This solution is actually onWikipedia, with the claim that it is good for times t > years.)It’s worth doing a little more with this expression. Quite soon inthe history of the universe, the exponential expansion will swamp thedensity term: κρ m H = αa a < − when a > (100 α ) / a i.e. a > . a , and then we can expand S in (56) and integrate for τ : Z ∞ t a − dt = a − Z ∞ t S − dt = a − Z ∞ t α − / e − Ht dt = a − α − / H − e − Ht = Z τ F τ dτ = ( τ F − τ ) , so that late on, with a > a or so, we have the approximation e − Ht = α / a H ( τ F − τ ) . (57)This tells us that everything from about t > t is bunched up veryclose to I + in conformal time. We saw above that ( τ F − τ ) /τ F , thefraction of conformal time ahead of us now, is about 0 .
26. If we replace τ , which corresponds to t , by τ corresponding to 10 t this fractiondrops to about 10 − .As a concrete instance of this, the Higgs mechanism is supposed tokick in between 10 − and 10 − seconds after the Bang; from (58)below that is a Hτ in the range 10 − − − . The same conformaltime before τ F corresponds to t = 4 − × years, which is aboutthirty times the current age of the universe. • at early times we can neglect the cosmological term as compared to thedensities and follow [7]. There is an early time when the two densitiesare equal: ρ m = ρ γ = ρ eq say, when 1 = BAa = βα . a a ,
24o with β/α ∼ × − this is when a/a ∼ × − (redshift about3000). Suppose this happens when t, τ and a are respectively t eq , τ eq and a eq . From the Friedmann equation, omitting H and using confor-mal time we calculate d adτ = a ¨ a + a ˙ a = κA , which is constant, so that this integrates to a = κA τ + Cτ, for a constant of integration C which can be fixed by the Friedmannequation (cid:18) dadτ (cid:19) = a ˙ a = κ ( Aa + B ) , whence C = κB .At equality we have a eq = BA , ρ eq = Aa eq = A B so that the expression for a ( τ ) can be rewritten a ( τ ) = a eq (cid:18) ττ ∗ (cid:19) + 2 (cid:18) ττ ∗ (cid:19)! , (58)with τ ∗ = ( κρ eq a eq / − / . Note that τ eq is defined by a ( τ eq ) = a eq so (cid:18) τ eq τ ∗ (cid:19) + 2 (cid:18) τ eq τ ∗ (cid:19) = 1and τ eq = τ ∗ ( √ − t to τ in this case,since dt = adτ so that t = τ ∗ a eq (cid:18) ττ ∗ (cid:19) + (cid:18) ττ ∗ (cid:19) ! . • Finally it’s worth noting that the ranges of applicability of thesetwo approximations overlap. Neglecting ρ γ /ρ m to obtain the firstis equivalent to neglecting ( βa ) / ( αa ) and this is less than 10 − for a > × − a . Neglecting H / ( κρ m ) to obtain the second is equiva-lent to neglecting a / ( αa ) and this is less than 10 − for a < − a .25 .1.2 Conformally rescaling FLRW models It is convenient to conformally-rescale an FLRW metric with Ω = a − . Thenˆ g = a − g = dτ − dσ k , where dτ = dt/a , which we continue to take as the definition of conformaltime. This metric is evidently everywhere regular. It’s conformally-flat(since FLRW is known to be) and is conformal to part of the Einstein staticcylinder (this is clear for k = 1 and not hard to see for k = 0 or − a ( t ) − is integrable at t = 0, we may choose the origins of τ and t to coincide. By (49) this holds for γ > / γ ≤ / λ , a ( t ) − is integrable towards infinity by (50) (evidentlythis holds for other matter models too). Thus there will always be a futureboundary I + and assuming the origins of t and τ coincide, it will be locatedat a final value τ = τ F = Z ∞ dta ( t ) . These models have a finite life-time in conformal time.Metrics like this are widely referred to as ‘asymptotically de Sitter’ eventhough the spatial metric, which becomes the metric of I + , contains k whichcan have any value (while for de Sitter k = 1). The process of conformal rescaling, provided the initial singularity was notconformally-infinitely remote, extended the space-time to a smooth surface t = 0 = τ at which the extended metric was smooth. This is despite thefact that the scale factor a ( t ) ∼ t / γ is never differentiable for the physicalrange of γ as a function of t at t = 0. In terms of τ , a ∼ τ / (3 γ − . This isdifferentiable with zero derivative for γ < /
3, differentiable with nonzeroderivative for γ = 4 /
3, the radiation value, and not differentiable for γ > / The example of FLRW extends at both ends. Motivated by this, we mightseek to pose an initial value problem for a cosmological model either with26ata at I + or with data at an initial singularity. This would need an exten-sion of the EFEs as these surfaces are not in space-time but are added asboundaries. This extension will be the Conformal Einstein equations , whichare the Einstein equations for g written as equations for ˆ g and either Ω orΩ − , whichever extends to the data surface under consideration. With a positive cosmological constant one expects, and in many cases canprove, that there is automatically a conformal rescaling that adds I + asa future boundary. It is therefore natural to contemplate posing an IVPwith data at I + and this was studied thirty years ago by Friedrich, [15]for the vacuum equations (with λ >
0) and [16] for the Einstein-Yang-Mills equations with λ > I + and equivalent to a symmetric hyperbolic system, for which existence anduniqueness theorems are available.In the simpler case of vacuum, the data at I + consist of two symmetrictensors ( h ij , E ij ) where h ij is a Riemannian metric on I + and E ij , which istrace-free and divergence-free in the metric covariant derivative determinedby h ij , is the derivative normal to I + of the electric part of the Weyl tensorat I + (recall that the Weyl tensor itself is necessarily zero at I + , but E ij can be thought of as the gravitational radiation data at I + ). There is a‘gauge freedom’: ( h ij , E ij ) → (˜ h ij , ˜ E ij ) = ( θ h ij , θ − E ij ) , (59)for any real, positive function θ on I + which automatically preserves theconditions on E . This corresponds to a freedom to change the conformalfactor Ω which is being used to add I + as a boundary. In cases with matterthere will also be data for the matter variables at I + .A proposal by Starobinsky [58], proved in some cases of interest by Ren-dall [53], was that, in the presence of a cosmological constant λ = 3 H , thespace-time metric should have an expansion of the form g = dT − e HT ( a ij + e − HT b ij + e − HT c ij + . . . ) dx i dx j , (60)where the spatial metrics a ij , b ij , . . . are time independent. This kind ofexpansion resembles the expansion of the Poincar´e metric in the ambientmetric construction of Riemannian geometers [14]. The correspondence isnot exact: in the ambient metric expansion only even powers of the defining27unction of the boundary appear. If we take e − HT to be the defining functionof the boundary, so that I + is at T = ∞ , then a ij is the metric of I + , b ij islinearly related to the Ricci tensor of I + : H b ij = − ( R ( a ) ij − R ( a ) a ij ) , (61)where R ( a ) ij , R ( a ) are respectively the Ricci tensor and Ricci scalar of themetric a ij , and c ij is proportional to the tensor E ij which is free data inFriedrich’s construction. The freedom (59) corresponds to a freedom toshift the origin in the T -coordinate: T → ˜ T = T − H log θ, which must be accompanied by a change in the comoving coordinates x i . As a simple example with data at the initial singularity, we consider radia-tion fluid solutions with cosmological constant λ and metric of the Bianchi-III form. This is one of the class of spatially-homogeneous but anisotropicmetrics for which the EFEs therefore reduce to ODEs. For diagonal Bianchi-III the space-time metric can be parametrised in terms of two functions oftime ( a ( t ) , b ( t )) as follows: g = dt − ( ab ) dz − ( ab − ) ( dθ + sinh θdφ ) . (62)The space-sections have a time-dependent product metric on R × H . Notethat the determinant of the spatial metric is a function of θ times a so a has the character of the FLRW scale-factor, and b measures anisotropy ofthe spatial metric. We introduce conformal time τ via dτ = dt/a and rescalethe metric: ˆ g := a − g = dτ − b dz − b − ( dθ + sinh θdφ ) . The conservation equation with a radiation fluid integrates as before to give ρ = m πG a − for some positive real constant m . The Einstein equations canbe written as the system3 b ′′ b + 6 a ′ b ′ ab − (cid:18) b ′ b (cid:19) + b = 0 (63)3 a ′′ a + 3 (cid:18) b ′ b (cid:19) − b − λa = 0 (64)28here prime is d/dτ and these have been simplified with the aid of theHamiltonian constraint, which in turn can be written as3 (cid:18) a ′ a (cid:19) = 3 (cid:18) b ′ b (cid:19) + b + ma + λa . (65)With ( m, λ ) fixed, these equations give a well-posed IVP with data ( a, b, a ′ , b ′ )subject to (65), which is preserved by the evolution. However we want togive data at the Bang . We shall see that these equations also have solutionswith initial data ( a, a ′ , b, b ′ ) = (0 , p m/ , b ,
0) at (say) τ = 0. Since a = 0there, this is an initial singularity, and we are working with the conformalEinstein equations (since the Einstein equations are not defined at the sin-gularity) but the situation is different from data at I + . Also, since we wishto give data at the Bang, we are constrained to give less of it.From (65), the cosmology expands forever (assuming it does so initially)so that a ′ > Q = 3 (cid:18) b ′ b (cid:19) + b , (66)which is manifestly non-negative. Then by (63) Q ′ = − a ′ a (cid:18) b ′ b (cid:19) ≤ ≤ Q ≤ b . Therefore b and b ′ b are bounded for all time. Write (65) as3 (cid:18) a ′ a (cid:19) = ma + λa + Q. (67)Thus a ′ a is bounded as long as a is and solutions exist until a diverges,which will define I + . This happens after finite conformal time, as we see bycomparing a with the solution L of the equation3( L ′ ) = m + λL , L (0) = 0 , L ′ (0) > . Then a ≥ L but L diverges in a time √ Z ∞ dL ( m + λL ) / , a goes to infinity at, say, τ = τ F < ∞ .We shall return to the asymptotic form after proving existence of solu-tions with data as claimed. For this, we put the system of Einstein equationsinto a first-order Fuchsian form. Set a = τ e U , b = e Σ , (68)when (63) and (64) becomeΣ ′ = ZU ′ = WZ ′ = − τ Z − W Z − e W ′ = − τ W − Z − W + 13 e + 2 λ τ e U while the Hamiltonian constraint becomes m = e U (cid:0) τ W + τ (3 W − Z − e − λτ e U ) (cid:1) , which can be interpreted just as the statement that the right-hand-side isconstant, which evaluation at τ = 0 shows to be positive. (In this setting,the momentum constraint is vacuously satisfied.)The system is of the form d X dτ = 1 τ MX + B ( X , τ ) , where X ( τ ) = (Σ , U, Z, W ) T is the vector of unknowns, M is a constantmatrix and B is smooth (in fact analytic) in its arguments.This system is singular as it has a pole in τ at the initial singularity.It is a first-order Fuchsian system and there is an existence theorem forsuch systems [54]: if the matrix M has no positive integer eigenvalues thenthe problem is well-posed for data annihilated by M . Here M clearly hasno positive eigenvalues and the allowed data take the form (Σ , U, Z, W ) =(Σ , U , ,
0) at τ = 0.Thus the solution exists at least for an interval in τ and then the dis-cussion above shows that existence continues until a diverges. There is a2-parameter family of solutions: the datum U is equivalently a ′ (0) and tiedto m by the Hamiltonian constraint; the datum Σ determines b = b (0)and therefore determines the metric on the initial singularity, which isˆ h = − b dz − b − ( dθ + sinh θdφ ) .
30s we go towards I + in this model, a diverges and from (67) we see that a ′ a = H + O ( a − ) , taking the positive square root (as is allowed). Thus there is a simple polein a : a ( τ ) = 1 H ( τ F − τ ) + O (1) , where τ → τ F as t → ∞ . Solving for a in proper-time t gives a = e Ht (1 + 0( e − Ht )) , and we are obtaining the Starobinski expansion (60). It’s an exercise tosolve (62)-(63) for the first few terms in a series in ( τ F − τ ) and confirm theexpansion (60). In particular the metric of I + is fixed by the O (1) term inthe expansion of b , and the O (( τ F − τ ) ) term in b is not fixed by the metricof I + but is the free data corresponding to Friedrich’s E ij .There are examples like this for all Class A Bianchi types in [61]. For the IVP at I + , the conformal Einstein equations of Friedrich [15] givea regular symmetric hyperbolic system requiring two tensors for the data,namely the metric of I + and the data for the gravitational radiation. Forthe IVP at the initial singularity, the conformal Einstein equations give asingular but Fuchsian system and require only one tensor for data, the metricof the Bang. Function counting therefore indicates that initial singularitiesat which one can pose an IVP have less free data and must be special, whichis confirmed by the observation that in particular they must have finiteWeyl tensor: in the rescaled space-time there is no singularity so ˆ C dabc inparticular is regular, but C dabc = ˆ C dabc so this must also be regular, thoughthis may be hard to detect. This will be the topic of a later section, butnote the consequence that in general the solutions with data given at I + will not evolve back to give initial singularities with finite Weyl tensor. Now we can introduce CCC, motivating it as an interaction of an earlierproposal of Penrose, the
Weyl Curvature hypothesis [48], with the discoverythat there is a positive cosmological constant in the world.31 .1 The Weyl Curvature Hypothesis
In [48] Penrose gave an argument that the Big Bang, viewed as a singularityof a Lorentzian manifold, was much more special than the Einstein equationsalone could explain. Very simply put, Penrose’s argument is that near theBang the matter content of the Universe was close to thermal equilibrium, inother words in a state of very high, possibly maximum, entropy; however theUniverse as a whole could not have been in a state of maximum entropy sinceit is an everyday experience that entropy continues to increase today – thereis a Second Law of Thermodynamics. Thus some other component of theUniverse must have been in a state of low entropy, equivalently in a specialstate, and this other component could only be the gravity or equivalentlythe geometry. Via the Einstein equations, the matter is point-wise tied tothe Ricci tensor so, Penrose argues, it must be the Weyl tensor that wasspecial at the Bang.Penrose gave quantitative force to the argument by a calculation of theentropy of the part of the Universe inside our past light cone and back tothe Bang. His estimate for this in 1979 was 10 k , k being Boltzmann’sconstant, and he later raised the estimate to 10 k when the consensustook hold that most galaxies have supermassive black holes at their corewhich contribute to this sum by their Bekenstein-Hawking entropy. Whilethis would seem to be an impressively large number, Penrose observes thatif all the matter inside our past light cone was collapsed into black holesthe corresponding figure would be 10 k , which is already vastly greater.Following Boltzmann, whose fondness for the formula S = k log W was suchthat he had it engraved on his tomb, Penrose observes that the volume W inphase space corresponding to the actual Universe is even more vastly smallerthan the apparently available volume: this is 10 as a fraction of 10 and is possibly the smallest number ever contemplated in physics , but thisis the fraction of phase space that the Creator’s pin had to hit to get theuniverse we have, a fact which needs explanation.There is no agreed measure of gravitational entropy but in [48] Penroseargued for a connection between it and the Weyl tensor, and recalled howthis worked for linear spin-2 theory in Minkowski space. After [48] appeared,various suggestions were made for definitions of gravitational entropy interms of scalar invariants of curvature but these have the wrong differentialorder (in terms of the number of derivatives of the metric arising in them) Although in [26] one finds an estimate of the chance of obtaining the observed uni-verse from inflation as one in 10 , which is about the same order of order of order ofmagnitude.
32o correspond to linear theory where one knows the answer. A more recentdiscussion of these definitions is given in [46]. An attempt to follow Penrose’ssuggestion more closely for cosmologies close to FRW was made in [39]. Thatdefinition has at least one good property but it isn’t clear how to extend thedefinition to cosmologies further away from FRW.Even without a universal definition of gravitational entropy, Penrosetook the view in [48] that the connection with the Weyl tensor was clearand proposed that the simplest conjecture to make was that the Weyl tensor C abcd is zero at any initial singularity .This is the Weyl Curvature Hypothesis . The force of ‘initial’ in theformulation is that this property could not hold for singularities formedin collapse to black holes since these are likely to have the character ofthe Schwarzschild singularity at which the Weyl tensor is certainly singular(since the Riemann tensor is singular but the Ricci tensor is zero). If theUniverse were to recollapse to a Big Crunch singularity then this would beat least as bad as the Schwarzschild singularity and could be more like thechaotic Mixmaster singularities, at that time conjectured (and now known[55]) to occur in vacuum Bianchi-IX collapse. Penrose in [48] wrote theWeyl tensor as here with all indices down, i.e. as C abcd , which seemed tosuggest that he would be content to have C dabc finite (when lowering theindex would lead it to vanish where the metric vanished). It is possible tomake this interpretation but at the time he told me that that was a finerdistinction than he wanted to make.Whether the Weyl Curvature Hypothesis is that C dabc vanishes or isfinite at the initial singularity is a smaller question than that of how oneis to tell: at the Bang, the Riemann tensor is singular as is the metric andtherefore the metric connection, so how is one to tell that the Weyl tensor,whose components are ten of the twenty components of the Riemann tensor,is finite? Scalar invariants of curvature bring in the metric or the volume-form so can mislead; likewise, components in a parallelly-propagated frameinvolve the metric connection.Another question that Penrose considered in [48] is how do you cause something, like vanishing of the Weyl tensor, which happens at the begin-ning ? We’ll see a nice answer to this below, but at the time he speculatedthat the ‘correct’ theory of quantum gravity, once it was found, might notbe invariant under time-reversal. 33 .2 Conformal gauge singularities There is a simple way to obtain singular space-times with a singularity atwhich the Weyl tensor is finite: start with a smooth space-time say M withmetric g ab and choose a smooth hypersurface Σ, most commonly space-like,and a function ˇΩ which vanishes at Σ; now rescale the metric – g ab → ˇ g ab =ˇΩ g ab . The rescaled space-time ˇ M say is certainly singular at Σ but its Weyltensor by (11) is not: ˇ C dabc = C dabc which is smooth. Singularities formedlike this have had different names over the years: isotropic in [19], sincea co-moving volume shrinks at the same rate in all directions approachingthe singularity, and conformally compactifiable , by analogy with behaviourof WAS space-times at I , but I think the best name is conformal gaugesingularities (used in [35] but due to Christian L¨ubbe) by analogy witha coordinate singularity, since a change of conformal gauge removes thesingularity.For the applications that we have in mind, we require Σ to be space-likein M but the example of FRW (Section 2.1.2) reminds us that we may notwant to assume that ˇΩ is smooth at Σ. Now two questions arise:1. Is there a well-posed initial value problem with data at Σ for the Ein-stein equations in ˇ M ? This will depend on the matter model consid-ered and raises a separate question: what happens if the Weyl tensoris zero rather than just finite?2. How general is this class of singularities with finite Weyl tensor? Arethere other classes of finite Weyl tensor singularities? Following the calculation in Section 2.2.2, we shouldn’t be surprised if thereis an IVP but we should expect the evolution equations to have a singularityin the time. I’ll do the radiation case as the simplest example so the physicalenergy-momentum tensor isˇ T ab = 13 ˇ ρ (4ˇ u a ˇ u b − ˇ g ab ) , where ˇ ρ is the fluid density, diverging at the Bang, and ˇ u a is the physicalfluid velocity, unit w.r.t. ˇ g . Here I’m introducing a convention that we’lladopt in this chapter: the physical metric after the Bang and associatedquantities carry a check (more accurately a h´aˇcek); unhatted and uncheckedmetrics and tensors will be unphysical.34t was shown in [19] that for large classes of perfect fluids with anisotropic singularity that the rescaled fluid flow must be orthogonal to thesingularity surface and must therefore be twist-free. Thus the flow definesa cosmic time τ , with the freedom to replace it by a function of itself, andwe may also introduce comoving space-coordinates.To have a conformal factor vanishing at the Bang we take the unphysicalmetric to be g ab = ˇΩ − ˇ g ab and then the unphysical fluid velocity is u a = ˇΩˇ u a .Guided by the FRW case we assume that ˇΩ is smooth at the Bang in theunphysical time-coordinate τ , and we’ll actually take ˇΩ = τ which ultimatelyis justified by deriving a solvable IVP.The surface Σ is at τ = 0. It has unit normal N a = V − τ ,a = u a for apositive scalar V and the unphysical metric in comoving coordinates is g = V − dτ − h ij dx i dx j . The second fundamental form K ab of the constant τ foliation and the un-physical acceleration A a are fixed by ∇ a N b = N a A b + K ab , with N a A a = 0 = N a K ab . Since also N a = V − τ ,a we have ∇ a τ b = V ( N a A b + K ab ) + V a N b , which must be symmetric so that V a = V ( A a + N a V τ ) . Recall (12):ˇ R ab = R ab + 2 ∇ a Υ b + g ab ∇ c Υ c − a Υ b + 2 g ab Υ c Υ c , then substituting for Υ and using the physical Einstein equations gives − π ˇ ρGτ c (4 u a u b − g ab ) = R ab − V τ (4 N a N b − g ab )+ Vτ ( K ab − Kg ab +2 N ( a A b ) + N a N b V τ ) . (69)Following [63], from the conservation equation in physical space-time andthe freedom to redefine τ we obtain a simple expression for the density: fora radiation fluid we may suppose8 π ˇ ρGc = V τ . ∂V∂τ = − K. (70)With the given expression for the density, (69) becomes R ab − V (1 − V ) τ (4 N a N b − g ab ) + Vτ ( K ab − Kg ab + 2 N ( a A b ) + N a N b V τ ) = 0 . (71)We decompose (71) in the standard (3 + 1) manner to obtain an evolutionequation and two constraints. The evolution equation is L N K ab = R ( h ) ab − KK ab +2 K ac K cb + ∇ ( h ) a A b − A a A b − Vτ K ab − h ab (cid:18) V K τ + V (1 − V ) τ (cid:19) , (72)where ∇ ( h ) a is the metric connection of the space metric h ab and R ( h ) ab is itsRicci tensor. The constraints are G = −
12 ( R ( h ) − K + K ab K ab ) = − V Kτ − V (1 − V ) τ (73)and G a = ∇ ( h ) b K ba − ∇ ( h ) a K = 2 τ ∇ ( h ) a V. (74)It is straightforward to check that the constraints are preserved by the evo-lution. The unknowns are ( V, h ab , K ab ) with evolution determined by (70),(72) and the definition L N h ab = 2 K ab . The system isn’t yet in Fuchsian form, since (72) has second-order poles in τ but this problem can be solved by some redefinitions of variables. Onecan read off constraints on data at τ = 0: from (74), V must be constant;from (72) K ab must vanish initially and the constant value of V must beone; nothing new comes from (73). The details are in [4] (see also [42], [44])and the conclusion is the solution exists, is unique and depends continuously on the data whichare just the 3-metric h (0) ab of the initial surface. One readily obtains the first few terms in power series in τ : h ab = h (0) ab + τ k ab + h.o. , K ab = τ k ab + h.o. , V = 1 − k τ + h.o. , k ab = 13 ( R (0) ab − R (0) h (0) ab ) , k = 16 R (0) . Inductively it’s clear that h ab and V , and therefore the space-time metric,are series in even powers of τ . Inhomogeneities in V and therefore in thedensity arise if the initial 3-Ricci scalar is nonconstant. Since the initial K ab must be zero, it follows that the magnetic part of the initial Weyl tensormust be zero. It was observed in [19] that the electric part of the initial Weyltensor is proportional to the trace-free part of the initial 3-Ricci tensor. Thusif the whole of the initial Weyl tensor is zero then the initial 3-metric hasvanishing trace-free Ricci tensor, so is Einstein and so is data for FLRW.Uniqueness of solution implies that the solution is then FLRW and in thiscase if the Weyl tensor is zero initially then it is always zero. In the radiation case this was considered in [43], and, along with otherpolytopic fluids, was shown in [4] still with this Weyl tensor property: if itvanishes initially then it is always zero. This is rather a strong property soit is natural to consider other matter models.
This case has trace-free physical ˇ T ab so we’ll again take ˇΩ = τ and (71) isreplaced by R ab − V τ (4 N a N b − g ab )+ Vτ ( K ab − Kg ab +2 N ( a A b ) + N a N b V τ )+ κVτ √ h Z f p a p b p d p = 0 . (75)Here κ = 8 πG/c , p = ( h ij p i p j ) / and h = det h ij . One also has the Vlasovequation (31) V ∂f∂τ − h ij p i ∂f∂x j − (cid:0) ( p ) ∂ i V − p m p n ∂ i h mn (cid:1) ∂f∂p j = 0 . (76)The Vlasov equation has no singularity at τ = 0 but the rescaled Einsteinequation (75) does and one needs to split it as before into constraints andevolution and obtain the first-order Fuchsian form. This can be done ([3],[5]) but one may obtain the Fuchsian conditions on the data by a simple-minded approach of seeking power-series solutions.First one can adjust the conformal gauge to set V = 1 at τ = 0; then(75) at O ( τ − ) gives conditions on the initial distribution function f ( x i , p j )37nd initial metric h ij : Z p i f ( x i , p j ) d p = 0 , h ij = 13 √ h Z p i p j f ( x k , p k )( h mn p m p n ) / d p. The first of these is a ‘vanishing dipole’ condition on the initial distributionfunction which has the effect of making the time-like eigenvector of thestress-tensor, which can be thought of as a mean matter velocity vector,orthogonal to the singularity surface. The second has the appearance of aconstraint relating f and h ij but in fact it’s an equation, determining h ij given suitable f . This can be seen by consideration of the problem: find h ij which minimises F ( h ij ) := ( det ( h mn )) − / R f ( x k , p k )( h ij p i p j ) / d p . It’s straightforward to show that this problem has a unique minimumgiven non-negative f , compactly supported in p , and the Euler-Lagrangeequations for the minimum are the desired constraint. From the minimisa-tion description, this initial metric can be seen to be smooth if f is.One has more Fuchsian constraints from (75) at O ( τ − ) and these de-termine the initial value of the second fundamental form, K ij from the thirdand fourth moments of f . Essentially one obtains an equation of the form M mnij K mn = N ij where M ijmn , N ij are obtained from χ ijmn = Z p i p j p m p n f ( h rs p r p s ) / d p and χ ijk = Z p i p j p k f ( h rs p r p s ) d p, (see [3, 5] for more detail).The data is strikingly different from the perfect fluid case – there onegave the initial metric with no extra data for the fluid while here one givesthe distribution function and this fixes the geometric data. It is also the casethat now the Weyl tensor can be zero initially but become nonzero later. Ina sense it emerges from the higher multipoles of f . Analysis leading to an equation corresponding to (71) or (75) has beendone for a range of other matter models [64]. Thus these cases have beentaken far enough to obtain Fuchsian conditions without reducing them com-pletely to the form where well-posedness can be proved. These cases in-clude Einstein-scalar-field with potentials, Einstein-Yang-Mills-Vlasov and38instein-Boltzmann. The last case is interesting as it must bridge betweenEinstein-perfect-fluid and Einstein-Vlasov depending on the behaviour ofthe scattering cross-section. Cases with massive Vlasov and nonzero scalarcurvature have been studied with spatially-homogeneous metrics, [65].
A question which needs to be addressed is how far one can deduce thata finite Weyl tensor singularity is necessarily a conformal gauge singular-ity. Equivalently what conditions on the Weyl tensor permit a conformalextension? This was considered in [35] (see also [33],[36]). The idea is tosuppose that one has a time-like conformal geodesic γ which is incompletebecause it runs into a singularity BUT along γ the Weyl curvature and asmany derivatives of it as are necessary are bounded in a suitable frame,suitably propagated. In [35] this was phrased in terms of the calculus oftractors, which won’t appear in these lectures. To do it without tractors,recall subsection 1.2.6 where a conformally-invariant way to decompose theWeyl tensor and its derivatives into components was found. With the aidof these we can give a definition for boundedness of the Weyl tensor andits derivatives up to any order k . This is the answer to the question raisedin section 3.1: how do you identify finite Weyl tensor when the Riemanntensor is singular?Now one can translate the result in [35] to a form without tractors: Let γ : [0 , τ F ) → M be the final segment of an incomplete time-like confor-mal geodesic in ( M, g ) , such that b a is bounded in [0 , τ F ) . Let W ⊂ M be aneighbourhood of γ [0 , τ F ) in which the strong causality condition holds. Let { e β } be a Weyl propagated orthonormal frame along γ .(i) If the Weyl tensor and its derivative have uniformly bounded norms along γ with respect to { e β } then there exists a neighbourhood U of γ [0 , τ F ) with U ⊂ W and a diffeomorphism ψ : V ⊂ R → U .(ii) Suppose that in U the norms of the derivatives of the Weyl curvatureup to order k are uniformly bounded then there exists a conformally relatedmetric g such that there exists U ∗ ⊃ U with a C k -extension of ( U, g ij | U ) into ( U ∗ , g ij k U ∗ ) .(iii) The Riemann curvature of g is C k − .Thus the conformal structure ( M, ˇ g ) is locally extendible. .3 The ‘outrageous suggestion’ Since about 1998 the belief that there is a positive cosmological constant hasreceived ever-strengthening support, including that of the Royal SwedishAcademy of Sciences. It was apparent early on to any relativist with aknowledge of the theory in subsection 1.2.4 that this indicated that theexistence of a space-like I + was therefore very likely. One would still needto show that solutions of the Einstein equations with positive Λ genericallyhave a smooth I + , something that was widely believed under the rubric ofthe ‘cosmic no-hair conjecture’ [18] and proved in certain cases.If there does exist a space-like I + then under weak conditions the Weyltensor will vanish there and the conformal metric will extend through. Onthe other hand, ever since 1979 Penrose had wanted a mechanism to causethe universe to have an initial singularity at which the Weyl tensor was zero.Now one presents itself, and this led Penrose in about 2005 to his ‘outrageoussuggestion’ which is CCC: that the initial singularity should be a conformalgauge singularity so that the conformal metric could be extended through itand, conformally, the Big Bang of one cosmology could be identified with the I + of an earlier one. He called the different phases ‘aeons’ and there wouldbe a well-defined conformal metric extending from aeon to aeon, while eachaeon would have a physical metric related to the overall conformal metricby a conformal factor that cycled from zero to infinity. The picture is notintended to be periodic but it does go through cycles.A very simple example is provided by the FLRW metric with source aradiation fluid. We dealt with this in section 2.1: the metric is g = dt − ( a ( t )) dσ k , where dσ k is one of the three 3-dimensional Einstein metrics. The conser-vation equation is integrated by ρa = 4 πµ/G where µ is a constant ofintegration and the Einstein equations reduce to the Friedmann equationwhich can be written in terms of conformal time τ as in (51): (cid:18) dadτ (cid:19) = µ − ka + 13 λa . (77)The RHS of (77) is never zero for k negative or zero, and for k > λµ > /
4. Suppose the RHS is never zero, than a expandsfrom a simple zero at say τ = 0 and will diverge at some final value τ F ,where in fact it will have a simple pole: a ∼ H ( τ F − τ ) − + O (1), where λ = 3 H , H >
0. 40he conformal metric ˜ g = dτ − dσ k is independent of τ so is oblivious to the passage through τ = τ F but (77)has an attractive symmetry: it is invariant if we replace a by ˜ a = c/a forsome constant c , and then ˜ λ = 3 µ/c and ˜ µ = λc /
3. If we take the viewthat λ is to be the same at each cycle then we should take c = 3 µ/λ and then µ is also the same at each cycle. This gives the simplest modelof a CCC: the scale factor runs from zero to infinity in a finite amount ofconformal time, and when it reaches infinity it is replaced by a constanttimes its reciprocal. In fact this model is periodic but its Weyl tensor isidentically zero (it vanishes initially and the source is a radiation fluid soit is always zero by section 3.2.1.). We’ll be guided by this model in whatfollows. Note that the model requires the matter to be radiation fluid in theprevious aeon: any admixture of dust will spoil the symmetry of (77). By and large we follow [49] in this subsection. In CCC then there are twospace-times representing successive aeons, ˆ M and ˇ M , respectively to thepast and the future of a common boundary Σ, and two space-time metrics,ˆ g ab for ˆ M and ˇ g ab for ˇ M . There is a third metric g ab on M = ˆ M ∪ ˇ M ∪ Σ forwhich Σ is smooth and space-like, and the metrics are conformally relatedby conformal factors ˆΩ and ˇΩ according toˆ g ab = ˆΩ g ab on ˆ M ; ˇ g ab = ˇΩ g ab on ˇ M .
Furthermore ˆΩ − and ˇΩ are smooth and tend to zero at Σ. We can call Σthe cross-over surface and g ab the cross-over metric .At this point we are free to rescale the cross-over metric, say g ab → Θ g ab when also ˆΩ → Θ − ˆΩ and ˇΩ → Θ − ˇΩ. This changes the product ˆΩ ˇΩ → Θ − ˆΩ ˇΩ so we refine the definition of the cross-over metric by imposing thecondition ˆΩ ˇΩ = − , (78)(we have − − and ˇΩ goes through zeroat Σ). This is Penrose’s reciprocal hypothesis imposed as a gauge condition.As a consequence ˇ g ab = ˆΩ − ˆ g ab , so that the metric of the later aeon is determined by metric of the earlieraeon and the conformal factor ˆΩ. In particular therefore the Einstein tensor41nd so the energy-momentum tensor in the later aeon are determined by themetric of the earlier aeon and ˆΩ. Consequently it becomes crucial to find aprescription for a unique preferred ˆΩ.This is a convenient place to introduce a piece of terminology from [47]:Penrose introduces the one-formΠ a = ∇ a ˆΩˆΩ − ∇ a ˇΩ1 − ˇΩ . (79)This is well-defined at the cross-over surface, as can be seen from the secondexpression, and Π a is proportional to N a = ∇ a ˇΩ, the normal to the cross-over surface. We’ll need Π a below.It’s part of Penrose’s view of CCC that, late in any aeon, the energy-momentum tensor of the matter should become trace-free. This isn’t nec-essary for the appearance of a boundary I + at which the Weyl curvaturevanishes – one can for example find dust cosmologies with that property –but for simplicity we’ll make the same assumption, although new physicsmay be needed to enforce it (see below, section 3.5). Thus the matter latein the previous aeon can be combinations of radiation fluid, Maxwell fields,massless Vlasov or some other massless field.With this assumption, Penrose suggested that one should restrict ˆΩ byrequiring that the cross-over metric g ab have the same scalar curvature asˆ g ab near to I + and this is just ˆ R = 4ˆ λ , with ˆ λ the cosmological constant inthe earlier aeon . We adapt (12) to this case: it can be directly written as (cid:3) ˆΩ + 16 R ˆΩ = 23 ˆ λ ˆΩ (80)when it is an equation applying the cross-over d’Alembertian to a functionwhich blows up at Σ and R is the scalar curvature of the cross-over metric,as yet unfixed; or by introducing φ = ˆΩ − , so that φ is smooth through I + with a simple zero, as ˆ (cid:3) φ + 23 ˆ λφ = 16 Rφ , (81) R as before. Note (80) and (81) have the form of (30), which is the fieldequation for the conformal scalar field. In this context, and with R = 4 λ ,Penrose [49] introduces the term the phantom field equation for this equation,and the phantom field for ˆΩ.Whatever we choose for R , we can expand φ in the manner of theStarobinski expansion (60): φ = Σ ∞ n =1 φ n ( x j ) e − nHT (82) I don’t want to make this assumption. λ = 3 H and solve (81) term by term. We find that φ and φ , thefirst two terms in the series (82), are freely specifiable and subsequent φ n are then uniquely determined. In particular2 H φ = ∆ ( a ) φ − H bφ + 16 Rφ (83)where ∆ ( a ) is the Laplacian of the metric a ij in the Starobinski expansion(60), and H b = R ( a ) by (61). Thus evaluating φ requires the choice of R , which we’ve deferred, to be made.Note the role of φ in fixing the metric of I + : from (60) and the definitionof φ the metric of I + is φ a ij . We’ll seek to fix φ shortly but comparisonof (50) and (82) shows that in the FLRW case necessarily φ = 0. Since weare taking the FLRW case for guidance, this suggests always taking φ = 0,which is allowed as it’s free data. Penrose makes this assumption in [49],where he calls it the delayed rest-mass hypothesis for a reason we come tonext. Given (78), the Einstein tensor post-bang, say ˇ G ab , can be calculated fromthe Einstein tensor pre-bang, ˆ G ab and φ = ˆΩ − . In particular this is true forthe Ricci scalar, but while the pre-bang Ricci scalar is ˆ R = 4 λ the post-bangRicci scalar ˇ R will have an extra term. This, from the trace of the post-bang energy-momentum tensor, will be an indicator of rest-mass appearingpost-bang. Penrose [49] writesˇ R = 4 λ + 8 πGµ. The relation between the metrics is ˇ g ab = φ ˆ g ab so that by (12)4 λ + 8 πGµ = ˇ R = φ − ( ˆ R + 6 ˆ (cid:3) φ /φ ) = φ − ( − λ + 2 Rφ + 12 φ − | ˆ ∇ φ | ) , using (81). Some manipulation turns this into the expression at the top ofp249 in [49]. From the expansion (82) we obtain8 πGµ = 24 H ( φ /φ ) e HT + O ( e HT ) . This is singular at the Bang (where T is infinite), as is to be expected, butwe see that the choice φ = 0 promoted above sets the leading term to zero– the choice φ = 0 delays the rate at which rest-mass appears after theBang, whence Penrose’s terminology.Coming back to the issue of fixing φ , Penrose in [49] offers severalpossible choices without settling for any one. These can be taken to be43 N a N b Φ ab = O ( ˇΩ ), • N a N b ∇ a N b = O ( ˇΩ ), • ∇ a Π a = O ( ˇΩ), or • a Π a − λ = O ( ˇΩ ).Here Φ ab is minus half the trace-free Ricci tensor.In [66] I suggested a different one, which is to choose φ so that themetric of I + , which is φ a ij , has constant scalar curvature. This amountsto solving the Yamabe problem for I + .The solution of this problem is veryoften unique, so that this prescription has at least that virtue, and it iseffectively what one does for FLRW, section 3.3, when one assumes that φ is a function only of t .In the long run, the choice between these possibilities should be physi-cally motivated. If there is to be a physically relevant scalar field after the Bang, one mightfeel that there should also be one in the previous aeon. In [34], L¨ubbedescribed a way to accomplish this. We recall the definition D ab [ φ, g cd , α ] := 4 φ a φ b − g ab g cd φ c φ d − φ ∇ a φ b + 2 φ L ab + 2 αφ g ab (84)from section 1.3.2. This is associated with the field equation Q ( φ, g ab , α ) := (cid:3) φ + 16 Rφ − αφ = 0 , (85)and there is the conformal invariance: if ˜ φ = Ω − φ then D ab [ ˜ φ, ˜ g cd , α ] = Ω − D ab [ φ, g cd , α ] , Q ( ˆ φ, ˆ g cd , α ) = Ω − Q ( φ, g ab , α ) , (86)and one also has g ab D ab = − φQ ( φ ) , ∇ a D ab = 4 Qφ b − φQ b . Now suppose one has the Einstein equations with source consisting of cos-mological constant λ , some extra massless fields T ex ab which are separatelyconserved, and the conformal scalar represented by stress-tensor D ab : G ab = − D ab [ φ, g cd , α ] − κT ex ab − λg ab . (87)44he trace of this gives R = 4 λ and L¨ubbe remarks that, when R = 4 λ wehave by (84) D ab [1 , g cd , λ/
4] = − G ab − λg ab . Therefore the EFEs (87) can be written D ab [1 , g cd , λ/
4] + D ab [ φ, g cd , α ] − κT ex ab = 0 . (88)Now rescale this:0 = Ω − ( D ab [1 , g cd , λ/
4] + D ab [ φ, g cd , α ] − κT ex ab )= D ab [Ω − , ˜ g cd , λ/
4] + D ab [Ω − φ, ˜ g cd , α ] − κ e T ex ab , where e T ex ab = Ω − T ex ab which is the correct rescaling for a trace-free stresstensor to preserve conservation.Now choose Ω = φ , then this is0 = D ab [ φ − , ˜ g cd , λ/
4] + D ab [1 , ˜ g cd , α ] − κ e T ex ab , which we recognise as the EFEs (88) but for the metric ˆ g with some otheradjustments. L¨ubbe’s result is therefore Given a solution ( φ, g ab , α, λ/ , T ex ab ) of the EFEs (88), there is anotherwith ( φ − , ˜ g = φ g, λ/ , α, e T ex ab = φ − T ex ab ) . To apply this to CCC we’ll take g to be ˆ g , the metric of the previousaeon, and ˜ g to be ˇ g , the metric of the current aeon. Then we need φ tovanish at the crossover. Thus it is present in the previous aeon but fadingto zero, while in the current aeon it starts very large (formally infinite). The choices we made in section 3.3 have to be changed slightly to accordwith the reciprocal hypothesis. We haveˆ g = d ˆ t − ˆ a dσ k = ˆΩ g, ˇ g = d ˇ t − ˇ a dσ k = ˇΩ g, so with ˆΩ = ˆ c ˆ a, ˇΩ = ˇ c ˇ a we have g = d ˆ t ˆ c ˆ a − c dσ k = d ˇ t ˇ c ˇ a − c dσ k , whence ˆ c = ˇ c = c say and dτ = d ˆ tc ˆ a = d ˇ tc ˇ a . − c ˆ a ˇ a Now by comparing with section 3.3, we’ll have successive aeons diffeomeophicif c = c = p µ/λ . This calculation has also given us the metric of I + as c dσ k so the scalar curvature of I + is 6 kc and the scalar curvature of thecross-over metric g is minus this. In [49] Penrose makes a speculation which I don’t think is essential to theCCC picture but is interesting and provocative in its own right. He con-templates the far future of a universe with positive λ . In a classic article[11] the remote future was considered by Freeman Dyson, before positive λ became the consensus, and much of what he said must still hold up. Histime-line for the far future had stars disappearing by 10 years, galaxies by10 years and black holes by 10 years (by Hawking evaporation – recallthat a black hole with mass N M ⊙ evaporates in a time of about 10 N years). After that, it depends if there is a lower limit to the mass of a blackhole: if the Planck mass is a lower limit then ordinary matter will spon-taneously collapse to black holes on a time-scale of 10 years, and thesewill then evaporate, but if there is no lower limit that process is more rapidand Dyson refers to [70] where a time-scale of 10 years for all matter tocollapse to black holes is suggested, and these then evaporate. In this lastcase of Dyson’s scenarios for the remote future, after these black holes havegone, one has a universe containing only massless particles and gravitation.This is also the future that Penrose argues for but by a different physicalmechanism, a kind of reverse Higgs mechanism: the Higgs mechanism issupposedly the process which gives rest mass to elementary particles atsome stage in the early universe, and Penrose’s speculation is that it turnsoff rest mass again at some late time. Evidently this needs ‘new physics’, soit is striking that Dyson had a similar picture by a different route.Once all particles are massless (or are once again massless) then therecan be no clocks, and proper time disappears from the world. This is partlyPenrose’s response to what he called the VBE : if the galaxies are gone by10 years or so then nothing much happens in the universe apart fromoccasional black hole mergers until the the supermassive black holes fromgalactic nuclei evaporate and this will take 10 years – this waiting char-acterises the Very Boring Era ([49]). If there are no clocks and thereforeno proper time then it won’t seem so long. This can also be seen as the46esolution of an apparent paradox: the proper time until I + is infinite buthow in the physical world can you have a completed infinity before the nextaeon? The resolution is that proper time loses its physical significance –there is a finite amount of conformal time before I + . If CCC is to be part of physics then it needs to make predictions which can beconfirmed. While CCC is agnostic about inflation, the default position wouldbe to seek to manage without it – Penrose has argued against the inflationconsensus for many years (see for example the discussion in [50]) and it iseasier these days to find mainstream articles critical of inflation (e.g. [27]).A good argument that one can manage without it would be to derive thedensity perturbation in the early universe within CCC and without callingon inflation. On the face of it this could be done since there is an epoch ofexponential expansion in CCC but it arises at the end of the previous aeonrather than very early in the current one – can one quantitatively supportthe assertion that ‘inflation happened before the Bang’ ? – but it hasn’t sofar been done. The emphasis has been on looking for other effects whichcome through from the previous aeon. The crossover surface is not a barrierto massless particles so that photons and gravitons should come through,and one might be able to detect the presence of a previous aeon by effectsin electromagnetic or gravitational radiation.We’ll briefly mention primordial magnetic fields. There are magneticfields at all scales in the universe [69]. Magnetic fields within galaxies areexplicable given primordial magnetic fields to act as seeds for dynamo am-plification processes [30]. Between galaxies there are known lower limits onmagnetic fields [25] and these intergalactic fields may be primordial. Theoccurence of primordial magnetic fields may be attributed to phase changesin the early universe or to inflation see e.g. [59], [69], but they could insteadcome through from the previous aeon.As a toy example, in the FRW metricˆ g = dt − a ( dx + dy + dz )consider the 2-form B = Bdx ∧ dy so that ∗ B = Ba dt ∧ dz. This 2-form is evidently closed and co-closed so it solves Maxwell equationsand corresponds to a Maxwell tensor F ab with F ab u b = 0 where u a (as usual)47s the fluid velocity or Hubble flow. Thus it’s a pure magnetic field, and it’sclearly smooth as a 2-form through the Bang, though for example, the norm F ab F ab = 2 B a − isn’t. Also one can calculate the energy-momentum tensorand obtain T ab u b = B πa u a . One might therefore define B πa as the energy density, which would go tozero faster than the volume diverges towards I + . However if one uses insteadthe conformal Killing vector X a = au a then there is a conserved current˜ J a = T ab X b = B πa u a , (foreshadowed in section 1.3.1) and the integral over a comoving 3-volume isconstant through the cross-over. Penrose has suggested looking for B-modesin the CMB in regions on the sky where the circles that we come to in thenext section are densest (?ref) since magnetic fields in the previous aeonwould be strongest inside superclusters and these regions of dense circlesmay be interpreted as superclusters hitting I + . Late in the previous aeon, all stars and galaxies should have gone but, be-fore they’ve evaporated, there should be a population of supermassive blackholes, relics of galactic clusters and superclusters. There will be mergersbetween these, releasing large bursts of gravitational radiation – the blackholes themselves could have masses of 10 M ⊙ and as much as 40% of themass could be radiated – and this burst will be of sufficiently short durationthat it could be treated as a δ -function wave supported on the light coneof the emission event. This wave will pass through the cross-over surfacebut immediately interact with the hot early universe in the next aeon. Thiswill diffuse out the energy into a region between two concentric spheres andproduce an inhomogeneity on the last-scattering surface of the CMB whichwill appear as an annulus of inhomogeneity on the intersection of the pastlight cone of an observer now with the last-scattering surface, which is essen-tially the CMB sky. This circular feature might be detectable by statisticalproperties: it might have detectably different mean temperature from thebackground; it might have detectably different temperature variance fromthe background.In a series of papers, Penrose and Gurzadyan have described a searchfor circles of lower variance, [20, 21, 22, 23]. The method is straightforward:48hoose a (large) set of centres, and for each centre a set of radii and widthsfor the annulus; then plot the temperature variance of the annuli, calculatedfrom published CMB surveys, with a threshold for significance. To assigna measure of statistical significance repeat the process with a model skyconstructed artificially but with the same statistical properties as the actualsky. Penrose and Gurzadyan have found large numbers of statistically sig-nificant circles, including sets of concentric circles where the same point onthe sky is the centre of as many as four. However the statistical significancehas been denied by several other groups: [10, 13, 24, 67]. The dispute cen-tres on the way the artificial, comparison skies are constructed. The sets ofconcentric circles were unexpected but can be argued to correspond in themodel to a supermassive black hole at rest with respect to the Hubble flowand into which a succession of smaller black holes falls.There is an interesting point emphasised in [23] about the centres ofconcentric sets: the circles are identified by having low variance in tempera-ture, but once identified their mean temperature can be calculated and thecentre indicated by a blue or red dot on the sky, according as this mean islower or higher than the ambient (see figure 2 in [23]). The distribution ofred and blue dots on the sky is strikingly inhomogeneous. To understandthe dots it is useful to think in 3 dimensions: our past light cone meets thelast-scattering surface in a 2-sphere and the future light cone of an imag-ined event in the previous aeon also meets the last scattering surface in a2-sphere, which one can think of as expanding; these two 2-spheres meet ina circle; if the centre of the expanding 2-sphere is inside our 2-sphere thenthe shock wave is moving away from us, the temperature will be red-shiftedand therefore lower and this event will get a blue dot; if the centre is outsidethen the shock wave is moving towards us, the temperature will be blue-shifted and therefore higher and this event will get a red dot. In the modelthen a clump of red dots represents a centre of activity outside the regionbounded by our 2-sphere and a clump of blue dots represents a centre ofactivity inside our 2-sphere. If the circles, and therefore the distributionof their centres, are statistical artefacts then it is hard to see why such acoherent picture emerges – why are there clumps of blue dots and clumpsof red dots at all, rather than a mixed jumble of red and blue dots?In a different series of papers, Meissner, Nurowski and collaborators [38,1, 2] have described a search for circles with anomalous mean temperature.The method of search is similar but the statistical analysis is quite differentand uses a nonparametric test due to Meissner [37]. In [2] the methodis modified: the statistic calculated is the difference between an average49emperature over an inner ring and an average temperature over an outerring, the two rings contiguous and forming a single wider ring. This followsa suggestion by Penrose that there should be a temperature profile of asharp rise and slow decline across these rings (so this test statistic should be negative . Both this series of papers and the Gurzadyan-Penrose series alsotested the effect of ‘twisting’ the observed sky. This, in terms of sphericalpolar coordinstes, is the (conformal) transformation( θ, φ ) → (˜ θ, ˜ φ ) = ( θ, φ + Sθ )of the sky, for varying choices of the real parameter S and would be expectedto disrupt genuine rings but not artefactual ones. Both sets of authors findthat the number of detected rings indeed declines sharply with increasing S . One difference between the two sets of work is that the rings (strictlyspeaking, annuli ) of the Polish group are significantly wider than those ofGurzadyan-Penrose. We’ll briefly describe a model for the generation of these circles by delta-function Ricci-curvature shock-waves from a source in the previous aeon,with the FLRW metric. Choose the source to be at the origin and put k = 0for convenience. We want to regard the spherically-symmetric shock-wave asa perturbation of the FLRW metric so that section 3.4.2 fixes conventions.The metrics in the two aeons areˆ g = d ˆ t − ˆ a dσ , ˇ g = d ˇ t − ˇ a dσ , with dσ = dr + r ( dθ + sin θdφ ) . The fluid content will be assumed to be radiation fluid with (unit, time-like)velocity vectors in the two aeons asˆ u = ∂ ˆ t , ˇ u = ∂ ˇ t . For a radiation fluid the conservation equation is solved for the densities by κ ˆ ρ γ = ˆ µ/ ˆ a , κ ˇ ρ γ = ˇ µ/ ˇ a as before, with κ = 8 πG/
3. 50he shock wave is supported on the light cone of the origin at a certaintime. This is best done in terms of the null coordinate u = τ − r where τ is conformal time. Suppose the energy momentum tensor of theshell in the previous aeon isˆ T ( S ) ab = δ ( u − u E ) ˆ F (ˆ t, r )ˆ ℓ a ˆ ℓ b , where the superscript S is to distinguish this from the background ˆ T ab whichis the radiation fluid, u E = τ E which is the conformal time at emission, andˆ ℓ a ∂ a = ∂ ˆ t + 1ˆ a ∂ r = 1ˆ a ( ∂ τ + ∂ r ) , which is the null generator of the light cone of the origin, normalised againstthe fluid velocity.The conservation equation implies r ˆ a ˆ F = ˆ m π = constantso that ˆ F can be thought of as proportional to energy or mass per unitarea on the spherically-symmetric expanding shock. There are two ways tointerpret the constant ˆ m . For one way, consider the conformal Killing vector X a = ˆ a ˆ u a (it’s easy to check that this is a conformal Killing vector) anddefine the current ˆ J a = ˆ T ( S ) ab X b = δ ( u − u E ) ˆ F ˆ a ˆ ℓ a . This is automatically conserved, and when integrated over a surface Σ ofconstant ˆ t gives Z Σ ˆ J a ˆ u a d Σ = Z Σ δ ( τ − r − u E ) ˆ F ˆ a r sin θdrdθdφ = ˆ m, so that ˆ m is the conserved quantity defined from this conserved current.This may not be the right definition of total energy in the shell and it maybe one should use ˆ u a in place of X a , in which case the energy in the shell isˆ E := Z Σ ˆ T ( S ) ab ˆ u a ˆ u b d Σ = 4 πr ˆ a ˆ F = ˆ m ˆ a E = ˆ m/ ˆ a E initially but tails away to zero as theprevious aeon expands, then jumps to infinity as the Bang is crossed andthen decreases again.The shell passes through to the next aeon and there will be correspond-ing checked quantities. How do we match them? The correct rescaling topreserve the conservation equation isif ˆ g ab = Θ ˇ g ab then ˆ T ab = Θ − ˇ T ab , and we’ll adopt this, which leads to ˆ F = Θ − ˇ F and then ˆ m = ˇ m . Now thechoice in the definition of energy in the shell noted above is crucial: if theenergy in the shell in the present aeon is ˇ m = ˆ m then it’s constant throughthe crossover; if rather it is ˇ E := ˇ m/ ˇ a = ˆ m/ ˇ a then it jumps up to infinityas it goes through, and then decays as the scale factor grows.We want to calculate the density perturbation at the last-scattering sur-face in the current aeon due to a shock like this. When the shock comesthrough I + it will interact with the ambient hot radiation fluid and spreadout. We look at this next. The last-scattering surface is located at redshift z = 1089 ([7]), or equiva-lently S = 1 / S = a/a as in section 2.1.1. We use this in (54)to obtain a Hτ LS = 0 .
052 and in (55) to obtain t LS = 3 . × years,which is the accepted value ([7]) (note here we’ve taken the origin in τ and t to be at the cross-over). The null cone of the spatial origin at t = 0 = τ is τ = r , which occupies a sphere of coordinate radius τ LS on the last-scattering surface. Our past light-cone meets the last-scattering surface ina sphere of coordinate radius τ − τ LS which will subtend an angle 2 δθ atus, where tan δθ = τ LS / ( τ − τ LS ) = 0 . / (2 . − . . δθ = tan δθ = 0 . horizon problem : .
04 radians is a little over 2 degrees sothat points on the CMB sky further apart than this have not been in causalcontact, in the sense that their causal pasts do not overlap, since the Bang .However, in CCC, their causal pasts do overlap but in the previous aeon.This is how CCC solves the horizon problem.The shock-wave envisaged in the last section meets the crossover in asphere of radius − τ E ( τ E is negative as it refers to an event in the previous52eon). This sphere spreads out in the next aeon but by causality it isconfined to the region between spheres of radii − τ E ± τ LS . Exactly howwide this ring is will depend on the details of the diffusion of energy in thehot early universe. Its actual width shouldn’t be more than .
04 radians butthe geometry of the intersection of this region with our past light cone canmake it appear wider.We obtain the size of the annulus on the last-scattering surface by using(57) and values from section 2.1.1:tan θ = τ LS − τ E τ − τ LS = a Hτ LS − a Hτ E a H ( τ − τ LS ) = 0 .
052 + 1 . e − Ht E . , where t E is proper-time of emission in the previous aeon, which we areassuming is essentially the same as the current one, and the factor 1 .
31 isthe approximate value of α − / .Recall that at the present time Ht ∼ .
82, and the term in t E can beneglected when Ht E ≥ . . o < θ < o will only be formed with 1 < Ht E < .
5. Later than that the circles maybe more like discs.To get an order of magnitude we consider the smallest possible circle.This has radius τ LS (since the light cone of the origin at τ = 0 has equation r = τ ) and the mass inside a sphere of this radius due to the backgrounddensity is M = π a LS ( τ LS ) ρ . The cosmological density, following section2.1.1 is ρ = Aa LS + Ba LS = ( α + β a a LS ) (cid:18) a a LS (cid:19) H κ . Also a Hτ LS = 0 .
052 as calculated above and( α + β a a LS ) = 0 .
45 + 1 . × − × . M = 4 π . × .
60 1 Hκ = 7 × M ⊙ . The most violent events imaginable late in the previous aeon (or indeedanywhere) are mergers of supermassive black holes. There are currentlybelieved to be black holes of mass 10 M ⊙ in the universe but these do seemto be the largest and there are arguments in the literature that these may bethe largest possible: see [41], [57]. If two black holes collide then as much as40% of the rest mass can be emitted as gravitational radiation so the mostextreme events should emit about 10 M ⊙ . If the shock wave considered53ere dumps about 10 M ⊙ of energy into the sphere under considerationthen that’s a fraction 7 × − of the background, while the scale of actualdensity perturbations at last-scattering is usually said to be δρ/ρ ∼ − ,so this is rather low. The fraction increases if the energy of the shock isconcentrated closer to the surface of the sphere, rather than distributedall across it. A detailed calculation of the scattering process in the earlieruniverse would be needed to get this right. The other view on what energyshould be is essentially ruled out as that would give a mass 10 a E /a LS M ⊙ dumped in the sphere, and the ratio a E /a LS can be vast. The outstanding problems with CCC seem to me to require more physicalcosmology rather than more mathematical cosmology. One wants a detailedmodel of the physical processes around the cross-over surface to answerquestions like: • can ‘circles in the sky’ be made to work? Are the events envisaged(namely super-massive black hole mergers in the previous aeon) of theright scale of energy and the right frequency of occurence to producethe circles? • can magnetic fields really come through or are thet damped out onthe ‘hot’ side? • is it possible to obtain the observed spectrum of density perturbationsfrom CCC? (This is widely regarded as the remaining great achieve-ment of theories of inflation.) • can Penrose’s suggestion of dark matter as ‘erebons’ [51] be justified? References [1] D. An, K. A. Meissner and P. Nurowski Structures in the Planck mapof the CMB, arXiv: 1307.5737 [2] D. An, K. A. Meissner and P. Nurowski Ring Type Structures in thePlanck map of the CMB, arXiv: 1510.06537 [3] K. Anguige Isotropic cosmological singularities. III. The Cauchy problemfor the inhomogeneous conformal Einstein-Vlasov equations. Ann. Physics (2000), 395–419 544] K. Anguige and K.P. Tod, Isotropic cosmological singularities. I. Poly-tropic perfect fluid spacetimes. Ann. Physics (1999), 257–293.[5] K. Anguige and K.P. Tod, Isotropic cosmological singularities. II. TheEinstein-Vlasov system. Ann. Physics (1999), 294–320.[6] T.N. Bailey and M.G. Eastwood, Conformal circles and parametrizationsof curves in conformal manifolds. Proc. Amer. Math. Soc.
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