Intrinsic vanishing of energy and momenta in a universe
aa r X i v : . [ g r- q c ] N ov Noname manuscript No. (will be inserted by the editor)
Intrinsic vanishing of energy and momenta in auniverse
Ramon Lapiedra · Juan AntonioMorales–Lladosa
Received: date / Accepted: date
Abstract
We present a new approach to the question of properly definingenergy and momenta for non asymptotically Minkowskian spaces in GeneralRelativity, in the case where these energy and momenta are conserved. In orderto do this, we first prove that there always exist some special Gauss coordinatesfor which the conserved linear and angular 3-momenta intrinsically vanish.This allows us to consider the case of creatable universes (the universes whoseproper 4-momenta vanish) in a consistent way, which is the main interest ofthe paper. When applied to the Friedmann-Lemaˆıtre-Robertson-Walker case,perturbed or not, our formalism leads to previous results, according to mostliterature on the subject. Some future work that should be done is mentioned.
Keywords
Energy and momenta of the Universe · Non asymptotic flatness · Intrinsic vanishing of momenta
PACS · PACS 98.80.Jk
Ramon Lapiedra · Juan Antonio Morales–LladosaDepartament d’Astronomia i Astrof´ısica,Universitat de Val`encia, E-46100 Burjassot, Val`encia, Spain.Tel.: +34-96-3543066Fax: +34-96-3543084E-mail: [email protected]: [email protected] not necessarily contradictory, perspectives. See [1] for an extensive and criticalreview on the current status of the problem.Diverse mathematical objects (energy-momentum pseudotensors and su-perpotentials, flat or curved background metrics, Killing vectors or other fieldsgenerating generalized symmetries, etc.) and several geometrical techniques(3+1 or 2+2 space-time splittings, initial data constraints and boundary con-ditions, etc.) seem appropriate to deal with this issue. See, for example, [2,3,4,5,6] for some detailed explanations and general comments on these subjects.However, nowadays, no consensus on a preferred approach nor any completeor definitive answer to the problem of how to associate linear and angular 4-momenta to a general space-time seem to have been reached by the relativisticcommunity. Of course, the existing points of view don’t exclude each otherand seem to point towards the correct understanding of the problem, whilethe possibility of new approaches remains still open.Nevertheless, the reader should be warned about the presence of a lot ofcriticisms in the current literature to the pseudotensor approach to definethe 4-momenta of a physical space-time, which is the approach adopted inthe present paper. These criticisms stress that the approach is by no meansa covariant one (see for example [7,8]), or argue against any definition ofenergy in General Relativity referring to 3-surface integrals, instead of beingquasi-local (referring to 2-surface integrals) from the very beginning [9], oreven accept the approach for asymptotically flat space-times but express somedoubts for the non asymptotically flat ones [10].Although the covariant approach to the definition of quasi-local conservedquantities in General Relativity followed in [7,8,9] and the results obtainedseem very interesting, we cannot fully share those criticisms.Before giving our personal opinion about it, let us begin remarking thatin [11] a particular “covariant Hamiltonian approach to quasi-local energy” ispresented. In this approach, each pseudotensor corresponds to a Hamiltonianboundary term, which brings the authors to the conclusion that “Hamiltonianapproach to quasi-local energy-momentum rehabilitates the pseudotensors”.Quoting this conclusion, Vargas [12] used again the pseudotensorial methodto calculate the energy of the universe in teleparallel gravity. The conclusionwas quoted in [13] too.Regarding the objection prescribing a quasi-local definition of energy inGeneral Relativity [9], we recall that our pseudotensorial method yields 4-momenta that are quasi-local quantities, in the sense that they can be ex-pressed subsequently as 2-surface integrals, even if their original definitionswere through 3-surface integrals.As far as the remarked [7,8] non covariance of the pseudotensorial methodis concerned, we recall that there is nothing invalidating in the fact that the en-ergy of a physical system can depend on the reference frame used and that, atthe same time, we look for some natural special frame in order to define some“proper” energy and momenta: at least, nothing, apparently, that from thevery beginning prevents us from approaching this problem. The same framedependence is present in classical mechanics, as it is mentioned in [14] in re- lation to the general problem of defining energy in a covariant way. But, for aparticle, for example, we can select a “natural” special frame (the one wherethe 3-momentum vanishes) to define the “proper” energy of the particle. Ina similar way, but pointing to General Relativity, we must select a “natural”congruence of observers and a “natural” coordinate system related to it in or-der to get rid of the spurious energy and momenta associated to the fictitiousgravitational field related to “bad” observers (for example, observers which donot fall freely) and “bad” related coordinates, so that we can reach some kindof space-time “proper” 4-momenta (see section 2, in the paragraph beginningjust after Eqs. (1)-(4)). This is in fact what is performed in the non problem-atic case of asymptotically flat space-times, where the “proper” coordinatesare those which are asymptotically Lorentzian, such that the correspondingenergy is the “proper” energy. We can think about the pseudotensorial methoddeveloped in the present paper as a proper generalization of this procedure tothe non asymptotically flat case, which would be a certain response to theaforementioned doubts in [10]. In fact, in [1] Szabados summarizes the ques-tion by simply saying: “to use the pseudotensors, a “natural” choice for a“preferred” coordinate system would be needed”. This is just what we havedone in the present paper.1.2 Summary of some previous workIn a previous paper [15], we addressed the question of properly defining thelinear and the angular 4-momenta of a significant family of non asymptoticallyflat space-times. As it is well known, and as we have just commented, seefor example [16] or [17], this proper definition can be accomplished withoutdifficulty in the opposite case of asymptotically flat space-times, but not inthe general case (for a concise and readable account, see also [18]). The reasonfor this difficulty in the general case stays in the dramatic dependence of thesemomenta on the coordinate system used. This fact is very well known but veryfew times has properly been taken into account in the literature of the field,where some authors use a given coordinate system to calculate some of themomenta, without any comments on the rightness of the coordinate selectionthat has been done. For related questions on this subject see, for instance, [2,19,20,21,22] and references therein.The family of space-times that we are going to consider in the present paperis the family of all non asymptotically flat space-times where these well definedmomenta are conserved in time. We call these particular space-times universes ,since it is to be expected that any space-time which could represent the actualuniverse should have conserved momenta, provided that these momenta beproperly defined, which is the goal achieved in the present paper.Then like in [15], we call creatable universes the universes which havevanishing 4-momenta, since again this is what could be expected to happen ifthe considered universe raised from a quantum fluctuation of the vacuum [23,24]. In fact, the question of the creatable universes is our main motivation to consider the subject of properly defining the momenta of non asymptoticallyflat space-times. Demanding the vanishing of the momenta can be a way ofsaying something relevant about how our actual Universe looks like eithernow or in the preinflationary phase. Thus, for example, in [25], perturbedflat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) universes according tostandard inflation, and also the perturbed open universes, were found to benon creatable. Therefore, among the inflationary perturbed FLRW universes,only the closed ones would be left as good candidates to represent the actualUniverse.In the present paper we present a new approach to the subject of properlydefining the two 4-momenta of a universe , as compared with the one presentedin the above reference [15] whose results we summarize here:In [15] we considered a given space-time, not necessarily asymptoticallyflat, with its general expressions for the linear 4-momentum and the angular4-momentum obtained from the Weinberg complex. Then, we assumed thatthe “intrinsic” values of these 4-momenta are conserved, that is, the valuescorresponding to some “proper” coordinate system that has to be consequentlydetermined. As mentioned above, space-times endowed with such conserved 4-momenta are called universes in the referred paper. We argued why thesecoordinates, { t, x i } , are, to begin with, Gauss coordinates referred to somespace-like 3-surface, Σ , whose equation then becomes t = t . We proved thatthe corresponding 3-space metric, dl , that is, dl = g ij dx i dx j for t = t , isasymptotically conformally flat over the 2-surface boundary, Σ , of Σ , andwe used 3-space coordinates x i adapted to this circumstance, i.e., dl | Σ = f δ ij dx i dx j , with f some function defined on Σ . Finally, looking for universeswith vanishing 4-momenta, we assumed that the metric components g ij go tozero fast enough when we approach Σ . In this way, we were able to definea family of universes whose 4-momenta vanish irrespective of the selected Σ and of the conformal coordinates used in the corresponding boundary Σ . Thefamily covers in particular the FLRW universes, for which we obtain the 4-momenta values previously obtained by some authors but not by all of them(see section 6 for some comments about these agreements and disagreements).The new approach is by no means a minor variation of the ancient one, aswe explain in the next subsection:1.3 Outline of the paperIn the present paper, given a universe , when trying to select the appropriatecoordinate systems in order to properly define its two 4-momenta, P α and J αβ ,we impose alternatively to [15] that both 3-momenta, P i and J ij , vanish, thelast one irrespective of the origin of momentum. However, according to whatwe have just explained about [15], we rest on Gauss coordinates based onsome space-like 3-surface, Σ , such that the corresponding 3-space metric canbe written in a conformally flat way on the boundary of Σ . Such Gausscoordinate systems, where both 3-momenta vanish (the last one irrespective of the origin), which at the same time are coordinates satisfying the aboveconformally flat property, will be called here intrinsic coordinate systems.Obviously, we first prove here that these intrinsic coordinate systems alwaysexist for any universe , which is a capital new result.However, in [15], in order to have vanishing 4-momenta, we had to assumethat the metric and its first derivatives went fast enough to zero when weapproach the boundary of Σ . In the present paper we do not need to makesuch an ad hoc assumption, and so our present approach, as compared withthe one in [15], stresses the intrinsic character, and so the physical meaning,of the given definition of the universe 4-momenta.The paper is organized as follows: In Sect. 2, given a space-like 3-surface, Σ , we give the corresponding family of coordinate systems where to choosethe right coordinate systems to properly define the linear and the angular4-momenta associated to this Σ . In section 3, we consider all 3-surfaces Σ showing the same boundary Σ . Then, by defining what we have called in-trinsic coordinates, we select the 3-surfaces Σ for which the linear and theangular 3-momenta vanish, after proving that this result is valid for some Σ .In Sect. 4, we define the notion of creatable universe and we discuss brieflyits goodness. In Sect. 5, we invoke some previous results to check the creat-ibility of the perturbed FLRW models in the new scheme, reproducing theknown conclusions also obtained in [25] on these models. Had we not beenable to confirm these results in the present approach, we should consider themas actually non valid, since we find now that the vanishing of P i and J ij inthe coordinates used is mandatory (although not sufficient) to confer physicalmeaning to the 4-momenta definition used. Finally, in Sect. 6, we comment onthe, sometimes, different values of the 4-momenta for FLRW universes, foundby different authors, including our work, and we point out which is, in ouropinion, the main interest of the paper, and in relation to this we refer tosome future work.We still add three appendices where some calculations are given in detail.A short report containing some results, without proof, of this work waspresented at the Spanish Relativity Meeting ERE-2009 [26]. universe , associated to a givenspace-like 3-surface In order to define the linear and angular 4-momenta of a universe we will usethe Weinberg complex [16].It remains to be checked wether the final results obtained in the presentpaper keep still valid for other complexes that, like the Weinberg one, aresymmetric in their two indices, which allows us to build the correspondingangular 4-momentum. This criterion leads to discard other pseudotensors asthe ones by Einstein, Bergmann or Møller, but not by Papapetrou or Landau-Lifshitz. Any case, the Weinberg complex is a very natural one, as it is veryconvincintly argued in [16], letting aside the interesting well known fact that
Weinberg complex gives also the correct 4-momenta in a Schwarzschild metric.As far as the Golberg pseudotensor is concerned, it is a very general one towhich the Weinberg one belongs as a particular case. About these differentpseudotensor see, for example [11].Before going to the notion of the 4-momenta of a general space-time, someprevious definitions and considerations.Metric signature: we use signature +2, that is, dτ ≡ − ds = − g αβ dx α dx β is the square of the corresponding elementary proper time when ds <
0. Thus,Greek indices take values from 0 to 3, and Latin indices from 1 to 3.Gauss coordinates: we can define them as coordinates in which ds = − dt + dl with dl ≡ g ij dx i dx j being positive defined. Although not globally,we can build such a coordinate system by referring the space-time metric toa congruence of observers that fall freely, by endowing each observer with acanonical (physical) clock, and finally by synchronizing (see next the notionof synchronization) all these different clocks [27]. In these coordinates, the3-surface t = t , with t any constant time, is a space-like 3-surface, Σ ,orthogonal to the congruence, in whose neighborhood the Gauss coordinatesystem is defined.Clock synchronization: given such a congruence of observers, each one en-dowed with his canonical clock, we can synchronize all them with the samemethod used in special relativity (using come and back light beams: see again[27]). Then, one finds that all events belonging to Σ , that is, all the events t = t , are simultaneous according to this definition. Thus, t is a physical anduniversal time (like time in the Minkowski space is, for example).Then, to properly define the notion of 4-momenta of a universe , associ-ated to some space-like 3-surface, Σ , we will take Gauss coordinates asso-ciated to this 3-surface, Σ , in the neighborhood of it (we explain next whywe make this choice). In the Weinberg approach [16], the linear and angularmomenta of the gravitational field are incidentally defined by integrating on Σ . The main Weinberg pursuit is to obtain an integral balance relation foreach momentum component such that, as it is standard, the time derivativeof a 3-volume Σ integral for this component density equates (using Gausstheorem) the minus correspondent flux through the 2-surface boundary Σ ofthe 3-volume Σ . These integrated balance equations come straightly from thevanishing of the ordinary (not covariant) divergence of the pseudotensor plusthe energy-momentum tensor and, by construction, the 3-volume integrals in-corporate a non geometric volume element. That is, this 3-volume elementis just dx dx dx independently of the meaning of this coordinates (see thedetails in [16]). As a result, these volume integrals can be expected to havea physical meaning only for some kinds of physical coordinates. On the otherhand, these 3-volume integrals, using again Gauss theorem, can be written as2-surface integrals over the 3-volume boundary, Σ . Then, according to [16],we have for the corresponding energy, P , linear 3-momentum, P i , angular3-momentum, J ij , and components J i of the angular 4-momentum, of the universe : P = κ Z ( ∂ j g ij − ∂ i g ) dΣ i , (1) P i = κ Z ( ∂ gδ ij − ∂ g ij ) dΣ j , (2) J jk = κ Z ( x k ∂ g ij − x j ∂ g ki ) dΣ i , (3) J i = P i t − κ Z [( ∂ k g kj − ∂ j g ) x i + gδ ij − g ij ] dΣ j , (4)where we have used the following notation: κ − ≡ πG , G is the Newtonconstant and we have taken c = 1 for the speed of light, g ≡ δ ij g ij , ∂ is thepartial derivative with respect to x ≡ t , and dΣ i is the surface element of Σ , the boundary of Σ . Further, indices i, j, ... are raised or lowered with theKronecker δ and angular momentum has been taken with respect to the originof coordinates. Why Gauss coordinates? We expect any well behaved universe, V , to havewell defined energy and momenta, i. e., P α and J αβ , α, β, ... = 0 , , ,
3, suchthat they are finite and conserved in time (a universe in our notation). So,for this conservation to make physical sense, we need to use a physical and universal time, according to the definition introduced at the beginning of thepresent Section. Then, still in accordance with these definitions, we are con-veyed to use a Gauss coordinate system with its universal time to properlydefine the universe simultaneous densities, as it must be from a physical point ofview.Then, as defined above, we will have for the line element of V : ds = − dt + dl , dl = g ij dx i dx j , (5)and we can write t = t = constant for the equation of Σ .The area of the 2-surface boundary Σ could be zero, finite or infinite.Let us precise that in the first case, when the area is zero, the 4-momentado not necessarily vanish, unless the metric and its first derivatives remainconveniently bounded when we approach Σ .Obviously, we have as many local families of Gauss coordinates as space-like 3-surfaces, Σ , we have in V . Then, P α and J αβ will depend on Σ , whichis not a drawback in itself (the energy of a physical system in the Minkowskispace-time also depends on the Σ chosen, i.e., on the Lorentzian coordinates In the Weinberg book [16], the case of an asymptotically flat space-time is the onlyconsidered. Nevertheless, it is straightforward to see that the displayed treatment also coversthe case of non asymptotically flat ones, provided that P α and J αβ as defined in (1)-(4)exist, i.e., provided that the corresponding integrals converge. chosen). But the problem is that, given a space-like 3-surface, Σ , we can stillhave many different 4-momenta, according to the particular Gauss coordinatewe choose, associated to the same Σ .Let us begin suppressing a part of the arbitrariness left in the choice ofGauss coordinates. In order to do this, we will choose Gauss coordinates suchthat the equation of Σ becomes x = 0, and dl on Σ reads dl ( t = t , x = 0) ≡ dl | Σ = f ( x a ) δ ij dx i dx j , (6)with f some given function, a, b, ... = 1 ,
2, and furthermore g a ( t = t ) = 0 . (7)That can always be done (see [15] and the last paragraph of the presentsection). Therefore, the induced 3-volume element dx dx dx used in our 3-volume integrals to define energy and momenta (see the just previous para-graph to Eqs. (1)-(4)) becomes physically sound.Furthermore, since t = t , x = 0, is now the equation of the 2-surface Σ ,the expressions (1)-(4) for P α and J αβ simplify to: P = − κ Z ∂ g aa dx dx , (8) P a = − κ Z ∂ g a dx dx , (9) P = κ Z ∂ g aa dx dx , (10) J ij = κ Z ( x j ∂ g i − x i ∂ g j ) dx dx , (11) J a = P a t + κ Z x a ∂ g bb dx dx , (12) J = P t − κ Z g aa dx dx (13)where g aa = g + g . Notice that, since the 3-volume element was dx dx dx ,corresponding to the 3-metric δ ij dx i dx j , and since Σ is x = 0, the induced2-metric on Σ is δ ab dx a dx b , whose metric determinant value is 1, and so the2-surface element dΣ i has become dx dx .Let us point that Σ could also be made of different sheets. Thus, in theabove Gaussian coordinates, these sheets could be the six faces of a cube thatincreases without limit. Its corresponding six equations would be ∀ i, x i = ± L ,for L → ∞ . These equations could be written x ′ i = 0, by defining the newcoordinates x ′ i = x i ∓ L and putting x ′ i = ±| ǫ | , with L → ∞ and ǫ → L and | ǫ | , and then we will take the above limits. The new coordinates x ′ i are Gauss Expression (6) will not always be valid globally. In this case we will have to cover Σ with different charts, performing each Σ –integration over each chart, and summing up thedifferent non overlapping chart contributions. coordinates with dl ′ conformally flat on Σ as it must be. All this means, inparticular, that the right hand side of, for example, (8) would actually be thesum of six similar integrals, one for each cube face. Nevertheless, in case wewould have taken x i = ± L , L → ∞ , as the equation of Σ , it can be easilyseen that essentially nothing would change in the present paper. universe , intrinsic coordinates always exist We start with a Gauss coordinate frame, { x α } , such that (6) and (7) aresatisfied. Let us prove that, from this coordinate frame, we always can moveto an intrinsic coordinate frame as defined in the Introduction. Let it be acoordinate transformation x α → x ′ α such that in the neighborhood of Σ wecan write the expansion in x ′ and t ′ − t t − t = ξ x ′ + ξ ( t ′ − t ) + ... ,x ≡ x = ξ x ′ + ξ ( t ′ − t ) + ... , (14) x a ≡ x a = ξ a + ξ a x ′ + ξ a ( t ′ − t ) + ... , where the expansion coefficients n ξ m and n ξ mi , with n, m = 0 , , , ... , are func-tions of x ′ a . Notice that this coordinate transformation is completely generalexcept for the fact that ξ = ξ = 0 . (15)To begin with, we will require that the new coordinates { x ′ α } be Gausscoordinates for V , associated to the space-like 3-surface Σ ′ , i.e. to t ′ = t .Actually, we will only require that the { x ′ α } be Gauss coordinates in theneighborhood of Σ ′ , the boundary of Σ ′ . Reducing our original requirementin this way is worth since it is known that Gaussian coordinates, sooner orlater, develop singularities under appropriate physical conditions (focussingtheorem, see for exemple [28]).On the other hand, since the equation of the boundary Σ is t = t , x = 0, this means by definition of boundary that the metric, g ij , and its firstderivatives, all them for t = t , exist only for, let us say, x >
0, at least insome elementary interval around x = 0. Then, since g ′ ij = − ∂t∂x ′ i ∂t∂x ′ j + ∂x k ∂x ′ i ∂x l ∂x ′ j g lk (16) Σ will still be the boundary of Σ ′ , provided that the functions x α ( x ′ β ) and itsderivatives, up to second order included, be well defined coordinates whereverthe metric g ij and its first derivatives are well defined in the neighborhood of Σ .Notice that, from Eqs. (14), the equation of Σ in the new coordinates { x ′ α } reads t ′ = t , x ′ = 0. Thus, if we name Σ ′ the 2-surface t ′ = t , x ′ = 0,we can say that Σ ′ = Σ . Then, besides requiring that { x ′ α } be Gauss coordinates for V in theneighborhood of Σ , the boundary of Σ ′ , we will require that, according to(6), dl ′ ( t = t , x = 0) ≡ dl ′ | Σ = f ′ ( x ′ a ) δ ij dx ′ i dx ′ j . (17)Furthermore, we will still require that the new linear and angular 3-momenta, P ′ i and J ′ ij (see (9), (10) and (11)), vanish, the last one irrespective of theorigin. That is to say, we want the new coordinate system { x ′ α } to be an intrinsic coordinate system as defined in the Introduction.From Eq. (11) we can see very easily that a necessary and sufficient con-dition to have J ij = 0, irrespective of the momentum origin, is that Z ∂ g i dx dx = 0 , ∀ i, (18)which for i = a leads to P a = 0. On the other hand, the three components of J ij can be more explicitly written J = κ Z ( x ∂ g − x ∂ g ) dx dx , (19) J a = κ Z x a ∂ g dx dx . (20)Then, aside (19) and (20) we also have (18). A sufficient condition to have allthis at the same time is that the g i metric components be such that Z ∂ g dx = Z ∂ g dx = 0 , (21) Z ∂ g a dx ( a ) = 0 , (22)where putting the a -index between parenthesis means that the index is notsummed up.In all: we start from a coordinate system, { x α } , where we have g = − , g i = 0 , (23) g a ( t = t ) = 0 , g ij ( t = t , x = 0) = f ( x a ) δ ij , (24)and we want to prove that a coordinate transformation (14) exists such thatthe new components of the metric satisfy g ′ = − , g ′ i = 0 , (25) g ′ ij ( t ′ = t , x ′ = 0) = f ′ ( x ′ a ) δ ij , (26)and that, according to (9), (10), (18), (19) and (20), we have: Z ∂ ′ g ′ aa dx ′ dx ′ = 0 , Z ∂ ′ g ′ i dx ′ dx ′ = 0 , (27) Z ( x ′ ∂ ′ g ′ − x ′ ∂ ′ g ′ ) dx ′ dx ′ = 0 , (28) Z x ′ a ∂ ′ g ′ dx ′ dx ′ = 0 , (29)where ∂ ′ means time derivative with respect the new time t ′ .What all these conditions (25)-(29) say about the functions n ξ m and n ξ mi which are present in the coordinate transformation (14)?In order to answer this question let us first write in the neighborhood of Σ : g ij = g ij + g ij x + g ij ( t − t ) + ..., (30)where, according to the notation used in (14), we have: g ij = g ij ( t = t , x = 0) , (31) g ij = ∂ g ij ( t = t , x = 0) , (32) g ij = ∂ g ij ( t = t , x = 0) , (33)and so on. This means that the expansion coefficients n g mij in (30) are functionsonly of x a .Then, Eqs. (27), (28) and (29) read Z g ′ aa dx ′ dx ′ = 0 , Z g ′ i dx ′ dx ′ = 0 , (34) Z ( x ′ g ′ − x ′ g ′ ) dx ′ dx ′ = 0 , (35) Z x ′ a g ′ dx ′ dx ′ = 0 , (36)where, similarly to (31), (32) and (33), we have put g ′ a = ∂ ′ g ′ a ( t ′ = t , x ′ = 0) = ∂ ′ g ′ a ( t = t , x = 0) , (37) g ′ = ∂ ′ g ′ ( t = t , x = 0) , (38)since, according to (14), t ′ = t , x ′ = 0 ⇔ t = t , x = 0.Similarly, Eq. (26) reads now: g ′ ij = f ′ ( x ′ a ) δ ij . (39)Thus, with the new notation n g ′ mij , the conditions (25)-(29) become (25),(34)-(36) and (39).Let us first consider conditions (25). To zero order in t ′ and x ′ (that is,strictly on the boundary Σ ) these conditions become( ξ ) − f ( ξ ) = 1 , ξ a = 0 , ξ ξ = f ξ
03 0 ξ , (40)from g ′ = − g ′ a = 0 and g ′ = 0, respectively. On the other hand, conditions (39) become f ′ δ ab = f δ cd ∂ ξ c ∂x ′ a ∂ ξ d ∂x ′ b , ξ a = 0 , f ( ξ ) − ( ξ ) = f ′ , (41)from g ′ ab = f ′ δ ab , g ′ a = 0 and g ′ = f ′ , respectively.It can be seen that the general solution of the system (40) and (41) is ξ a = ξ a = 0 . (42) ξ = s ff ′ ξ = cosh ψ, (43)1 √ f ′ ξ = p f ξ = sinh ψ, (44)plus M ab ≡ ∂ ξ a ∂x ′ b = λ (cid:18) cos θ sin θ − sin θ cos θ (cid:19) , λ ≡ p f ′ /f , (45)the Jacobian matrix of the conformal transformation in two dimensions. In(43), (44) and (45) the functions ψ , λ and θ are arbitrary functions of x ′ a .Notice that (45) says that in the integrals (34)-(36) we can put dx ′ dx ′ = λ − dx dx .We still must have: g ′ a = ( f ξ b + g b ξ ξ ) M ba + f ξ
13 1 ξ ,a − ξ ξ ,a , (46) g ′ = 2( f ξ
13 1 ξ − ξ ξ ) + g
033 1 ξ ( ξ ) , (47) g ′ aa = ( g bc ξ + g bc ξ ) M ba M ca = λ ( g aa ξ + g aa ξ ) , (48)where g ′ a , g ′ and g ′ aa are functions of x ′ a such that (34), (35) and (36)are satisfied. The derivative with respect x ′ a is denoted by , a (for instance, ξ ,a ≡ ∂ ξ ∂x ′ a ).In Eqs. (46) and (47) new expansion coefficients ξ i and ξ appear, whichare not included in (42)-(45). But they appear in Eq. (25) when it is taken tozero order in t ′ and order one in x ′ (remember that up to now we have onlyconsidered the lowest order of this equation), which becomes: g ′ a = ( f ξ b + g b ξ ξ ) M ba + f ξ
03 0 ξ ,a − ξ ξ ,a = 0 , (49) g ′ = f ( ξ
03 0 ξ + ξ
13 0 ξ ) − ξ ξ − ξ ξ + g
033 0 ξ ξ
03 0 ξ = 0 , (50) g ′ = 2( f ξ
03 1 ξ − ξ ξ ) + g
033 0 ξ ( ξ ) = 0 . (51)Therefore, we must fit the new expansion coefficients, ξ i and ξ , plus thearbitrary functions λ , θ , and ψ , of Eqs. (43)-(45), in order to satisfy the system(46)-(48) plus (49)-(51). Let us show that this can always be done.First, since the Jacobian matrix M ab is regular, we can always fit the ξ b such that the two Eqs. (46) be satisfied. Second, since f = 0, ( dl isstrictly positive) and (see Eq. (43)) ξ = 0, we can fit ξ such that Eq. (47) be satisfied too. Furthermore, it can be seen (see Appendix A) that, to get P ′ = 0, ψ can always be fitted such that Eq. (48) becomes satisfied.Next, we consider the three remaining Eqs. (49)-(50). Since (see again (43)) ξ = 0 we can fit ξ such as to have (50). Similarly for Eq. (51) by fitting ξ . Finally, it can be proved (see Appendix B) that the Jacobian matrix (45)can always be fitted in order to have Eq. (49) satisfied.In all, we have just proved that for any universe there always exist in-trinsic coordinate systems, that is Gaussian coordinates, { x ′ α } , satisfying thesupplementary conditions (39), and such that P ′ i = 0 and, irrespective of theangular momentum origin, J ′ ij = 0. Let it be a universe that we have referred to intrinsic coordinates { x ′ α } . Then,we will call that universe a creatable universe if in these coordinates we alsohave: P ′ = 0 , J ′ i = 0 . (52)This means, according to Eqs. (8), (12) and (13), that P ′ = − κ Z g ′ aa dx ′ dx ′ = 0 , (53) J ′ a = κ Z x ′ a g ′ bb dx ′ dx ′ = 0 , (54) J ′ = − κ Z g ′ aa dx ′ dx ′ = − κ Z f ′ dx ′ dx ′ = 0 . (55)that is, g ′ aa and f ′ must be such that the above four integrals vanish.On the other hand, we find after some calculation g ′ aa = ( g bc ξ + g bc ξ ) M ba M ca = λ ( g aa ξ + g aa ξ ) (56)which can be compared with (48). Notice that here we are left with no morefreedom to fit a given value of g ′ aa in order to have (53) and (54): in fact, both,the Jacobian matrix M ab , plus ξ and ξ (that is to say, plus ψ , according to(43) and (44)), have already been fitted such as to have intrinsic coordinates.This means, that:A universe is not necessarily a creatable universe, which even if expectedis a very remarkable result.Now, before we can continue, we must say something about Eq. (55), thatwould have to be satisfied if, according to our definition, we have a creatableuniverse. Since f ′ is strictly positive it seems at first sight that (55) can only besatisfied in any one of the two following cases: first, if the area of Σ vanishes(in which case f ′ should remain conveniently bounded when we approach Σ ;notice that the boundary Σ could not belong to Σ ′ , in which case f ′ could goto infinite when we approach Σ ); second, if f ′ goes to zero when we approach Σ ′ , which means again that Σ does not belong to Σ ′ . But, actually, these are not the only cases where we can have (55), since,according to what is said at the end of section 2, Σ could have several differentsheets, and it could happen that the different contributions from these differentsheets compensate among them to give a vanishing value for R f ′ dx ′ dx ′ .Thus, in Minkowski space, M , in Lorentzian coordinates (which are intrinsic coordinates) we have f ′ = 1. But, Σ is made from six sheets, the six faces of acube that increases without limit. Then, the two contributions correspondingto two opposite faces cancel each one to the other.Anywise, some one could argue that we could only define a given universe as a creatable universe if P α = J αβ = 0 for ANY intrinsic coordinate sys-tem. But this would be an exceeding demand since not even the case of theMinkowski space-time, M , would satisfy such a strong requirement. Actually,one type of intrinsic coordinates for this universe are the standard Lorentzcoordinates. Furthermore, in these coordinates, all 4-momenta, P α and J αβ vanish, so that this universe is a creatable universe according to the definitionwe have just given. Nevertheless, it can be easy seen (see Appendix C) thatstarting from Lorentz coordinates, one can always make an elementary coor-dinate transformation leading to new, non Lorentzian, intrinsic coordinates,such that the new energy P ′ does no more vanish. Obviously, according tosection 3, this elementary coordinate transformation has to be one where theinfinitesimal version of the coefficients ξ and ξ do not vanish, that is Eq.(15) does not more occur.The reason for this non vanishing energy, P ′ , in M is that, by doingthe above elementary coordinate transformation, we have left a coordinatesystem (the Lorentzian one) which was well adapted to the symmetries ofthe Minkowskian metric: the ones tied to the ten parameters of the Poincar´egroup.Thus, given a universe which has P α = J αβ = 0 for some intrinsic coor-dinate system, if there are other intrinsic coordinates where this vanishing isnot preserved, we should consider that this non preservation expresses the factthat the new intrinsic coordinates are not well adapted to some basic metricsymmetries. To which symmetries, to be more precise? In general terms, tothe ones which allow us to have just vanishing linear and angular 4-momentafor some intrinsic coordinate system.In other words: in spite of the apparent freedom in the choice of the co-ordinate frame, we have characterized in an intrinsic way if a universe has orhas not vanishing 4-momenta. In our framework, in order to have this vanish-ing, we only need to find ONE intrinsic coordinate frame where P α and J αβ vanish, which, as we have just explained, can only been found in some special universes . In Ref. [25] the creatibility of perturbed FLRW universes was addressed. Themain result of that paper which concerns us here is that in the flat case it is found that the energy is infinite, P = ∞ , for inflationary scalar perturba-tions plus arbitrary tensor perturbations. This seems to say that inflationaryperturbed flat FLRW universes are not creatable. Nevertheless, as it has beenalready stressed at the end of the Introduction, this assessment needs to bevalidated in the new framework we have developed in the present paper, wherecreatibility can only be considered for intrinsic coordinate systems, i. e., sys-tems where, in particular, the linear and angular 3-momenta, P i and J ij ,vanish.Then, we prove next that both momenta vanish in the coordinate systemwhere it was obtained that P = ∞ . Therefore, we conclude that, in thenew framework of the present paper, the non creatibility of the inflationaryperturbed flat FLRW universe remains unchanged.Let us prove first that P i vanish. According to Ref. [25] we write theperturbed 3-space metric dl as dl = a ( t )(1 + k4 r ) ( δ ij + h ij ) dx i dx j , (57)where a ( t ) is the cosmic expansion factor.In the flat case, k = 0, when considering inflationary scalar perturbations,the perturbed 3-space metric, h ij , reads h ij ( x , τ ) = Z exp( i k · x ) h ij ( k , τ ) d k (58)with the following expression for the Fourier transformed function h ij ( k , τ ): h ij ( k , τ ) = h ( k , τ )ˆ k i ˆ k j + 6 η ( x , τ )(ˆ k i ˆ k j − δ ij ) . (59)Here h ≡ h kk and η are convenient functions, ˆ k i ≡ k i /k , k ≡ √ k i k i , and τ isdefined such that dt/dτ ≡ a .According to Eq. (2): P i = lim r →∞ r πG Z I i d k (60)where I i ≡ Z exp( i k · x )[ ˙ h kk ( k , τ ) δ ij − ˙ h ij ( k , τ )] n j dΩ = Z exp( i k · x )[ ˙ h ( k , τ )( δ ij − ˆ k i ˆ k j ) + 6 ˙ η ( x , τ )( 13 δ ij − ˆ k i ˆ k j )] n j dΩ. (61)Here, the dot stands for the time, t , derivative and with dΩ the integrationelement of solid angle.Notice that here we have taken as Σ the 2-surface t = t , r = R → ∞ ,instead of the six faces of the over growing cube reported at the end of section2. Of course, the 3-volume element remains dx dx dx since these are thecorresponding intrinsic coordinates. We can take r = R → ∞ for Σ because of the choice of the above cube had only the function of making easier theproof of the existence of intrinsic coordinates.On the other hand, one easily finds Z exp( i k · x ) n i dΩ = 4 πikr ( sin krkr − cos kr )ˆ k i ≡ Φ ( k, r )ˆ k i (62)where what is important for us here is that Φ does not depend on ˆ k i . Then I i = Φ [6 ˙ η ( x , τ )( 13 ˆ k i − ˆ k i )] = − Φ ˙ η ( x , τ )ˆ k i . (63)But, as it has been quoted in [25], in the case of inflationary scalar pertur-bations, in which we are interested here, η ( k , τ ) does not actually depend onˆ k . Then, by symmetry, R I i d k = 0, and so, P i = 0 for any time.Next, we consider general tensor perturbations and we see that P i vanishtoo. As quoted again in Ref. [25], the above Fourier transformed function h ij ( k , τ ) reads now: h ij ( k , τ ) = H ( k, τ ) ǫ ij (ˆ k ) , (64)where the symmetric matrix ǫ ij is transverse and traceless: ǫ ij k i = 0 , ǫ ii = 0 . (65)The above I i integral becomes now I i = − Z exp( i k · x ) H ( k, τ ) ǫ ij n j dΩ, (66)which according to (62) and the first equation in (65) becomes I i = 0. Thisis, we have again P i = 0.Thus, when inflationary scalar and general tensor perturbations are bothpresent we have P i = 0, as we wanted to prove.The next step will be to prove that, for any time, J jk vanish too for bothtypes of perturbations. Let us first consider inflationary scalar perturbations,that is, Eq. (59).According to Eq. (11): J jk = lim r →∞ r πG Z I jk d k, (67)where I jk = Z exp( i k · x )[ n k ˙ h ij ( k , τ ) − n j ˙ h ki ( k , τ )] n i dΩ. (68)But, obviously: Z exp( i k · x ) n i n j dΩ ∝ δ ij , ˆ k i ˆ k j , (69)that is, the calculation of this integral must give a contribution which goes like δ ij , and another one which goes like ˆ k i ˆ k j . Then, it is easy to verify that when these two kinds of contributions are introduced in (68) we obtain identically I jk = 0, and so J jk = 0.Finally, we will consider general tensor perturbations, that is, h ij ( k , τ )given by Eqs. (64) and (65). In this case (68) becomes I jk = ˙ H ( k, τ ) Z exp( i k · x )( n k ǫ ij − n j ǫ ki ) n i dΩ. (70)But having in mind (69) and the first equation of (65) it is straightforwardto see that I jk and then J jk vanish.All in all, for any time, P i and J ij vanish in the same coordinate systemwhere it was proved (see Ref. [25]) that P = + ∞ . Then, we can assert thatour perturbed flat FLRW universe is really a non creatable one.On the other hand, it can be easily seen that in the present new approach,as in [25], perturbed closed FLRW universes are creatable, while perturbedopen FLRW universes are not. The energy of Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) cosmologieshas been calculated by different authors using divers procedures, like pseu-dotensorial methods based on specific choices of coordinates [29,30,31,32,33],or Hamiltonian methods imposing boundary conditions [13,21], or by choos-ing an appropriate background configuration [2,8], or even by other procedures[10]. Quasi-local approaches have also been extensively considered, providingdistinct results because of the different used definitions [9,34]. Many authors(us including) agree with the following statement: the total energy vanishesboth for closed and flat FLRW universes, but diverges to −∞ for the classof open models (negative curvature index, k = − [2] under these assumptions: (i) the considered background is the Minkowskispace-time, (ii) the conservation laws are referred to the background isometries,and (iii) the perturbed metric and the energy content are considered in somesynchronous gauge (by taking Gauss coordinates).Now, before ending the paper we would like to point out that the maininterest of it could be to give a criterion to discard from the very beginning asmuch as possible space-times as candidates to represent our actual Universe.The criterion could be that good initial candidates must be creatable universes.Thus, as commented above, in [25] it was claimed that, within the inflationaryperturbed FLRW universes, only the closed case corresponds to a creatable universe. Of course, the criterion is not a consequence of the theory of theGeneral Relativity when applied to cosmology. It is only a guess, one appear-ing in the literature at last since 1973 [23,24] that we find so appealing, in ouropinion, as to deserve that its consequences be explored, as we have continuedto do in the present paper. This result, obtained in [25] in a non conclusiveway, has been fully validated in the framework of the present paper, as it hasbeen proved in Sect. 5. Similarly, since some other space-times have latelybeen considered as candidates to represent our Universe (see for example, [36],[37]), we could check them to see if they fulfill the above criterion of creatibil-ity. When making this checking, in the case we obtained P α = 0 and J αβ = 0for a given t = t , we still had to verify that the result does not depend of thevalue of t , that is, we would have to verify a posteriori that we were dealingwith a space-time which is a universe . All this would deserve some future work. Acknowledgements
This work has been supported by the Spanish Min-isterio de Ciencia e Innovaci´on MICINN-FEDER project No. FIS2009-07705.
A Fitting the function ψ to get P ′ = 0 We must fit ψ such that g ′ aa , given by (see (48)) g ′ aa = λ ( g aa ξ + g aa ξ ) , (71)gives P ′ = 0. Notice that according to Eq. (10) we have P ′ = κ Z g ′ aa dx ′ dx ′ . (72)On the other hand, from (43) and (44), the equation (71) can be written as a = b cosh ψ + c sinh ψ , (73)where a ≡ g ′ aa , b ≡ λ g aa , c ≡ λ p f g aa (74)Then, putting cosh ψ ≡ x , we obtain the algebraic second order equation( b − c ) x − abx + a + c = 0 , (75)9that only has real solutions if a + c ≥ b . (76)But we can ensure it by taking a large enough. This can always be made since if a ≡ g ′ aa = 0is such that R a dx ′ dx ′ = 0, then we also will have R Kadx ′ dx ′ = 0, with K a constantwhose absolute value, | K | , is as large as we wanted. Furthermore, if | K | is large enough, wecan easily see that for the new coefficient a , that is, for Ka , one at least of the x solutionsis larger than one, as it must be. B Fitting conveniently the functions λ and θ or the functions λ and ψ According to what is said at the end of section 3, we must fit the functions λ and θ suchthat Eq. (49) be satisfied. Taking in account (46), the Eq. (49) becomes: g ′ a = ( ξ ξ − ξ ξ ) g b M ba + f ( ξ
13 1 ξ ,a − ξ
03 0 ξ ,a )+ ξ ξ ,a − ξ ξ ,a (77)where ξ ,a ≡ ∂ ξ ∂x ′ a , and so on. Furthermore, having in mind (43), (44) and the definitionof λ in (45), Eq. (77) becomes: g ′ a = λ ( M ba g b + X a ) , (78)where we have put X a ≡ p f ∂ψ∂x ′ a . (79)Then, from (45), we obtain the system λ ( g cos θ − g sin θ ) = − λX + g ′ (80) λ ( g cos θ + g sin θ ) = − λX + g ′ . (81)Notice that, in this system, the functions g ′ a are defined modulus an arbitrary constantfactor K (as it was, above, the case with g ′ aa ). This means that, in (80) and (81), wecan take g ′ a as small as we want, provided that the original g ′ a remain bounded (theunbounded special case will be considered next), which in turn means that we can take asthe system to solve λ ( g cos θ − g sin θ ) = − X (82) λ ( g cos θ + g sin θ ) = − X , (83)whose unique solution, out of the singular case g a = 0, is λ cos θ = − g X + g X ( g ) + ( g ) ≡ Y , (84) λ sin θ = g X − g X ( g ) + ( g ) ≡ Y , (85)that is to say λ = p Y + Y , tan θ = Y Y . (86) The singular case a ≡ g ′ aa = 0, would give as a solution for (73) tanh ψ = − b/c , whichonly exists if | b/c | < g ′ a goes toinfinite when we approach Σ . (Obviously this will have to be compatible with the vanishingof the integrals R g ′ a dx ′ dx ′ ). In this case, the system (80), (81), becomes: λ ( g cos θ − g sin θ ) = g ′ (87) λ ( g cos θ + g sin θ ) = g ′ , (88)with g ′ a going to infinite, whose solution is λ = ∞ , tan θ = lim g ′ a →∞ g
031 1 g ′ − g
032 1 g ′ g
031 1 g ′ + g
032 1 g ′ . (89)We could still consider the remaining two special cases where, only one of the twofunctions g ′ a goes to infinite, but the reader can see easily than also in both cases asolution exists for λ , θ .To end with this Appendix B, let us consider the above singular case g a = 0. It seemsthat now the four Eqs. (46) and (49) cannot always be satisfied by fitting ξ b and M ab since these four unknown functions appear now through only two quantities ξ b M ba .Nevertheless, let us proceed along the following lines:As far as Eq. (49) is concerned, we always can satisfy it by fitting some convenient valuesof ξ b , since f = 0 and M ab is a regular matrix.On the other hand, according to (78) and (79), Eq. (46) reads now g ′ a = 2 λ p f ∂ψ∂x ′ a . (90)Using λ as an integrating factor, we always can find a family of solutions ψ of these twoequations. Then, we must fit this family of solutions such that the Eq. (48) we are left with, g ′ aa = λ ( g aa cosh ψ + g aa p f sinh ψ ) , (91)becomes satisfied. To see that this is also possible, in (90) we will choose g ′ a = ǫ a g , with ǫ a = 1, ∀ a , and g a function such that R g dx ′ dx ′ = 0. In this case we have ∂ψ∂x ′ = ∂ψ∂x ′ ,that is ψ is a function of x ′ + x ′ ≡ y , but not of y ≡ x ′ − x ′ : ∂ψ∂y = 0 . (92)Then, let us integrate (91) along y over Σ . We will have a = b cosh ψ + c sinh ψ , (93)with a = Z g ′ aa dy , b = Z λ g aa dy , c = Z λ p f g aa dy , (94)where, like ψ , the coefficients a , b , c , depend only on y . On the ground of what was saidfor the coefficient a of Appendix A, the present coefficient a is also as greater as we want.Then, we can conclude that (93) always have a solution for ψ for any function g ′ aa suchthat R g ′ aa dx ′ dx ′ = 0. That is to say, Eqs. (46), (48) and (49) can all be satisfied at thesame time, as we wanted to prove in the present singular case g a = 0.1 C The counter example of Minkowski space
In section 4, we claim that if we have a universe such that its ten 4-momenta vanish forsome given intrinsic system of coordinates, we cannot hope to keep this ten-fold vanishingagainst any coordinate change going to new intrinsic coordinates. The reason of this is thateven Minkowski space, M , have not such a property.In order to see this, refer M to Lorentzian coordinates. These are obviously intrinsiccoordinates, in the sense of the present paper. Furthermore, all ten 4-momenta vanish in thisLorentzian frame. Thus, according to our definition, M is an example of creatable universe.Then, let us make some general infinitesimal coordinate transformation: x α = x ′ α + ǫ α ( x ) , (95)where the old coordinates, { x α } , are Lorentzian coordinates. Let us subject the functions ǫ ( x ) to the condition that the new coordinates { x ′ α } be intrinsic coordinates. That is, thenew metric components g ′ αβ = η αβ + η αρ ∂ β ǫ ρ + η βρ ∂ α ǫ ρ (96)has to satisfy on the one hand, Eqs. (25) and (26) (the first one up to zero order in t ′ − t and order one in x ′ ). On the other hand, the time derivatives ∂ ′ g ′ i , ∂ ′ g ′ aa , must fulfill theconditions (34)-(36) Z g ′ aa dx dx = 0 , Z g ′ i dx dx = 0 , (97) Z ( x ′ g ′ − x ′ g ′ ) dx dx = 0 , (98) Z x ′ a g ′ dx dx = 0 , (99)which mean that P ′ i = 0 and that, irrespective of the origin of the angular momentum, J ′ ij = 0 (notice that to first order we can put dx dx instead of dx ′ dx ′ ).After some elementary calculations, all these conditions are written: ε a = ∂ a ε , ε = ε , ε = 0 , (100) ε a = ∂ a ε , ε = ε , (101) ε a = − ∂ a ε , ε = (1 − f ′ ) / , (102) g ′ a = ∂ a ε + ε a , g ′ = 2 ε , g ′ aa = 2 ∂ a ε a , (103)where we have used the notation ε i ≡ ε i .A particular solution of this system is ε = ε = 0 , ε i = ε i = 0 , ε a = − ∂ a ε , (104) ε = (1 − f ′ ) / , ∂ aa ε = 0 . (105)On the other hand, we similarly obtain: g ′ aa = 2 ∂ a ε a (106)which, according to the corresponding equation in (104), becomes g ′ aa = − ∂ aa ε . (107)Thus, since ε is small, but otherwise arbitrary, we always can choose ε so as to have Z g ′ aa dx dx = 0 , (108)that is, so as to have P ′ = 0. Then, as we have announced, we cannot preserve the van-ishing of P ′ α and J ′ αβ when making a general coordinate transformation from an intrinsiccoordinate system to another intrinsic one.2 References
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