The Friedmann-Lemaitre-Robertson-Walker Big Bang singularities are well behaved
TTHE FRIEDMANN-LEMAˆITRE-ROBERTSON-WALKER BIG BANGSINGULARITIES ARE WELL BEHAVED
OVIDIU CRISTINEL STOICA
Abstract.
We show that the Big Bang singularity of the Friedmann-Lemaˆıtre-Robertson-Walkermodel does not raise major problems to General Relativity. We prove a theorem showing that theEinstein equation can be written in a non-singular form, which allows the extension of the spacetimebefore the Big Bang. The physical interpretation of the fields used is discussed.These results follow from our research on singular semi-Riemannian geometry and singular Gen-eral Relativity.
Contents
1. Introduction 12. The main ideas 43. The Big Bang singularity resolution 64. Physical and geometric interpretation 75. Perspectives 8References 81.
Introduction
The universe.
According to the cosmological principle , our expanding universe, although itis so complex, can be considered at very large scale homogeneous and isotropic. This is why wecan model the universe, at very large scale, by the solution proposed by A. Friedmann [1, 2, 3].This exact solution to Einstein’s equation, describing a homogeneous, isotropic universe, is ingeneral called the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric, due to the rediscoveryand contributions made by Georges Lemaˆıtre [4], H. P. Robertson [5, 6, 7] and A. G. Walker [8].The FLRW model shows that the universe should be, at a given moment of time, either inexpansion, or in contraction. From Hubble’s observations, we know that the universe is currentlyexpanding. The FLRW model shows that, long time ago, there was a very high concentration ofmatter, which exploded in what we call the
Big Bang . Was the density of matter at the beginningof the universe so high that the Einstein’s equation was singular at that moment? This questionreceived an affirmative answer, under general hypotheses and considering General Relativity tobe true, in Hawking’s singularity theorem [9, 10, 11] (which is an application of the reasoning ofPenrose for the black hole singularities [12], backwards in time to the past singularity of the BigBang).Given that the extreme conditions which were present at the Big Bang are very far from whatour experience told us, and from what our theories managed to extrapolate up to this moment, wecannot know precisely what happened then. If because of some known or unknown quantum effectthe energy condition from the hypothesis of the singularity theorem was not obeyed, the singularity
Date : June 3, 2015. Horia Hulubei National Institute for Physics and Nuclear Engineering, Bucharest, Romania.E-mail: [email protected], [email protected]. a r X i v : . [ g r- q c ] J un ight have been avoided, although the density was very high. One such possibility is explored inthe loop quantum cosmology [13, 14, 15, 16, 17, 18], which leads to a Big Bounce discrete model ofthe universe.Not only quantum effects, but also classical ones, for example repulsive forces, can avoid theconditions of the singularity theorems, and prevent the occurrence of singularities. An importantexample comes from non-linear electrodynamics, which allows the construction of a stress-energytensor which removes the singularities, as it is shown in [19] for black holes, and in [20] for cosmo-logical singularities.We will not explore here the possibility that the Big Bang singularity is prevented to exist byquantum or other kind of effects, because we don’t have the complete theory which is supposed tounify General Relativity and Quantum Theory. What we will do in the following is to push thelimits of General Relativity to see what happens at the Big Bang singularity, in the context of theFLRW model. We will see that the singularities are not a problem, even if we don’t modify GeneralRelativity and we don’t assume very repulsive forces to prevent the singularity.One tends in general to regard the singularities arising in General Relativity as an irremediableproblem which forces us to abandon this successful theory [21, 22, 23, 15]. In fact, contrary to whatis widely believed, we will see that the singularities of the FLRW model are easy to understandand are not fatal to General Relativity. In [24] we presented an approach to extend the semi-Riemannian geometry to the case when the metric can become degenerate. In [25] we applied thistheory to the warped products, by this providing means to construct examples of singular semi-Riemannian manifolds of this type. We will develop here some ideas suggested in some of theexamples presented there, and apply them to the singularities in the FLRW spacetime. We will seethat the singularities of the FLRW metric are even simpler than the black hole singularities, whichwe discussed in [26, 27, 28].1.2. The Friedmann-Lemaˆıtre-Robertson-Walker spacetime.
Let’s consider the 3-space atany moment of time as being modeled, up to a scaling factor, by a three-dimensional Riemannianspace (Σ , g Σ ). The time is represented as an interval I ⊆ R , with the natural metric − d t . At eachmoment of time t ∈ I , the space Σ t is obtained by scaling (Σ , g Σ ) with a scaling factor a ( t ). Thescaling factor is therefore given by a function a : I → R , named the warping function . The FLRWspacetime is the manifold I × Σ endowed with the metric(1) d s = − d t + a ( t )dΣ , which is the warped product between the manifolds (Σ , g Σ ) and ( I, − d t ), with the warping function a : I → R .The typical space Σ can be any Riemannian manifold we may need for our cosmological model,but because of the homogeneity and isotropy conditions, it is in general taken to be, at least atlarge scale, one of the homogeneous spaces S , R , and H . In this case, the metric on Σ is, inspherical coordinates ( r, θ, φ ),(2) dΣ = d r − kr + r (cid:0) d θ + sin θ d φ (cid:1) , where k = 1 for the 3-sphere S , k = 0 for the Euclidean space R , and k = − H .1.3. The Friedmann equations.
Once we choose the 3-space Σ, the only unknown part of theFLRW metric is the function a ( t ). To determine it, we have to make some assumptions about thematter in the universe. In general it is assumed, for simplicity, that the universe is filled with a fluidwith mass density ρ ( t ) and pressure density p ( t ). The density and the pressure are taken to dependon t only, because we assume the universe to be homogeneous and isotropic. The stress-energy ensor is(3) T ab = ( ρ + p ) u a u b + pg ab , where u a is the timelike vector field ∂ t , normalized.From the energy density component of the Einstein equation, one can derive the Friedmannequation (4) ρ = 3 κ ˙ a + ka , where κ := 8 π G c ( G and c being the gravitational constant and the speed of light, which we willconsider equal to 1 for now on, by an appropriate choice of measurement units). From the trace ofthe Einstein equation, we obtain the acceleration equation (5) ρ + 3 p = − κ ¨ aa . The fluid equation expresses the conservation of mass-energy:(6) ˙ ρ = − aa ( ρ + p ) . The Friedmann equation (4) shows that we can uniquely determine ρ from a . The accelerationequation determines p from both a and ρ . Hence, the function a determines uniquely both ρ and p . From the recent observations on supernovae, we know that the expansion is accelerated, corre-sponding to the existence of a positive cosmological constant Λ [29, 30]. The Friedmann’s equationswere expressed here without Λ, but this doesn’t reduce the generality, because the equations con-taining the cosmological constant are equivalent to those without it, by the substitution(7) (cid:26) ρ → ρ + κ − Λ p → p − κ − Λ . Therefore, for simplicity we will continue to ignore Λ in the following.
Figure 1.
The standard view is that the universe originated from a very dense state,probably a singularity, and expanded, with a short period of very high acceleration (theinflation).
The current standard view in cosmology is that the universe started with the Big Bang, whichis in general assumed to be singular, and then expanded, with a very short period of exponentiallyaccelerated expansion, called inflation (Fig. 1). . The main ideas
The solution proposed here is simple: to show that the singularities of the FLRW model don’tbreak the evolution equation, we show that the equations can be written in an equivalent formwhich avoids the infinities in a natural and invariant way. We consider useful to prepare the readerwith some simple mathematical observations, which will clarify our proof. These observations canbe easily understood, and combined they help us understand the Big Bang singularity in the FLRWspacetime.2.1.
Distance separation vs. topological separation.
Let’s consider, in the space R parametrizedby the coordinates ( u, v, w ), the cylinder defined by the equation v + w = 1. The transformation(8) (cid:40) x = uy = uvz = uw makes it into a cone in the space parametrized by ( x, y, z ), defined by(9) x − y − z = 0 . Figure 2.
A cylinder may have the metric of a cone, but from topological viewpoint,it still remains a cylinder. Similarly, it is not necessary to assume that, at the Big Bangsingularity, the entire space was a point, but only that the space metric was degenerate.
The natural metric on the space ( x, y, z ) induces, by pull-back, a metric on the cylinder v + w =1 from the space ( u, v, w ). The induced metric on the cylinder is singular: the distance betweenany pair of points of the circle determined by the equations u = 0 and v + w = 1 is zero. But thepoints of that circle are distinct.From the viewpoint of the singularities in General Relativity, the main implication is that justbecause the distance between two points is 0, it doesn’t mean that the two points coincide. We cansee something similar already in Special Relativity: the 4-distance between two events separatedby a lightlike interval is equal to 0, but those events may be distinct.In fact, as we have seen in section § I × Σ, so there is no way toconclude that the space Σ t = { t } × Σ reduces at a point, just because a ( t ) = 0. However, we triedto make this more explicit, for pedagogical reasons.2.2. Degenerate warped product and singularities.
The mathematics of General Relativityis a branch of differential geometry, called semi-Riemannian (or pseudo-Riemannian) geometry (see e.g. [31]). It is a generalization of the Riemannian geometry, to the case when the metric tensoris still non-degenerate, but its signature is not positive. In this geometric framework are defined otions like contraction, Levi-Civita connection, covariant derivative, Riemann curvature, Riccitensor, scalar curvature, Einstein tensor. These are the main ingredients of the theory of GeneralRelativity [32, 31, 33].The problem is that at singularities these main ingredients can’t be defined, or become infinite.The perfection of semi-Riemannian geometry is broken there, and by this, it is usually concludedthat the same happens with General Relativity.In [24] we introduced a way to extend semi-Riemannian geometry to the degenerate case. There isa previous approach [34, 35], which works for metric of constant signature, and relies on objects thatare not invariant. Our need was to have a theory valid for variable signature (because the metricchanges from being non-degenerate to being degenerate), and which in addition allows us to definethe Riemann, Ricci and scalar curvatures in an invariant way, and something like the covariantderivative for the differential forms and tensor fields which are of use in General Relativity. Afterdeveloping this theory, introduced in [24], we generalized the notion of warped product to thedegenerate case, providing by this a way to construct useful examples of singularities of this wellbehaved kind [25].From the mathematics of degenerate warped products it followed that a warped product likethat involved in a FLRW metric (equation 1) has only singularities which are well behaved, andwhich allow the extension of General Relativity to those points. At these singularities, the Riemanncurvature tensor R abcd is not singular, and it is smooth if a is smooth. The Einstein equation canbe replaced by a densitized version, which allows the continuation to the singular points and avoidsthe infinities.2.3. What happens if the density becomes infinite?
In the Friedmann equations (4), (5),and (6), the variables are a , the mass/energy density ρ and the pressure density p . When a → ρ appears to tend to infinity, because a finite amount of matter occupies a volume equal to 0.Similarly, the pressure density p may become infinite. How can we rewrite the equations to avoidthe infinities? As it will turn out, not only there is a solution to do this, but the quantities involvedare actually the natural ones, rather than ρ and p . As present in the equations, both ρ and p arescalar fields. They are the mass and pressure density (see equation (3)), as seen by an observermoving with 4-velocity u = ∂ t | ∂ t | , in an orthonormal frame. However, there is no orthonormal framefor a = 0, since the metric is degenerate at such points. Because of this, it makes no sense to use ρ and p , which can’t be defined there.On the other hand, to find the mass from the mass density, we don’t integrate the scalar ρ , butthe differential 3-form ρ d vol , where(10) d vol ( t, x, y, z ) := √ g Σ t d x ∧ d y ∧ d z = a √ g Σ d x ∧ d y ∧ d z is the volume form of the manifold (Σ t , g Σ t ). Since the typical space is the same for all momentsof time t , det g Σ is constant.The volume element , or the volume form is defined as(11) d vol := √− g d t ∧ d x ∧ d y ∧ d z = a √ g Σ d t ∧ d x ∧ d y ∧ d z. It follows that(12) d vol = i ∂ t d vol . The values ρ and p which appear in the Friedmann equations coincide with the components ofthe corresponding densities only in an orthonormal frame, where the determinant of the metricequals −
1, and we can omit √− g . But when a →
0, an orthonormal frame would become singular,because det g →
0. A frame in which the metric has the determinant − a ( t ) = 0. In a non-singular frame, det g has to be variable, as it is in the comoving coordinatesystem of the FLRW model. e shall see in Theorem 3.1 that, unlike the (frame-dependent) scalars ρ and p , the differentialforms ρ d vol , p d vol , ρ d vol , and p d vol are smooth, even if a vanishes.3. The Big Bang singularity resolution
As explained in section § √− g .Consequently, we make the following substitution:(13) (cid:26) (cid:101) ρ = ρ √− g = ρa √ g Σ (cid:101) p = p √− g = pa √ g Σ We have the following result:
Theorem 3.1. If a is a smooth function, then the densities (cid:101) ρ , (cid:101) p , and the densitized stress-energytensor T ab √− g are smooth (and therefore nonsingular), even at moments t when a ( t ) = 0. Proof.
The Friedmann equation (4) becomes(14) (cid:101) ρ = 3 κ a (cid:0) ˙ a + k (cid:1) √ g Σ , from which it follows that if a is a smooth function, (cid:101) ρ is smooth as well.The acceleration equation (5) becomes(15) (cid:101) ρ + 3 (cid:101) p = − κ a ¨ a √ g Σ , which shows that (cid:101) p is smooth too. Hence, for smooth a , both (cid:101) ρ and (cid:101) p are non-singular.The four-velocity vector field is u = ∂∂ t , which is a smooth unit timelike vector. The densitizedstress-energy tensor becomes therefore(16) T ab √− g = ( (cid:101) ρ + (cid:101) p ) u a u b + (cid:101) pg ab , which is smooth, because (cid:101) ρ and (cid:101) p are smooth functions. (cid:3) Remark 3.2.
We can write now a smooth densitized version of the Einstein Equation:(17) G ab √− g + Λ g ab √− g = κT ab √− g. This equation is obtained from the same Lagrangian as the Einstein Equation, since the Hilbert-Einstein Lagrangian density R √− g − √− g already contains √− g . Hence, we don’t have tochange General Relativity to obtain it. What we have to do is just to avoid dividing by √− g whenit may be zero.Since d vol = √− g d t ∧ d x ∧ d y ∧ d z (11), we can rewrite the densitized version of the EinsteinEquation:(18) G ab d vol + Λ g ab d vol = κT ab d vol , where(19) T ab d vol = ( ρ d vol + p d vol ) u a u b + p d vol g ab , and all terms are finite and smooth everywhere, including at the singularity. Remark 3.3. If a (0) = 0, the equation (14) tells us that (cid:101) ρ (0) = 0. From these and equation (15)we see that (cid:101) p (0) = 0 as well. Of course, this doesn’t necessarily tell us that ρ or p are zero at t = 0, they may even be infinite. Figure 3 A. shows how the universe will look, in general. Anotherinteresting possibility is that when a (0) = 0, also ˙ a (0) = 0. In this case we may have a ( t ) ≥ t = 0, for example if ¨ a (0), and obtain a Big Bang represented schematically in Fig. 3 B.This is very similar to a Big Bounce model, except that the singularity still appears. . B. Figure 3. A. A schematic representation of a generic Big Bang singularity, correspondingto a (0) = 0. The universe can be continued before the Big Bang without problems. B. Aschematic representation of a Big Bang similar to an infinitesimal Big Bounce, correspondingto a (0) = 0, ˙ a (0) = 0, ¨ a (0) > Physical and geometric interpretation
We compare the approach to the FLRW singularity proposed here with the one we proposed in[36]. An important advantage of the present approach is that the fundamental quantities, whichremain smooth at the singularity, have a more natural physical interpretation.The idea from [36] is based not on the densitized version of the Einstein equation (17), but on aversion proposed in [37], named the expanded Einstein equation ,(20) ( G ◦ g ) abcd + Λ( g ◦ g ) abcd = κ ( T ◦ g ) abcd . In this equation we used the
Kulkarni-Nomizu product of two symmetric bilinear forms h and k ,(21) ( h ◦ k ) abcd := h ac k bd − h ad k bc + h bd k ac − h bc k ad . The idea is based on the Ricci decomposition of the Riemann curvature tensor,(22) R abcd = S abcd + E abcd + C abcd , where S abcd = R ( g ◦ g ) abcd , E abcd = ( S ◦ g ) abcd , S ab := R ab − Rg ab , and C abcd is the Weyl tensor.The expanded Einstein equation (20) can be rewritten explicitly as(23) 2 E abcd − S abcd + Λ( g ◦ g ) abcd = κ ( T ◦ g ) abcd , In [36], we showed that on a FLRW spacetime, all the terms in equation (23), and consequentlythe expanded Einstein equation (20), are finite and smooth even at the singularity. This suggeststhat the Ricci part of the Riemann curvature, although contains the same information as the Riccitensor away from singularities, is more fundamental. Another advantage of quasi-regular singu-larities is that, in dimension four, they satisfy automatically Penrose’s
Weyl curvature hypothesis [38], as we have shown in [39], while this is not known to be true for more general semi-regularsingularities.The extension of the FLRW solution through the singularity proposed in this article is isometricwith the one proposed in [36]. However, the solution proposed here has more advantages, becauseit is more general, and it leads to better physical interpretations. It is expressed in terms of thedensities ρ √− g and p √− g , or equivalently ρ d vol and p d vol , which are finite and smooth everywhere,including at the singularity. Unlike ρ and p , which are usually called densities and in fact arescalars, and which are singular, ρ √− g and p √− g have the correct physical meaning of densities.For example, to obtain mass, one integrates the density. But on manifolds, one does not integratescalars, but differential forms like ρ d vol and p d vol . It is the differential form which has both thegeometric meaning, and the physical meaning. This is why ρ d vol and p d vol make more sense than ρ and p . In addition, they lead to the stress-energy tensor density T ab √− g from (16), which is ctually what we obtain from the matter Lagrangian, not the tensor T ab . This is because theLagrangian density is the density L matter √− g , and not the scalar L matter , and(24) T ab √− g = 2 δ ( L matter √− g ) δg ab . In addition, the Lagrangian density from which the Einstein equation follows is, in terms of theHilbert-Einstein Lagrangian density R √− g , the matter Lagrangian density L matter √− g , and thecosmological constant Λ, is(25) 12 κ (cid:0) R √− g − √− g (cid:1) + L√− g. In the solution presented here, although the scalar curvature R is singular at a ( t ) →
0, all termsfrom the Lagrangian density are finite and smooth. In the process to obtain the Einstein equation,the densitized one (17) is obtained. As long as we are not sure that √− g (cid:54) = 0, it is not allowed todivide by it, and the densitized Einstein equation is the one one should use.Hence, the approach presented here in terms of densities (which are but the components ofdifferential forms) has some advantages over the one presented in [36], for being more general, andhaving better geometrical and physical meaning.5. Perspectives
We have seen that, when expressed in terms of proper variables, the equations of the FLRWspacetime are not singular. Is this a lucky coincidence, or it reflects something more general? Herewe argue that this is a particular case of a more general situation, and this result is part of a largerseries.In [24, 25], it is developed the geometry of manifolds endowed with metrics which are not neces-sarily non-degenerate everywhere. A special type of singularities, named semi-regular , are shown tohave nice properties. They admit covariant derivatives or lower covariant derivatives for the fieldsthat are important, and the Riemann curvature tensor R abcd is smooth. They satisfy a densitizedEinstein equation .In the case of stationary black holes , the singularity r = 0 is due to a combination of the factthat the coordinates are singular (just like in the case of the event horizon), and the metric isdegenerate. In [26, 27, 40] it is shown how we can remove the coordinate singularity, making g ab finite, and analytic at r = 0. In [28] it is shown that black hole singularities are compatible with global hyperbolicity , which is required to restore the conservation of information .In [37], a class of semi-regular singularities, having good behavior is identified. They are namednamed quasi-regular . In [39], it is shown that quasi-regular singularities satisfy Penrose’s Weylcurvature hypothesis [38]. A large class of cosmological models with big-bang that is not necessarilyisotropic and homogeneous is identified. In [41, 42] is shown that semi-regular and quasi-regularsingularities are accompanied by dimensional reduction , and connections with various approachesto perturbative quantum gravity are presented.
Acknowledgments.
The author thanks the reviewers for valuable comments and suggestions toimprove the clarity and the quality of this paper.
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