Crítica de la razón teórica
PRÓLOGO 4 5 representaciones más o menos explícitas de las que el último se arma Recusa
3. F is at least C3 on 3M, and
4. fαβ(T, xγ) is at least C3 and non-degenerate on [0, δ = T(t1)], for some t1 ∈
R+, where T(t)≡ lim →0+ Rt 1 a(u)du.
Analogously to the proof of the FRW result, Ericksson and Scott were able to prove that the deceleration parameter q was, indeed, the characterising feature of the IPS in these RM models, which describe a wide class of possible cosmologies [11, ch 9].
Theorem 5.25
Consider an RM space-time (M,g) with comoving fluid flow. (M,g) admits an IPS, with λ = 1−β, at which the fluid flow is regular if and only if there exists a β ∈ R+ ∪ {+∞} such that along every flow-line lim
t→0+q = β, where q = − ¨ ll ˙ l2 as t→0+.
As was also mentioned in [11, ch 9], however, qitself cannot be the characterising feature of the IPS for all models if it needs to satisfy the requirements of Theorem 5.25, as counter-examples are known. Due to counter-examples it is also argued that neither the weak nor the strong energy condition, nor a limiting γ-law equation of state, nor a point-like singularity could be the general characterising feature of the IPS [11, ch 9]. Thus, the precise characterising property of the IPS for all possible models still needs to be found.
5.7
Summary of example models
After the review of known results concerning the IPS, it is now natural to raise the question of what specific models actually admit such an IPS. Several models haven been shown to possess an IPS at which the fluid flow is regular. The following collection of models is based on [9, 11] in which a complete list, references and a discussion of these models may be found.
The majority of the example models with an IPS are, in fact, perfect fluid cosmologies, all of which are given in table 5.1 categorised according to their physical characteristics - the fluid vorticity, shear, and acceleration; the Weyl tensor and its electric and magnetic parts† and the equation of state of the perfect fluid. This
table may be compared to the results regarding perfect fluids which were presented in section 5.5. It clearly reflects the GVR [47] which states that barotropic perfect fluids which admit an IPS must be irrotational, i.e. have zero vorticity. One can also see that the Mars models have non-geodesic fluid flow, but still admit an IPS. Thus, geodesicity of the fluid flow cannot be a necessary condition for a barotropic perfect fluid to allow the existence of an IPS.
Interestingly, all the listed non-FRW models in table 5.1 possess an exact γ-law equation of state, which raises the question of whether there can actually exist non-
44 5. Previous results and implications of isotropic past singularities
Model ωab σab u˙a Cabcd Eab Hab p=p(µ) λ
FRW 0 0 0 0 0 0 yes (a subclass) 2− 32γ Kantowski-Sachs 0 (a) 0 (b) (b) 0 yes (p= 13µ) 0 Szekeres (subclass) 0 (a) 0 (b) (b) 0 yes (p= 0) 12 Bondi 0 (a) 0 (b) (b) 0 yes (p= 0) 12 Tabensky-Taub 0 (a) 0 (b) (b) 0 yes (p=µ) -1
Collins 71 0 (a) 0 (b) (b) (a) yes (p= (γ−1)µ) 2− 32γ Mars 95 0 (a) (a) (b) (b) (a) yes (p=µ) -1 Table 5.1: Perfect fluid cosmological models with an IPS: (a) means that the relevant tensor is non-zero away from the IPS, but vanishes as the IPS is approached, (b) means that the relevant tensor components are bounded as the IPS is approached, with some components having a non-zero limit (after Ericksson and Scott [9, 11]).
FRW barotropic perfect fluids which admit an IPS and which donot have an exactγ- law equation of state. The table furthermore shows that the only known perfect fluid cosmologies which allow an isotropic singularity and satisfy the strongest version of the WCH, i.e. which have Cabcd = 0 at the IPS, are the FRW models. This lends
weight to the FRW conjecture stated in section 5.4.
Of the example space-times in table 5.1 we will discuss the structure of the IPS in the Kantowski-Sachs, Szekeres and Mars models as examples in chapter 7 before investigating cosmological futures in these cosmologies.
It is not only instructive to look at models which admit an IPS, but also to inves- tigate perfect fluids which do not allow an IPS. Table 5.2 shows some of these models categorised according to their physical characteristics. The Collins-Wainwright and
Model ωab σab u˙a Cabcd Eab Hab p=p(µ)
FRW 0 0 0 0 0 0 yes (a subclass)
Collins-Wainwright 0 0 (c) (c) (c) 0 yes
Wyman 0 0 (c) (c) (c) 0 yes
Barnes-Stephani 0 (c) 0 (c) (c) 0 (d)
Table 5.2: Perfect fluid cosmological models without an IPS: (c) means that the relevant quantity is non-zero, (d) means that the behaviour of the relevant quantity is uncertain (after Ericksson [11]).
Wyman models are irrotational and shear-free, but have a non-geodesic fluid flow and, by the ZAR, are hence precluded from the class of space-times which admit an IPS.
In conclusion, much is already known about barotropic perfect fluid cosmologies and IPSs therein. As has been shown in section 5.2 the expansion scalar of the fluid must diverge at an IPS, i.e. models which do not satisfy this cannot admit an IPS. The GVR shows that barotropic perfect fluids which have non-zero vorticity
5.7. Summary of example models 45
can also not admit an IPS and the ZAR implies that models which are furthermore shear-free do not have an IPS if the fluid flow is non-geodesic. For the case of a shear-free, irrotational and geodesic fluid flow there are cases with and without IPS, since the FRW result proved that not all FRW models admit an IPS. The only unknown cases are irrotational barotropic perfect fluid cosmologies with non- zero shear. There are examples with non-zero shear and an irrotational, geodesic fluid flow (e.g. Kantowski-Sachs models) and also irrotational examples with non- zero shear and non-zero acceleration (e.g. Mars models) which admit an IPS. This warrants some more research.
There are certainly also imperfect fluid cosmological models known which admit an IPS at which the fluid flow is regular. These are the Mimoso-Crawford models with an anisotropic fluid source without heat flux and the Carneiro-Marugan model whose matter source can be interpreted as a superposition of an anisotropic scalar field with radiation and dust [11]. The structure of the IPS of the latter will also be discussed in chapter 7 before its future metric singularity will be investigated.
Even though there is already much known about the implications of an IPS - especially in barotropic perfect fluid cosmologies - there are still many questions to be answered. A characterising feature of the IPS would provide us with the fundamental answer, but as mentioned in the previous section, it still needs to be found.
Chapter 6
Example space-times with
vanishing conformal factor
Specific example space-times provide valuable guidance in the quest for physi- cally reasonable definitions of future singularities. In this chapter, we investigate the existence of space-time singularities in four cosmological models which admit a vanishing conformal factor, three of which, in fact, are FRW models. Motivated by the Weyl Curvature Hypothesis (WCH), we will analyse the behaviour of the scalar K, as defined in section 4.4, throughout the evolution of the non-FRW cos- mology†. It proves to be an example in whichK steadily decreases. In the light of
the quiescent cosmology concept and the WCH we will regard this model as physi- cally unrealistic. Nevertheless, it is technically important to analyse these types of models to see what distinguishes them from physically reasonable models.
The use of proper time of the fluid flow will become essential in probing whether the vanishing conformal factors in this chapter, and especially the metric singular- ities found for particular future values of the cosmic time in the next chapter, can actually correspond to physical space-time singularities, i.e. whether they occur at finite or infinite values of the proper time. We can take advantage of the normal- isation of the fluid flow‡; the proper time for a timelike curve with tangent vector
ua and parametrisation s is given by [21, p 44]
τ =
Z
(−uaua)1/2ds. (6.1)
Thus, since the integrand is one in our case, the comoving coordinate time t cor- responds to the proper time of the fluid flow, if we choose s = t. The proper time between two metric pathologies at comoving coordinate times ts1 and ts2, where
ts2 > ts1, is therefore simply given by
τ =ts2 −ts1. (6.2)
The focus will lie on the analysis of big crunch singularities in FRW models in sections 6.1.1, 6.1.2 and 6.2.1, before we investigate a singularity in section 6.2.2
†K is trivially zero in the FRW space-times. ‡see section 4.3.1.
48 6. Example space-times with vanishing conformal factor
which can be either a past or a future singularity, depending on a constant. In the case of a past singularity, it is shown to be an IPS, according to Definition 4.1.
Several scalar quantities have been analysed with GRTensorII. Since the calcu- lations are quite extensive, only the results will be presented here. Further details of the calculations can be found in Appendix B. In order to avoid confusion with conformal structures for past singularities, we equip the relevant quantities of the conformal structure for future evolution with a ¯.