Nov 17, 2009 - FRW Universe Models in Conformally Flat Spacetime Coordinates. Ãyvind Grønâ .... dη/dt < 1 for t>t0 in expanding universe models. Hence the ...
FRW Universe Models in Conformally Flat Spacetime Coordinates Øyvind Grøn∗ † and Steinar Johannesen∗ ∗ Oslo University College, Department of Engineering, Cort Adelers gate 30, N-0254 Oslo, Norway
arXiv:0911.3310v1 [gr-qc] 17 Nov 2009
† Institute of Physics, University of Oslo, P.O. box 1048, N-0316 Oslo, Norway
Abstract The 3-space of a universe model is defined at a certain simultaneity. Hence space depends on which time is used. We find a general formula generating all known transformations to conformally flat spacetime coordinates, and work out the physical interpretation of conformal coordinate systems in different universe models. We show that continual creation appears when conformal time is used in Friedmann-Robertson-Walker models with negative spatial curvature, and in universe models where the inflationary era started as a quantum fluctuation at t > 0, for example at the Planck time. A general procedure is given for finding coordinates to be used in Penrose diagrams.
1. Introduction. Space-time and 3-space have very different characters according to the general theory of relativity. Spacetime is absolute, meaning that the properties of spacetime do not depend upon the reference frame. The 3-space, on the other hand is relative. Further on we will use the word space for the 3-space defined by a certain simultaneity. A.J.S.Hamilton and J.P.Lisle [1] have recently presented a new way to conceptualize space in their river model of black holes. Similarly, in a cosmological context, we here define a flow of inertial reference particles as a continuum of free particles with a velocity field approaching asymptotically the large scale velocity field of the distant matter in the universe. We see that there are two different ways of thinking about space: as a simultaneity space in 4-dimensional spacetime or as a flow of inertial frames. It may be useful to introduce separate names for these conceptions. We therefore make the following definitions. Coordinate space is a continuum of events taking place at a constant coordinate time. Inertial flow is a continuum of local inertial frames with vanishing velocity at a specified position. These frames consist of a flow of freely falling particles. The spacetime outside a black hole provides an example of both. The coordinate space is defined by constant Schwarzschild time coordinate. The inertial flow is a continuum of inertial frames with vanishing velocity infinitely far from the black hole. In cosmology one uses comoving coordinates in which freely moving reference particles have constant spatial coordinates. The coordinate time shown on locally Einstein synchronized clocks carried by the reference particles is called cosmic time, and the coordinate space defined by constant cosmic time is called the cosmic space. The cosmic 1
inertial flow is usually called the Hubble flow. Just as one can cut many different surfaces through a 3-dimensional body, one can define many different coordinate spaces in 4-dimensional spacetime. Flat spacetime, for example, can be separated in time and space in many different ways. Two of them are: · The space in Minkowski spacetime. This is the coordinate space of a coordinate system comoving with free reference particles that constitute a non-expanding inertial flow. · The private and public space of the Milne universe model. The private space of this universe model is just the Euclidean space in Minkowski spacetime, and the public space is the negatively curved coordinate space of a coordinate system comoving with a set of free particles constituting an expanding inertial flow, i.e. it is the cosmic space of an empty universe model. Similarly, in the non-empty Friedmann-Robertson-Walker (FRW) universe models we shall define two types of coordinate space, · cosmic space defined by constant cosmic time. · conformal space defined by constant conformal time. 2. The line element of the FRW universe models We shall here study FRW-universe models and introduce a dimensionless radial coordinate χ. In the case of a universe model with curved space the length unit is equal to the present value r0 of the spatial curvature radius, c p r0 = , (1) H0 |1 − Ω0 |
where H0 is the present value of the Hubble parameter, and Ω0 6= 1 is the total energy density relative to the critical density at the point of time t0 for which the scale factor a(t0 ) = 1 (see equation (2)). This expression for the spatial curvature follows from the Friedmann equations [2]. In a universe model with flat space, i.e. with Ω0 = 1, the unit of length is equal to the Hubble length c/H0 of the universe. We also introduce a dimensionless cosmic time t. The unit of time is equal to the time taken by light to move a unit of length. This implies that from now on (except in equations (5) and (205) - (207)) we use units so that c = 1. The line element of a FRW-universe then takes the form ds2 = −dt2 + a(t)2 [ dχ2 + Sk (χ)2 dΩ2 ] .
(2)
where dΩ2 = dθ2 + sin2 θ dφ2 . Here χ is called the standard radial coordinate. As a consequence of curvature isotropy in space the function Sk obeys [3,4] Sk′ 2 + kSk2 = 1 . where ’ denotes differentiation with respect to χ. Hence , 0 0, the horizon radius is given by Z t dt χH = = η − η(ti ) (10) ti a(t) 3
The present velocity of a reference particle with χ = constant with respect to the observer at χ = 0 is given by Hubble’s law
where
b 0 a(t0 )χ = H b0χ , vH = H b 0 = l0 H0 H
(11) (12)
is a dimensionless Hubble parameter. Note that H0 has the dimension inverse time. This means that with our dimensionless time coordinate da H0 = 1 (13) l0 a(t0 )
while
b0 = H
1 a(t0 )
dt
da dt
t=t0
t=t0
b0. The velocity vH of the Hubble flow is greater than 1 for χ > 1/H b 0 = 1 for a flat universe. Then From equations (5) and (12) it follows that H
(14)
b= H H
(15)
r(t) = r0 a(t)
(16)
H0
i.e. the dimensionless Hubble parameter of a flat universe is equal to the ratio of the ordinary Hubble parameter at an arbitrary point of time and its present value. However, in a curved universe this normalization of the dimensionless Hubble parameter is not compatible with the corresponding normalization of the scale factor a(t0 ) = 1. Note also that since all distances are scaled by a(t), the curvature radius of curved space at an arbitrary point of time is
Hence in a curved universe, the scale factor is equal to the dimensionless curvature radius a(t) = r(t)/r0 . 3. Conformal coordinates in FRW universe models The FRW universe models have vanishing Weyl curvature tensor, and hence conformally flat spacetime. It is therefore possible to introduce coordinates (T, R) so that the line element has a conformally flat form [6-9]. These types of coordinates were introduced in the description of relativistic universe models by L.Infield and A.Schild [6], who sacrificed the advantage of the standard description where freely moving reference particles have constant coordinates, for a metric conformal to Minkowski spacetime, where the speed of light is constant. The coordinates (T, R) are called conformally flat spacetime (CFS) coordinates [10-12]. They have recently been applied by J.Garecki [13] to calculate the energy of matter dominated Friedmann universes. He has argued that in this connection there is a definite advantage in using CFS coordinates. Furthermore, G.U.Varieschi [14] has described universe models based on so called conformal gravity using CFS coordinates. 4
We shall here investigate the FRW universe models with reference to these coordinates. Then the line element has the form ds2 = A(T, R)2 (−dT 2 + dR2 + R2 dΩ2 ) = A(T, R)2 ds2M ,
(17)
where A(T, R) is the CFS scale factor, and ds2M is the line element of the Minkowski spacetime. M.Ibison [15], K.Shankar and B.F.Whiting [16] and M.Iihoshi et al. [9] have shown that a general coordinate transformation that takes the line element (8) into the form (17) is T = 1 [f (η + χ) + g(η − χ)] ,
R = 1 [f (η + χ) − g(η − χ)] ,
2
2
(18)
where f and g are functions that must satisfy an identity deduced below. The transformation (18) can be described as a composition of three simple transformations. The first transforms from the coordinates η and χ in the line element (8) to light cone coordinates (null coordinates) u=η+χ , v =η−χ . (19) √ This rotates the previous coordinate system by −π/4 and scales it by a factor 2. The scaling is performed for later convenience. The second transforms u and v to the coordinates u˜ = f (u) , v˜ = g(v) . (20) Finally, we scale and rotate with the inverse of the transformation (19), R = u˜−˜v .
T = u˜+˜v , 2
2
(21)
Note that T 2 − R2 = u˜v˜ .
(22)
Taking the differentials of T and R we get − dT 2 + dR2 = −d˜ ud˜ v = −f ′ (u)g ′(v) du dv = f ′ (u)g ′(v)(−dη 2 + dχ2 ) .
(23)
Comparing the expressions (8) and (17) for the line element and using the previous formula, we find A(T, R)2 =
a(η)2 f ′ (u)g ′ (v)
(24)
and f ′ (u)g ′(v)Sk (χ)2 = R2 . By (18) and (19) the last equation may be written as 2 f ′ (u)g ′(v) Sk u−v = 1 [f (u) − g(v)]2 . 2
4
(25)
(26)
Substituting v = u and utilizing that Sk (0) = 0, this equation gives g(u) = f (u). Hence equation (26) reduces to u−v 2 ′ ′ (27) f (u)f (v) Sk = 1 [f (u) − f (v)]2 , 2
4
5
and equation (18) takes the form T = 1 [f (η + χ) + f (η − χ)] ,
R = 1 [f (η + χ) − f (η − χ)] .
2
2
(28)
Inserting χ = 0 gives T = f (η) and R = 0. Hence the physical interpretation of the function f is that it represents the transformation from parametric time to conformal time at χ = R = 0. Different choices of the function f generate different types of conformal coordinates. The function f is assumed to be increasing, meaning that the conformal time proceeds in the same direction as the parametric time at χ = R = 0, i.e. in the same direction as the cosmic time. We shall later introduce several generating functions f satisfying this relationship. Inserting equation (25) in (24) and demanding that A(T, R) > 0, we obtain the following expression for the CFS scale factor A(T, R) =
a(η(T,R)) Sk (χ(T,R)) |R|
.
(29)
We shall now integrate equation (27), written in the form
Sk
1 u−v 2 2
=
4f ′ (u)f ′ (v) [f (u)−f (v)]2
.
(30)
For this purpose we introduce the function cot x for k = 1 1/x for k = 0 Ik (x) = coth x for k = −1
(31)
so that Ik′ (x) = −Sk (x)−2 . Inserting v = a where a is an arbitrary constant in equation (30) and integrating leads to f (x) − f (a) =
2f′ (a) x−a b+Ik 2
,
(32)
where b is a constant of integration. This generalises a corresponding expression deduced by Iioshi et al. who have put a = 0. As will be shown below, our more general choice gives us a new solution not obtainable with a = 0. Equation (30) has some interesting properties. It is invariant with respect to an additive and a multiplicative constant on the function f . Hence f (a) and f ′ (a) may be chosen as arbitrary constants. This gives the solution h i−1 f (x) = c b + Ik x−a +d (33) 2
for x 6= a, where c and d are arbitrary constants. Demanding that the function f is continuously differentiable, we must define f (a) = d. In appendix A we demonstrate that this function satisfies the equation (27). The expression (33) contains generating functions of many transformations to conformally flat coordinates considered previously, as well as new ones. We shall now show that the coordinate differentials transform as a combination of a 6
Lorentz transformation and a scaling. Consider a comoving particle PH in the Hubble flow, keeping χ constant. We shall relate this particle to one at rest in the conformal coordinate system, R = constant, by a local Lorentz transformation. The partial derivatives of the coordinates T and R are ∂T = ∂R = 1 [f ′ (u) + f ′ (v)] (34) ∂η
and
∂R ∂η
∂χ
2
= ∂T = 1 [f ′ (u) − f ′ (v)] . ∂χ
2
(35)
The recession velocity in cosmology is the coordinate velocity of particles with χ = constant, i.e. it is the coordinate velocity of the Hubble flow. Obviously, the recession velocity of the Hubble flow vanishes in the cosmic coordinate system. G.Endean has introduced the recession velocity in the CFS coordinate system [10]. It is given by f ′ (u)−f ′ (v) V = tanh θ = dR . (36) = ′ ′ dT
f (u)+f (v)
χ=constant
where θ is the rapidity of the particle PH in the CFS system. It may be noted from equation (36) that this velocity cannot exceed that of light. From this equation it also follows that the Lorentz factor is γ = cosh θ = and γV = sinh θ =
f ′ (u)+f ′ (v) 2 [f ′ (u)f ′ (v)]1/2 f ′ (u)−f ′ (v) 2 [f ′ (u)f ′ (v)]1/2
,
(37)
.
(38)
Hence [f ′ (u)f ′ (v)] −1/2 dT = cosh θ dη + sinh θ dχ
(39)
[f ′ (u)f ′ (v)] −1/2 dR = sinh θ dη + cosh θ dχ .
(40)
and The differentials of the coordinates used by the two observers are therefore related by dT B 0 cosh θ sinh θ dη = , (41) dR 0 B sinh θ cosh θ dχ which corresponds to a composition of a Lorentz transformation with rapidity θ and a scaling with the factor B = [f ′ (u)f ′ (v)]1/2 = a , (42) A
which we call the relative scale factor. It can also be expressed in terms of T and R by B=
|R| Sk (χ(T,R))
.
(43)
Note that the scale factor A(T, R) in the line element for the CFS system depends upon the radial coordinate R. Hence as described the CFS coordinates the universe looks inhomogeneous. This has the following explanation. Due to the relativity of simultaneity and the relative motion of the reference particles, R = constant, of the CFS system and 7
those of the cosmic system, χ = constant, the CFS space represents a different simultaneity space than the cosmic space. The universe is homogeneous, but time dependent in the cosmic system. Thus the time dependence of the cosmic system is transformed to a time and space dependence in the CFS system. By means of the expression (33) for the generating function, one may deduce the following general formula for the recession velocity in conformally flat space (see Appendix C), f ′ (f −1 (T + R)) − f ′ (f −1 (T − R)) V = dR (44) = ′ −1 dT χ=constant f (f (T + R)) + f ′ (f −1 (T − R)) where
f ′ (f −1 (x)) = 1 [{b(x − d) − c}2 + k(x − d)2 ] .
(45)
2c
Inserting the expression (45) into equation (44), we obtain V =
2R[(b2 + k)(T − d) − bc] . (b2 + k)[(T − d)2 + R2 ] − c[2b(T − d) − c]
(46)
From equation (14) and the relation dt = adη it follows that b = 1 da . H 2
(47)
HR = 12 ∂A .
(48)
a dη
It will later be shown that the parametric time η may be used as a CFS time coordinate for flat universe models. This motivates the following definition of the Hubble parameter in an arbitrary CFS coordinate system A ∂T
This definition means that HR represents the expansion or the contraction of the CFS flow defined by reference particles with R = constant. It should be noted, however, that these particles are not freely falling. They do not constitute an inertial flow. From equation (36) we find that the Doppler shift factor due to the recession velocity is r r f ′ (u) 1+V D= = . (49) ′ 1−V
f (v)
Using the expression (33) for the generating function we arrive at the following general expression of the Doppler factor D(T, R) =
k( T +R−d )2 + [ b ( T +R−d )−c ]2 k( T −R−d )2 + [ b ( T −R−d )−c ]2
1/2
.
(50)
The total redshift z of an object at R emitting light at a time T as observed at R = 0 at a time T0 is given by [6] 1 + z = D(T, R)
A(T0 ,0) A(T,R)
where R = T0 − T , since light moves with constant velocity in the CFS system. 8
(51)
4. Universe models with negative spatial curvature 4.1. Conformal coordinates in negatively curved universe models M.J.Chodorowski [17] has recently presented an interesting discussion of the concept space in a cosmological context. He has deduced the form of the Robertson-Walker line element for the case of a negatively curved space as expressed in terms of conformal time [4], and pointed out that space is a coordinate dependent concept. In this case S−1 (χ) = sinh χ. The CFS coordinates used by Chodorowski may be obtained by choosing the values a = 0, b = −1, c = 2Ti and d = Ti in (33), where Ti is the conformal time corresponding to t = η = 0 at χ = 0. This gives the generating function f (x) = Ti ex .
(52)
Hence the transformation (28) between the coordinates (η, χ) and the conformal coordinates (T, R) is T = Ti eη cosh χ , R = Ti eη sinh χ , (53) where Ti is the conformal time corresponding to t = η = 0 of the universe model at χ = 0 . This is the initial time for universe models with radiation and dust. The inverse transformation is √ Ti eη = T 2 − R2 , tanh χ = R , (54) T
where R > 0 and T 2 > R2 . At χ = R = 0, the clocks showing conformal time goes exponentially faster than the clocks showing parametric time. Equations (4), (53) and (54) lead to 2 S−1 (χ(T, R)) = 2R 2 . (55) T −R
From equation (55) and (29) it follows that the line element for a universe model with negative spatial curvature, as expressed in terms of the conformal coordinates (T, R), takes the conformally flat form a(η(T, R))2 2 dsM , (56) T 2 − R2 √ showing that the relative scale factor is B = T 2 − R2 . In the (T, R)-system, each reference particle with χ = constant in the cosmic coordinate system has a constant recession velocity ds2 =
V = R = tanh χ T
(57)
which is less than 1. According to this equation χ is the rapidity of a reference particle with radial coordinate χ. The question of superluminal expansion of space using the line element (56) has been discussed by Lewis et al.[18]. From equations (11), (36) and (57) it follows that the rapidity χ is (58) χ = vH . b0 H
Figure 1 shows the cosmic coordinate system (η, χ) in a Minkowski diagram referring 9
to the conformal coordinate system of the observer at χ = 0. It follows from equations (54) that the world lines of the reference particles with χ = constant are straight lines, and the curves of the cosmic space η = constant are hyperbolae with centre at the origin as show in the diagram in Figure 1.
T T0
Ti
χ = χH t = 0
6 P @ @ H @ @
O
R
Figure 1. Minkowski diagram for universe models with negative spatial curvature with reference to the conformal coordinates (T, R). Here the line OH is the world line of a reference particle with χ = constant. The line PH represents the backwards light cone. The hyperbola represents the cosmic space at cosmic time t = 0 of the observer.
Equation (53) shows that the velocity of the (T, R)-system relative to the (η, χ)system, i.e. of a particle with R = constant, is given by dχ dη
= − tanh χ .
(59)
Hence the (T, R)-system contracts relative to the (η, χ)-system, i.e. relative to the Hubble flow. In a (η, χ)-diagram the world lines of the reference particles with R = R1 are given by R sinh χ = 1 e−η , (60) Ti
and the simultaneity curves T = T1 of the conformal space by cosh χ = T1 e−η . Ti
These curves are shown in Figure 2.
10
(61)
η
6 R = const
T = const -
χ
Figure 2. Minkowski diagram for universe models with negative spatial curvature with reference to the (η, χ)coordinate system. The diagram shows world lines R = constant and simultaneity curves T = constant . We see that the (T, R)-system contracts relative to the (η, χ)-system, i.e. relative to the Hubble flow.
4.2. The Milne universe model The Milne universe is a model of an empty universe with Ω0 = 0. This model has negative spatial curvature. The scale factor is a(t) = t
(62)
The normalisation condition a(t0 ) = 1 is fullfilled since the unit of time is equal to the present age t0 of the Milne universe, using that Ω0 = 0 in equation (1). Equation (7) then gives eη = t (63) with η ∈ when t ∈ < 0, ∞ >. For the Milne universe the transformation between the cosmic coordinates and the conformal coordinates is T = t cosh χ , R = t sinh χ , (64) with inverse transformation t=
√
T 2 − R2 ,
tanh χ = R . T
(65)
In this case T = t at χ = 0, i.e. the conformal clock at χ = 0 goes at the same rate as the cosmic clocks. However, while the cosmic clocks go at the same rate at all positions, the conformal clocks go faster for larger values of χ. Inserting equation (65) into equation (56) gives ds2 = ds2M , (66) which shows that in the case of the Milne universe model the conformal coordinates are the same as the coordinates of flat spacetime in a static reference frame in which the line element takes the Minkowski form. Hence the Hubble parameter HR in the conformal coordinate system vanishes in this case. 11
For this universe model the relative velocity v as given in equation (57), between the (η, χ)-system and the (T, R)-system, is equal to the velocity vH of the Hubble flows. This means that for the Milne universe, the conformal coordinate system is comoving with an observer at χ = 0. The conformal space is then identical to the private space of this observer. This is not so for the other universe models we consider. In the Milne universe, the public space at t = 0 is not represented by a hyperbola in Figure 1, but by the light cone R = T . This means that the private space has a finite extension at the time T . However, the public space at t = t1 is represented by the hyperbola T 2 − R2 = t21 in the (T, R)-diagram, and hence has an infinite extension. 4.3. Particle horizon in negatively curved universe models using conformal time. We here consider universe models with negative spatial curvature, and where the time coordinate η → 0 when the cosmic time t → 0. Then the universe extends only out to the hyperbola in Figure 1, and not out to the light cone. As an interesting application of the conformal coordinates, we will discuss the existence of a particle horizon using these coordinates. In Figure 1 we have drawn a Minkowski diagram where the hyperbolae represent the simultaneity space at constant cosmic time measured by clocks moving along the straight world lines from the origin. We shall now show how equation (9) can be deduced directly from the figure. Consider an observer at the point P having coordinates (0, T0 ). The line P H given by R = T0 − T represents the backwards light cone of this observer. A straight line χ = constant through the origin represents a spherical surface in space at different times. The particle horizon of this observer is the spherical surface within which he may receive information emitted after cosmic time t = 0. The space at t = 0 is represented by the hyperbola HTi given by T 2 − R2 = Ti2 . The position of the horizon is given by the intersection between P H and HTi having coordinates TH =
T02 + Ti2 2T0
,
RH =
T02 − Ti2 . 2T0
(67)
The time coordinate ηP of the observer is found by inserting R = 0 in the first of equation (54) giving T0 . (68) eηP = Ti The radial coordinate of the horizon χH is found from the transformation (53) which leads to T +R T +R eχH = cosh χH + sinh χH = = , (69) η H Ti e Ti since the time coordinate ηH = 0 at the point H in Figure 1. Inserting the coordinates of the point H from equation (67) into this equation, we arrive at eχH =
T0 Ti
.
(70)
Combining equations (68) and (70), we obtain equation (9). A comoving object in the universe has a position given by a fixed coordinate χ. In the 12
present case this corresponds to a straight worldline in Figure 1. The physical significance of the coordinate χH is that an observer at R = 0 cannot observe objects with χ > χH at the point of time T0 . 4.4. Continual creation in negatively curved universe models with dust and radiation using conformal time. The scale factor of a universe model with dust and radiation and with negative spatial curvature, is given in parametrical form by [2,19] a = α(cosh η − 1) + β sinh η
(71)
t = α(sinh η − η) + β(cosh η − 1) ,
(72)
and where α= and
1 b2 H Ω 2 0 m0
b0 β=H
p
=
Ωγ0 =
k Ωm0 2(Ω0 −1) 1 Ω 2 m0
r k Ωγ0
Ω0 −1
p Ωγ0
for k 6= 0 for k = 0 for k 6= 0
Ωm0 2(1−Ω0 )
and β =
(74)
for k = 0
with Ω0 = Ωm0 + Ωγ0 . In the present case with k = −1, we have r α=
(73)
Ωγ0 1−Ω0
.
(75)
Here η ∈< 0, ∞ > when t ∈< 0, ∞ >. Transforming to conformal coordinates by means of equation (53) and choosing Ti = 1 (α + β) , (76) 2
the line element (56) takes the form 2 2 α−β α+β 1− √ ds2M , ds = 1 − √ 2 2 2 2 2 T −R 2 T −R 2
(77)
where T 2 − R2 > Ti2 . Note that when β = 0, we obtain the line element of a dust dominated universe 4 α 2 ds = 1 − √ ds2M (78) 2 T 2 − R2
in accordance with [13,16]. Putting α = 0 we obtain the line element of a radiation dominated universe with negative spatial curvature in conformal coordinates 2 β2 2 ds2M . (79) ds = 1 − 4(T 2 − R2 ) 13
The relationship between the cosmic time and the conformal time at R = χ = 0 is dt = A(T, 0)dT .
(80)
From equation (77) it then follows that α+β α−β dt = 1 − 1− dT , 2T
(81)
2T
which shows that dT /dt > 1, i.e. the clocks showing conformal time go at a faster rate than those showing cosmic time. The expression (81) may be integrated with an integration constant determined by the condition t = 0 for T = Ti . The result is, however, already contained in the transformation equation (28). With R = 0 we have from equation (54) T = Ti eη .
(82)
Then, using equation (72) we obtain T t = T − α ln T − 1 (α − β) i − β , Ti
2
(83)
T
showing that t = 0 for T = Ti = (α + β)/2. In a universe model dominated by dust equation (83) reduces to t=
T 2 −Ti2 T
− 2Ti ln T ,
(84)
Ti
and in a radiation dominated universe model, t=
(T −Ti )2 T
χ1 that did not exist at the point of time T = T1 . The enlargement of conformal space is therefore due partly to expansion, and partly to creation of new space, matter and radiation. Although the conformal space is finite, a space traveller can never reach the boundary of the universe since it moves outwards with a velocity greater than the velocity of light. Differentiation of equation (54) while keeping η = 0 gives dR = √ 12 2 (86) dT
1−Ti /T
η=0
for T > Ti . On the other hand, the expansion of the conformal space represented by the velocity of the Hubble, flow, is dR = tanh χ . (87) dT
χ=constant
Note that (dR/dT )η=0 > 1 and (dR/dT )χ=constant < 1. This means that the boundary of the universe moves faster than the inertial flow representing the expansion of space. Hence new space is continually created at the boundary. However, using cosmic time no new space is created. This apparent contradiction is solved by noting that constant cosmic time and constant local time represent different simultaneities. So the conformal space defined at constant conformal time is different from the cosmic space defined at constant cosmic time. An exception among the negatively curved universe models is the Milne universe. Due to the special form of the scale factor in this universe model, the parametric time η → −∞ when the cosmic time t → 0 . Hence the space t = 0 is not represented by a hyperbola, but by the light cone in Figure 1. This means that there is no new space created using local time in this universe model. Continual creation is not in conflict with the concept of a Big Bang. Interpreted as a singularity, the Big Bang is not a part of the spacetime. What the theory can describe, is the events happening after the Big Bang singularity. The cosmic space at t = 0 corresponds to η = 0 and is again represented by the hyperbolae in the Figures 1 and 2. Hence there is continual creation of matter and radiation at the boundary of the conformal space in these universe models. The Kretschmann curvature scalar for the models with dust and with radiation are 15
respectively 245760 (T 2 − R2 )3 α2 1572864 (T 2 − R2 )4 β 4 √ Kd = . , Kr = [4(T 2 − R2 ) − β 2 ]8 [2 T 2 − R2 − α]12
(88)
These expressions show that there is a physical singularity with infinitely great spacetime curvature at the boundary T 2 − R2 = Ti2 with continual creation. 4.5 Negatively curved universe models with vacuum energy The inflationary era is a brief period dominated by vacuum energy with accelerated expansion at the beginning of the universe. This era is often said to make space flat. That cannot, however, be the case. The inflationary era cannot change a universe model with curved space, k 6= 0 , to a model with flat space, k = 0 . It can only make space approximately flat. The curvature decreases exponentially. Such a space is still curved, although the curvature may be so small that we are not able to measure it. So if our universe entered the inflationary era with curved space, the space will still be curved today. The region of initial conditions for universe models with curved space is much larger than that for flat space. Hence, if the universe entered the inflationary era by some sort of quantum processes, the probability that the space is curved is much larger than that it is flat. One may therefore conclude that we probably live in a universe with curved space, but that the space was inflated so much in the inflationary era that we are not able to measure the curvature. Although the difference between such a curved universe and a flat one is negligible small as far as observed properties of the universe is concerned, there are some very interesting conceptual differences between curved universe models and flat ones. For example a flat universe dominated by Lorentz Invariant Vacuum Energy (LIVE) has a steady state character and may be infinitely old, while a corresponding universe with negatively curved space is evolving and has a finite age. In this case the scale factor is
for t > 0, where
b Λ t) , a(t) = 1 sinh(H bΛ H
b Λ = l0 H
p
(89)
Λ/3 ,
(90)
l0 is given in equation (5), and Λ is the cosmological constant [8]. The dimensionless b = l0 a/a, Hubble parameter, H ˙ is b =H b Λ coth(H b Λ t) . H
(91)
According to equation (16) the dimensionless curvature radius of space at an arbitrary point of time is rˆ = r = a (92) r0
which increases exponentially during most of the inflationary era, and the curvature decreases exponentially. Using equation (7), the parametric time η is [20] bΛt b Λ t)) = ln(tanh H ). η = −arccoth(cosh(H 2
16
(93)
Note that η → −∞ when t → 0 , and that η → 0 when t → ∞ . This means that the conformal space extends out to the light cone R = T . Hence, there is no continual creation in such a universe model. For this universe model the limit t → ∞ at χ = 0 corresponds to a conformal time T = Ti according to equation (66). Hence it is natural in this case to replace the initial time Ti with a final time Tf . The conformal time at R = 0 is therefore b t H T = Tf eη = Tf tanh Λ < Tf , (94) 2
where Tf is the final conformal time. Hence the LIVE dominated universe with k = −1 has a finite conformal age. Again this is a coordinate effect since dT =
b Λ Tf /2 H dt b Λ t/ 2) cosh2 (H
,
(95)
which shows that the rate of the conformal time decreases exponentially compared to the rate of the cosmic time. This model is, however, not realistic since the general theory of relativity is valid only after the Planck time, tP l = 5.4 · 10−44 s. Before this time the universe may have existed in a quantum era which cannot be described without a quantum theory of gravity. We here assume that the universe entered a vacuum dominated inflationary era at the Planck time. This implies that the conformal space only extends out to a hyperbola given by t = tP l , which represents the frontier against a quantum era with properties that we cannot describe with out present theories. At this frontier there is continual creation of new conformal space. Equations (89) and (93) imply that the scale factor is given in terms of the parametric time as 1 . (96) a(η) = − b Λ sinh η H
Together with equation (56) this relation implies that that the line element for the present universe model, as expressed by conformal coordinates, takes the form #2 " 1 2T f ds2 = ds2M . (97) 2 T 2 − (T 2 − R2 ) b Ω0 H0 f This expression shows that the relationship between cosmic time and conformal time is dt =
1 √
2Tf 2 T −(T 2 −R2 ) 0 f
b0 Ω H
dT .
(98)
The coordinate t is the proper time of the freely moving observers. The expression shows that the clocks showing conformal time slows down towards a vanishing rate of time as we approach the boundary T 2 − R2 = Tf2 of the conformal space. The conformal space has an interesting behaviour in this universe model. It is annihilated as we see in Figure 4. At the point of time T = Tf , conformal space starts vanishing at R = 0. Then a spherical hole develops which does not belong to the conformal space. Note that space as defined in the introduction consists of simultaneous events in spacetime, and the points inside the hole does not correspond to events in the FRW-universe. 17
An observer can never reach the boundary of the hole since the cosmic time approaches infinity at this boundary. At the local time T1 the hole in the universe is represented by the horizontal line segment P1 Q1 . At the time T2 this hole has expanded so that it is now represented by the line segment P2 Q2 . The part of the line T = T2 outside P2 Q2 and inside the hyperbola represents new emptiness which has appeared after the time T1 . The enlargement of the hole in conformal space is therefore due partly to expansion, and partly to annihilation of space.
T
χ = χ1 η = 0
6
Q2
P2 Q1
P1
T = T2 T = T1
O
R
Figure 4. The hatched region represents a succession of conformal spaces in a LIVE dominated universe at different points of time. The horizontal line segment P1 Q1 represents a hole in the conformal space at the conformal time T1 . At the conformal time T = T2 the hole P1 Q1 that existed at T = T1 has expanded to P2 Q2 . In addition the figure shows that conformal space has been annihilated on the line T = T2 outside the line segment P2 Q2 and inside the hyperbola.
At the present time the universe is filled with radiation, matter and vacuum energy. If the vacuum energy is of the LIVE type, the energy density will remain constant in the future. But the density of radiation and matter will decrease. Hence the universe will approach a vacuum dominated state. This means that in conformal coordinates the final destiny of our universe will be as described above. 4.6. A second type of conformal coordinates for universe models with negative spatial curvature For universe models with negative spatial curvature one may introduce a second type of b by choosing a = 0, b = 0, c = 1 and d = 0 in equation (33). conformal coordinates (Tb, R) This gives the generating function f (x) = tanh(x/2) .
(99)
The transformation (28) then takes the form Tb =
sinh η , cosh η + cosh χ 18
b= R
sinh χ . cosh η + cosh χ
(100)
If the universe model begins at η = 0, this transformation maps the first quadrant χ > 0, b < 1 − Tb. On the other hand, if the universe η > 0 onto the triangle 0 < Tb < 1 and 0 < R is infinitely old, the fourth quadrant χ > 0, η < 0 is mapped onto the triangle −1 < Tb < 0 b < 1 + Tb. The inverse transformation is and 0 < R coth η =
b2) 1+(Tb2 −R 2Tb
,
coth χ =
b2) 1−(Tb2 −R b 2R
.
(101)
From equation (29) it then follows that the line element now takes the form ds2 =
b 2 4a(η(Tb, R)) ds2M 2 2 2 2 b b b [1 − (T − R )] − 4R
b spacetime diagram in As drawn in the (Tb, R) points with χ = χ0 and the simultaneity curves by b − a1 )2 − Tb2 = a2 − 1 and (R 1
(102)
Figure 5 and 6, both the world lines of η = η0 are hyperbolae, given respectively
b2 = b2 − 1 , (Tb − b1 )2 − R 1
(103)
where a1 = coth χ0 and b1 = coth η0 . These equations are valid both for universe models dominated by radiation and dust and by LIVE, the only difference being that η0 > 0 with b radiation and dust, and η0 < 0 with LIVE. The velocity in the (Tb, R)-system of a particle with χ = constant is given by b b sinh η sinh χ 2TbR V = dR = =− . (104) dTb
1+cosh η cosh χ
χ=constant
b2) −1+(Tb2 +R
In this coordinate system the initial recession velocity vanishes. Note that v < 0 for b > 0 and v > 0 for TbR b < 0 since Tb2 + R b2 < 1. TbR b The reference particles of the (Tb, R)-system and the (T, R)-system have different b = constant is motions. The velocity in the (η, χ)-system of a particle with R dχ dη
=
sinh η sinh χ 1+cosh η cosh χ
.
(105)
b Hence the (Tb, R)-system expands relative to the (η, χ)-system while the (T, R)-system b contracts for T > 0. Tb
6
1
@ @ @ @ @ @ η = const @ @ @ @ χ = const 1
19
-
b R
Figure 5. Minkowski diagram for universe models with negative spatial curvature containing dust and radiation with reference to the conformal coordinate system of type II defined in equation (100). The diagram shows world lines χ = constant and simultaneity curves η = constant . We see that the (η, χ)-system contracts b relative to the (Tb, R)-system.
Figure 5 gives an illusion of a contracting universe of finite extension with a finite age, since the world lines of particles with χ = constant, which define the Hubble flow, all b = (1, 0). However, according to equation (101), approaching terminate at the point (Tb, R) this point means that the parametic time η, and hence also the cosmic time t and the scale factor, approaches infinity. Furthemore, this equation also implies that χ approaches b infinity at the line Tb = 1 − R. Tb
6
χ = const
η = const
1
-
b R
−1
Figure 6. Minkowski diagram for universe models with negative spatial curvature containing vacuum energy with reference to the conformal coordinate system of type II defined in equation (100). The diagram shows world lines χ = constant and simultaneity curves η = constant . We see that in this case the (η, χ)-system b expands relative to the (Tb, R)-system.
Figure 6 gives an illusion of a universe of finite extension with a finite age, and with attractive gravitation. Since the world lines of the particles with χ = constant curve towards the Tb-axis, the coordinate velocity of these particles decelerates. According to equations (93) and (101), the conformal time Tb = −1 when the cosmic time t = 0 , and Tb → 0 when t → ∞ , meaning that this universe will become infinitely old. From equab − 1. Furthermore, tion (101) it also follows that χ approaches infinity at the line Tb = R equation (89) implies that a¨ > 0, which means that the Hubble flow has accelerated expansion. An important difference between the CFS systems of type I and II is that in the (T, R)-system the free particles defining the Hubble flow move with constant velocity b along straight world lines, while in the (Tb, R)-system their world lines are hyperbolae. For a negatively curved universe with radiation and dust the line element with type
20
II conformal coordinates takes the form √ " 2 2 ds2 =
b − 2α 1+Tb −R
b 2 )2 −4Tb2 (1+Tb2 −R
b 2 )2 −4Tb2 (1+Tb2 −R
+4β Tb
#2
ds2M .
b = 0 is related to the cosmic time by The type II conformal time at R b2 2Tb 2Tb + β 2T . − arcsin t=α 1−Tb2
1−Tb2
1−Tb2
(106)
(107)
The corresponding relationship for a LIVE dominated universe is b Λ t = ln Tb . H
(108)
The Tb-clocks goe faster than the cosmic clocks in this universe model. From equations (93), (101) and (102) it follows that with the type II conformal coordinates, the line element of a negatively curved universe model dominated by LIVE takes the form (109) ds2 = 21 ds2M . b Tb2 H Λ
The coordinate transformation between the present conformal coordinates and those introduced in equation (53) is (T 2 −Ti2 )−R2 Tb = , 2 2 (T +Ti ) −R
b= R
2Ti R (T +Ti )2 −R2
.
(110)
4.7. A third type of conformal coordinates for universe models with negative spatial curvature For universe models with negative spatial curvature one may also introduce a third type e this time by choosing b = coth(a/2), c = sinh−2 (a/2) of conformal coordinates (Te, R), and d = tanh(a/2) in equation (33). This gives the generating function f (x) = − coth(x/2) .
(111)
Then the transformation (28) takes the form Te =
sinh η , cosh χ − cosh η
e= R
sinh χ . cosh η − cosh χ
(112)
If the universe model begins at η = 0, this transformation maps the region 0 < η < χ onto e and the region 0 < χ < η onto the region 0 < R e < −1 − Te. the region 0 < Te < −1 − R, On the other hand, if the universe is infinitely old, the region −χ < η < 0 is mapped onto e < Te < 0, and the region 0 < χ < −η onto the region 0 < R e < Te − 1. the region 1 + R The inverse transformation is coth η =
(Re2 −Te2)−1 2Te
, 21
coth χ =
(Te2 −Re2)−1 e 2R
.
(113)
b From equation (24) it follows that the line element again takes the form (102) with (Tb, R) e As drawn in the (Te, R) e spacetime diagram in Figure 7 and 8, both replaced by (Te, R). the world lines of points with χ = χ0 and the simultaneity curves η = η0 are hyperbolae, given respectively by e + a1 )2 − Te2 = a2 − 1 and (R 1
e2 = b2 − 1 , (Te + b1 )2 − R 1
(114)
where a1 = coth χ0 and b1 = coth η0 . Again these equations are valid both for universe models dominated by radiation and dust and by LIVE, the only difference being that η0 > 0 with radiation and dust, and η0 < 0 with LIVE.
χ = const
η = const
@ @ @ @ @ @ @
Te
6
@−1 e R @ @ −1 @ @ @ @ @ @ @ @ χ = const
η = const
Figure 7. Minkowski diagram for universe models with negative spatial curvature containing dust and radiation with reference to the conformal coordinate system of type III defined in equation (112). The diagram shows world lines χ = constant and simultaneity curves η = constant .
22
χ = const Te
η = const
6
1 −1 η = const
e R
χ = const
Figure 8. Minkowski diagram for universe models with negative spatial curvature containing vacuum energy with reference to the conformal coordinate system of type III defined in equation (112). The diagram shows world lines χ = constant and simultaneity curves η = constant .
The CFS picture of type III of the evolution of a universe with radiation and dust is rather pathological. The universe starts at Te = 0 with a hole in a finite region around e = 0 where the conformal space does not exist, surrounded by a region with radiation R and dust of infinitely great extension. The conformal space then expands and so does the e hole. As Te approaches infinity, the clocks are reset to come from minus infinity, and R e can be removed by changchanges from negative to positive values. A negative sign of R ing the coordinates θ and φ corresponding to a reflection through the origin in space. This means that the expansion is replaced by a contraction. The conformal space then contracts to a vanishing extension at Te = −1. The corresponding picture for a LIVE dominated universe is as follows. At the point of time Te = 1 the conformal space appears with vanishing extension. The conformal space then expands towards infinite extension as the conformal time approaches infinity. At this e changes from positive to point the clocks are reset to come from minus infinity, and R negative values, so that the expansion is replaced by a contraction. The conformal space then contracts to a final state at Te = 0 similar to the initial state with radiation and dust. In the present case the line element is still given by (109). 5. Flat universe models 5.1. Conformal coordinates in flat universe models In flat universe models η and χ already are conformal coordinates, corresponding to a = 0, b = 0, c = 2 and d = 0 in equation (33). This gives the generating function f (x) = x.
23
However, η = constant defines the same cosmic space as that given by t = constant. But when k = 0, the relationship (27) permits us to introduce a second type of conformal coordinates defined by a = 1, b = 2, c = 2 and d = −1 in equation (33). This gives the generating function f (x) = −1/x . (115) The transformation (28) then takes the form T =
η χ2 −η 2
and R =
χ η 2 −χ2
.
(116)
If the universe model begins at η = 0, this transformation maps the region 0 < η < χ onto the region 0 < T < −R, and the region 0 < χ < η onto the region 0 < R < −T . On the other hand, if the universe is infinitely old, the region −χ < η < 0 is mapped onto the region R < T < 0, and the region 0 < χ < −η onto the region 0 < R < T . The inverse transformation has the same form, η=
T R2 −T 2
and χ =
R T 2 −R2
.
Since ∂T /∂η > 0, T increases in the same direction as η and t. The line element takes the form 2 a(η(T,R)) 2 ds = (−dT 2 + dR2 + R2 dΩ2 ) . T 2 −R2
(117)
(118)
From equation (46) it follows that a particle with χ = constant has a recession velocity in the (T, R)-system given by V = 2T R , (119) T 2 +R2
Again the initial recession velocity vanishes. Furthermore V > 0 for T R > 0 and V < 0 for T R < 0. Note that V < 0 corresponds to expansion and V > 0 to contraction when R < 0. A flat universe model with radiation and dust has [2,19] a = 1 αη 2 + βη , 2
t = 1 αη 3 + 1 βη 2 6
where α = Ωm0 /2 and
β=
2
p Ωγ0 .
(120) (121)
The relationship between the conformal time at R = 0 and the cosmic time is t=
3βT −α 6T 3
.
(122)
The conformal clocks go faster than the cosmic ones. Here η ∈< 0, ∞ > when t ∈< 0, ∞ >. From equation (116) it then follows that the conformal time T is negative for these universe models, and T → 0 when χ is constant and η → ∞. The line element as expressed by the (T, R)-coordinates takes the form #2 " p 2 2 (Ω T − 4 Ω (T − R )) T m0 γ0 ds2M . (123) ds2 = 4(T 2 − R2 )3 24
The Einstein-deSitter universe is a dust dominated, flat universe. It has β = Ωγ = 0. The line element for this universe in the T, R-coordinate system is 2 Ωm0 T 2 2 ds = ds2M . (124) 4(T 2 − R2 )3 The world lines of the reference particles in the cosmic coordinate system, χ = χ0 , in the conformal reference frame are given by (R + a2 )2 − T 2 = a22 ,
(125)
where a2 = (2 χ0 )−1 > 0. The simultaneity curves of the cosmic space, η = η0 , as given in the conformal system, are (T + b2 )2 − R2 = b22 , (126) where b2 = (2 η0 )−1 . These two sets of hyperbolae are drawn for a universe model with matter and radiation in Figure 9. T
6 χ = const @ @ @ @ @ η = const @ @ @ @ R @ @ @ @ @ @ @ @ η = const χ = const Figure 9. Minkowski diagram with reference to the conformal coordinate system for a flat universe model dominated by matter and radiation. The diagram shows world lines χ = constant and simultaneity curves η = constant . For this universe model the Hubble flow is expanding for T > 0 and contracting for T < 0 relative to the conformal system.
The Minkowski diagram in Figure 9 shows that the universe with radiation and dust appears at T = 0 having an infinitely great extension. Then a hole develops expanding with the velocity of light. For an infinitely large conformal time the Hubble flow expands relative to the conformal frame. As T approaches infinity, the conformal clocks are reset to come from minus infinity, and R changes from negative to positive values. The universe then contracts relative to the conformal frame to vanishing extension at T = 0. Hence 25
during the period with positive time the conformal frame contracts relative to the Hubble flow, and during the succeeding period with negative time the conformal frame expands faster than the Hubble flow. The world lines of the reference particles of the conformal coordinate system, R = R0 , in the (η, χ)-system are given by (see Figure 10) (χ + a3 )2 − η 2 = a23 ,
(127)
where a3 = (2R0 )−1 . The simultaneity curves of the conformal space, T = T0 , have the equation (η + b3 )2 − χ2 = b23 , (128) where b3 = (2T0 )−1 .
η
R = const 6
T = const R = const
T = const -
χ
Figure 10. Minkowski diagram with reference to the (η, χ)-system for a flat universe model dominated by matter and radiation. The diagram shows world lines R = constant and simultaneity curves T = constant . In this universe model the conformal system expands relative to the Hubble flow.
The world lines of the reference particles with R = constant in the conformal coordinate system is given by equation (36) which leads to dχ 2ηχ (129) = − tanh θ = 2 2 > 0 . dη
η +χ
R=R1
This means that the (T, R)-system expands relative to the (η, χ)-system. Hence the (η, χ)-system contracts relative to the (T, R)-system with a velocity dR R (130) = tanh θ = 2T 2 2 < 0 dT
T +R
χ=χ1
as shown in Figure 9. In a flat LIVE dominated universe the scale factor is b
a(t) = eHΛ (t−t0 ) , 26
(131)
b Λ is given in equation (90) and is constant. Then the parametric time η is where H b η = − 1 e−HΛ (t−t0 ) ,
bΛ H
(132)
which increases from −∞ to 0 as t increases from −∞ to ∞. The world lines χ = constant and the simultaneity curves η = constant are shown in a Minkowski diagram with reference to the conformal coordinate system for this universe model in Figure 11. The corresponding world lines R = constant and simultaneity curves T = constant are shown in Figure 12. In terms of the parametric time η or of the conformal coordinates R and T , the scale factor may be written 2 2 (133) a(η) = − b1 = T b−R . HΛ η
HΛ T
In this case the line element in conformal coordinates takes the form [20] ds2 = b 21 2 ds2M HΛ T
(134)
as in the negatively curved case. There is no continual creation in such a universe model. The evolution of the flat LIVE dominated universe in the CFS-system is as follows. The universe starts at T = 0 and expands from an initial state with vanishing extension. In this era the conformal space has finite extension, although it expands without limit. Again, as the conformal time T approaches infinity, the clocks are reset to come from minus infinity, and the sign of the radial coordinate is changed. The conformal space then has infinite extension and is contracting. The contraction slows down to a final state at rest at T = 0, in which the conformal space still has infinite extension. T
χ = const 6
η = const
-
R
η = const
χ = const
Figure 11. Minkowski diagram with reference to the conformal coordinate system for a flat universe model dominated by vacuum energy. The diagram shows world lines χ = constant and simultaneity curves η = constant . For this universe model the Hubble flow is expanding for T > 0 and contracting for T < 0 relative to the conformal system.
27
η
6 χ @ @ @ @ T = const @ @ @ @ @ R = const @ T = const R = const
Figure 12. Minkowski diagram with reference to the (η, χ)-system for a flat universe model dominated by vacuum energy. The diagram shows world lines R = constant and simultaneity curves T = constant . In this universe model the conformal system contracts relative to the Hubble flow.
5.2. Particle horizon in flat universe models using conformal time We shall here discuss the particle horizon in flat universe models from the perspective of the conformal coordinates T and R. Let us first consider a universe model dominated by vacuum energy. It extends backwards in time to t → −∞. In such a universe model there is no particle horizon. However, this universe model is not realistic. We cannot expect that the general theory of relativity can give a realistic description of spacetime before the Planck time. Hence we assume that the inflationary era, which may be described classically, starts at the Planck time ηP l . Then there is a particle horizon around an observer at P with coordinates T = T0 > 0 and R = 0. As in section 4.3 the horizon H is defined as the intersection between the past light cone at P , R = T0 − T , and the space at η = ηP l . In the present case this space is represented by the hyperbola R2 = T 2 + 2bP l T where bP l = (2ηP l )−1 . This gives TH =
T02 (T0 + 2bP l )T0 , RH = . 2(T0 + bP l ) 2(T0 + bP l )
(135)
Inserting these expressions into the transformation (117) gives χH = − 1 − 1 = − 1 − ηP l . T0
2bP l
T0
(136)
Since P has coordinates T = T0 and R = 0, the transformation (117) also gives ηP = − 1 . T0
(137)
Hence the equation of the particle horizon for a flat universe model starting at the Planck time is χH = ηP − ηP l . (138)
This has here been deduced by using the spacetime diagram in Figure 11 with respect to the conformal coordinates T and R. Equation (138) is in accordance with equation (10) with ti = tP l . 28
6. Universe models with positive spatial curvature 6.1. Conformal coordinates in positively curved universe models We shall now consider positively curved universe models. Then we can introduce conformal coordinates (T, R) by choosing a = 0, b = 0, c = 1 and d = 0 in equation (33). This gives the generating function f (x) = tan(x/2) . (139) The transformation (28) then takes the form [15,16] T =
sin η , cos η + cos χ
R=
sin χ , cos η + cos χ
(140)
transforming the triangle defined by 0 < χ < π and 0 < η < π − χ onto the first quadrant R > 0, T > 0, and the triangle defined by 0 < χ < π and π + χ < η < 2π onto the e Te) fourth quadrant R > 0, T < 0. We also have a conformal coordinate system (R, given by the same formulae (140), which maps the triangle defined by 0 < χ < π and e < 0. Note that the negative sign of R e can π − χ < η < π + χ onto the left half plane R be removed by changing the coordinates θ and φ corresponding to a reflection through the origin in space. These domains are those needed in universe models with dust and radiation. We will later specify the domains needed for universe models with vacuum energy. The inverse transformation is cot η =
1−(T 2 −R2 ) 2T
,
cot χ =
1+(T 2 −R2 ) 2R
.
(141)
The world lines of the cosmic reference particles, χ = χ0 , as given in the conformal system, are (R + a4 )2 − T 2 = a24 + 1 , (142) where a4 = cot χ0 . The simultaneity curves of the cosmic space, η = η0 , are (T + b4 )2 − R2 = b24 + 1 ,
(143)
where b4 = cot η0 , describing hyperbolae with centra at (−b4 , 0). These curves are shown in Figure 13 in the first quadrant.
29
T
P
6 χ = const
@ η = const @ @ @ @ @ @ R H
Figure 13. The first quadrant of the (T, R)-diagram for universe models with positive spatial curvature with reference to the conformal coordinate system defined in equation (140).
Note that all points on a simultaneity curve come arbitrary close to the R-axix in the limit η0 → 0. Hence, in this coordinate system, the Big Bang occurred everywhere at the moment T = 0. Therefore there is no continual creation of conformal space in this coordinate system. This is different from Big Bang as described with reference to a conformal coordinate system of type I in a negatively curved universe model where the Big Bang occurred along the light cone in Figure 3. The line element now takes the conformally flat form [4,13,15] ds2 =
4 a(η(T, R))2 ds2 , 4T 2 + [1 − (T 2 − R2 )]2 M
(144)
where the scale factor depends upon the matter and energy contents of the universe. For instance in a universe with radiation, the line element is 2 4βT 2 ds = ds2M . (145) 4T 2 + [1 − (T 2 − R2 )]2 From equation (46) we obtain the following expression for the recession velocity V =
2T R 1+(T 2+R2 )
.
(146)
Once more the initial recession velocity vanishes. 6.2. Particle horizon in positively curved universe models with radiation and dust We shall here give a geometrical discussion of the particle horizon in universe models with positive spatial curvature with reference to the conformal coordinate system defined in equation (140). In general, using cosmic coordinates, the particle horizon is given in equation (9). 30
Let us see how this can be deduced by a geometrical consideration based upon Figure 13 of an observer at P . The horizon is the intersection of his backwards light cone with the space defined by η = 0. This intersection is at the point H in Figure 13, and from equation (140) it follows that the horizon is represented by R-axis in the figure. The conformal coordinates of the point H are TH = 0 and RH = T0 . Inserting this into equation (142) of a hyperbola χ = χH gives cot χH =
1 − T02 . 2T0
(147)
We also need to find the η-coordinate of P . The conformal coordinates of P are TP = T0 and RP = 0. Inserting this into equation (143) of a hyperbola η = ηP gives cot ηP =
1 − T02 . 2T0
(148)
Hence χH = ηP in accordance with equation (9). In a positively curved universe with radiation and dust a = α(1 − cos η) + β sin η
(149)
t = α(η − sin η) + β(1 − cos η) ,
(150)
and where α=
Ωm0 2(Ω0 −1)
and β =
r
Ωγ0 Ω0 −1
.
(151)
For this universe model the relationship between the cosmic time and conformal time at R = 0 is 2 (152) t = α arcsin 2T 2 − 2T 2 + β 2T 2 . 1+T
1+T
1+T
The scale factor a(η) increases from zero at η1 = 0 to a maximal value at η2 = π − arctan
β α
,
(153)
and decreases to zero at η3 = 2η2 . In a radiation dominated universe α = 0 which gives η3 = π. In a dust dominated universe β = 0 giving η3 = 2π. In a closed universe the coordinate χ is defined in the interval < 0, π > which means that the whole universe is covered except some points where the coordinates are not uniquely defined. Imagine an observer at χ = 0. At a point of time η he can see objects at χ < χH where χH = η is the radius of his particle horizon. When χH = π the whole universe is inside the horizon. Then the observer can see all of the universe. In a universe existing at η > π there is no longer any particle horizon. In a radiation dominated universe with k = 1 the Big Crunch occurs at η = π. The whole univese can only be seen at this final moment. In a matter dominated universe the Big Crunch occurs at η = 2π. Then the whole universe can be seen at the moment of maximum extension. At this moment the particle horizon vanishes. 31
Consider a free particle with χ = χ0 in a universe with dust. When the parametric time approaches π − χ0 , then T → ∞ and R → ∞. The conformal clocks go increasingly fast relative to the cosmic and the parametric clocks. The relative rate approaches infinity as η → π − χ0 . Hence the spacetime can not be covered by a single conformal coordinate e covering the domains I system. We must use two coordinate systems (T, R) and (Te, R) and II in the (η, χ)-plane as shown in Figure 14. The particle with χ = χ0 enters the domain II at the parametric time π − χ0 , and comes back to domain I at the time π + χ0 . η
6
2π
I π
I
II
@ @ @ I @B @ @ χ0 π
-
χ0
II χ
Figure 14. The domains of the conformal coordinate systems defined in equation (140) for dust dominated universe models. The right hand part of the figure shows the spherical 3-space of the closed universe model, suppressing the angular coordinate θ. The coordinate χ has the value χ = 0 at the top of the figure which we call the North pole, and the value χ = π at the South pole. At χ = χ0 the point B in the left hand part of the figure corresponds to the circle between the regions I and II on the right hand part.
The motion of the particle in the conformal coordinate systems looks very strange. At the conformal time T = 0 the particle starts moving upwards from the R-axis in Figure 15 along a hyperbola in the first quadrant. As the parametric time approches π − χ0 , the conformal time and radius approaches infinity. The further motion of the particle must be described in the other conformal coordinate system. At the parametric time π + χ0 the particle enters the original conformal coordinate system at infinite past and from infinitely far away, arriving at the point of departure at a parametric time 2π corresponding to a conformal time T = 0.
32
Te
T 6
-
(a)
T 6
6
-
R
(b)
-
e R
R
(c)
Figure 15. Minkowski diagram showing the motion of a free particle in a dust dominated universe model with positive spatial curvature, with reference to the conformal coordinate systems defined in equation (140). The world line of the particle in the (η, χ)-system is shown in Figure 9. (a) Imagine an observer following the particle. Initially he observes clocks and meter sticks showing T and R. There is a Big Bang at T = 0. After the Big Bang the observer finds that T and R increase towards infinity. (b) The clocks and meter e They have values increasing from minus infinity. sticks are now replaced by new ones showing Te and R. e At T = 0 the expansion stops, and the universe starts contracting. Eventually Te approaches infinity and e minus infinity, corresponding to a reflection through the origin. (c) Now the clocks and meter sticks are R replaced by the old ones, but adjusted so that the time T shown on the clocks increases from minus infinity and the distance R shown on the meter sticks decreases from infinity. At the final moment, T = 0, there is a Big Crunch. The condition that the particle shall enter the region I again is that χ0 < π − 2 arctan(β/α).
In Figure 16 we have shown a backwards light cone of an observer at the North pole at the parametric time η 0 ∈ < π, 2π >. P is the observation event. The emission of the light is at ηA = 0 , χA = 2π − η 0 . At the parametric time (η 0 − π)/2 the light passes from the region I to the region II, and at time η 0 − π it passes the South pole on the sphere in Figure 9. This looks like a reflection in Figure 16. Then, at the time (η 0 + π)/2 it passes into the region I again, arriving at P at η = η 0 . The corresponding paths in the conformal coordinate systems are shown in Figure 17. The conformal radius of the emission point is sin η 0 RA = − (154) 1+cos η 0
which is positive. If the light had been emitted from A in the opposite direction, it would have moved directly towards the North pole, not around the universe via the South pole. This means that A would have been seen by the observer at the North pole much earlier than at η0 . Hence for π < η0 < 2π, all of the universe could have been seen, and there is no horizon. The closed universe model with radiation and dust has a horizon only for η0 < π. In a dust dominated universe the horizon exists during the expansion era, and vanishes as the universe reaches its maximum size.
33
η
6
2π P
I
@ @ II π @ @ @ @ @ @ @ I @ @ @ A π
-
χ
Figure 16. Past light cone of an event P in the (η, χ)-plane in a closed universe model.
Te
T 6
A
R
T 6
@ @ @ @ @
6
-
e R
P
R
@ @ @ @ @
Figure 17. Past light cone of an event P in the conformal coordinate systems in a closed universe model.
Again η = 0 for t = 0, and there is no continual creation. On the other hand there is continual annihilation of conformal space, as was explained by considering a similar situation in section 4.5. In the region χ3 < χ < π where χ3 = π − 2 arctan(β/α), a e spherical hole where space is not defined develops in the (Te, R)-system as shown in Figure e e 18(a). It first appears at R = 0 at the point of time T = α/β. However, a reference pare ticle with 0 < χ < χ3 escapes the hole in the (Te, R)-system and enters the (T, R)-system as shown in Figure 18(b). All world lines end on the hyperbola given by equation (143) with η0 = η3 , representing the Big Crunch. Hence a person at R = 0 will observe that the Big Crunch approaches and reaches his position at T = −β/α.
34
η = η3
Te
T 6
6
-
χ < χ3
-
e R
R
χ > χ3 η = η3 χ < χ3
(a)
(b)
Figure 18. The final fate of a dust and radiation filled universe with positive spatial curvature. (a) The curve e = 0 where conformal space η = η3 represents the boundary of an expanding spherical hole with center at R disappears. The figure shows world lines of particles with χ = constant. Reference particles with χ3 < χ < π hit the hole, and those with 0 < χ < χ3 avoid it. Note that the Te-axis corresponds to χ = π, while the T -axis corresponds to χ = 0. (b) As Te approaches infinity, the conformal clocks are reset to come from minus e changes from negative to positive values. The conformal space then has a finite volume with infinity, and R decreasing size. Again, the curve η = η3 represents the boundary where space is annihilated. The particles with χ = constant that avoided the hole in (a) come from infinitely far away and vanish when they arrive at this boundary.
6.3. LIVE dominated universe with positive spatial curvature In a universe model with k = 1 dominated by LIVE the scale factor is b Λ t) , a(t) = 1 cosh(H bΛ H
(155)
b Λ is given in equation (90). This is a bouncing universe where t goes from −∞ to ∞, and H model without a Big Bang. For this model the parametric time is b Λ t)) , η = arcsin(tanh(H (156) where −π/2 < η < π/2. The scale factor as expressed in parametric time is a(η) =
1 b Λ cos η H
.
(157)
For a LIVE dominated universe model with positive patial curvature we shall consider three different types of conformal coordinates. The coordinates of the first type are given in equation (140). The line element then takes the form [21] ds2 =
4 ds2M 2 2 2 2 b HΛ [1−(T −R )]
35
.
(158)
From equations (17) and (158) we find the conformal scale factor A(T, R) =
2 sgn(R) b Λ [1−(T 2 −R2 )] H
.
(159)
The sign factor is due to the requirement that A(T, R) > 0, and the fact that the denominator is positive for T 2 − R2 < 1 and negative for T 2 − R2 > 1. From Figure 20 we see that the first condition corresponds to R > 0 and the last condition to R < 0. The parametric time η is given in equation (156) with −∞ < t < ∞ corresponding to −π/2 < η < π/2. The coordinate region I in Figure 19 defined by −π/2 < η < π/2 and 0 < χ < π − |η| in the (η, χ)-system is transformed onto the the region given by R > 0 and T 2 < 1 + R2 in the (T, R)-system as shown in Figure 20a. The coordinate region II defined by −π/2 < η < π/2 and π − |η| < χ < π in the (η, χ)-system is transformed onto e < 0 and Te2 > 1 + R e2 in the (Te, R)-system e the region given by R as shown in Figure 20b. η
6
π 2
I
@ @ II @ @
-
π
χ
II
− π2
Figure 19. The (η, χ)-coordinate regions of a LIVE dominated universe with positive curvature that are transe formed into the (T, R)- and (Te, R)-coordinate regions of Figure 20. T
6
η=
-
η = − π2
π 2
Te
6
-
R
χ = const
η = − π2
η=
(a)
π 2
χ = const
(b) 36
e R
Figure 20. In the CFS space of a LIVE dominated universe with k = 1 there is initially an empty bubble of decreasing size in which space does not exist. New space is created, and at the conformal time T = −1 the bubble vanishes. A particle with χ = χ1 < π/2 comes from the boundary of the bubble at a point of time p T1 = − 1 + R12 . It moves towards the origin, comes instantaneously to rest at T = 0 and then accelerates outwards. As shown in part (a) of the figure a new bubble is created at T = 1 and expands. Conformal space p is annihilated at the boundary of the bubble. The particle hits the bubble at T = 1 + R12 . A particle with e χ = χ2 where π/2 < χ2 < π comes from a different region II in spacetime, covered by the (Te, R)-coordinate system shown in part (b) of the figure. A bubble of conformal space appears at Te = 1 and expands. The particle comes from the boundary of this bubble. It accelerates away from the origin and moves towards infinity. Then it enters the region I covered by the (T, R)-system. In this system it comes from spatial infinity at an infinitely past time, moves to a finite distance from the origin, and then accelerates back to spatial infinity in e the infinite future. Then it enters the region II again, reappearing at infinite past in the (Te, R)-system from e a direction opposite to the one it started its motion. Conformal space is annihilated and vanishes at T = −1. p The particle hits the boundary of space at Te = − 1 + R22 .
We shall describe the evolution of a region where 0 < χ < π/2 in a LIVE dominated universe with positive spatial curvature. Initially there is a hole in a conformal space of infinite extension. At the boundary of the hole, the cosmic time t approaches −∞. The hole decreases in size with continual creation of new space and vacuum energy at the boundary. This is similar to the situation described in section 4.4. At the conformal time T = −1 the hole vanishes. Then at T = 1 the hole reappears, and there is continual annihilation. In this region every reference particle with χ = constant is created at the boundary of the first hole, and is annihilated at the boundary of the second hole. In the region with π/2 < χ < π a conformal space of finite extension, initially created e e = 0, is expanding with continual creation. at T = 1 and R 6.4. New types of conformal coordinates for a LIVE dominated universe with positive spatial curvature
V.F.Mukhanov [8] has introduced a parametric time ηb = η − π/2 where −π < ηb < 0. In this way he obtains the same value ηb = 0 at the end of the universe for all cases k = −1, k = 0 and k = 1. However, our choice (156) is natural if one focuses on the symmetry of the function a(t), i.e. on the symmetry of the expansion history of the universe. V.F.Mukhanov uses the transformation (140) with η replaced by ηb. Expressed by our coordinate η this corresponds to choosing a = π/2, b = 0, c = 1 and d = 0 in equation (33). This defines a second type of conformal coordinates for LIVE dominated universe model with positive spatial curvature with generating function f (x) = tan(x/2 − π/4) =
tan(x/2)−1 tan(x/2)+1
.
(160)
so that transformation (28) takes the form Tb = −
cos η , sin η + cos χ
b= R
sin χ , sin η + cos χ
(161)
transforming the triangle defined by 0 < χ < π and −π/2 < η < χ − π/2 onto the second b < 0, Tb > 0, and the triangle defined by 0 < χ < π and χ − π/2 < η < π/2 quadrant R 37
b > 0, Tb < 0. The inverse transformation is onto the fourth quadrant R tan η =
b2 )−1 (Tb2 −R 2Tb
,
cot χ =
b 2 )+1 (Tb2 −R b 2R
.
(162)
One may wonder if it is possible to define a new type of conformal coordinates by choosing a = −π/2, b = 0, c = 1 and d = 0 in equation (33). Then the generating function would be 1+tan(x/2) . (163) f (x) = tan(x/2 + π/4) = 1−tan(x/2)
so that the transformation (28) would take the form Te =
cos η , cos χ − sin η
e= R
sin χ , cos χ − sin η
(164)
Replacing χ by π − χ and R by −R, this transformation has the same form as in (140). Changing the sign of R corresponds to replacing the coordinate θ by π − θ and φ by φ + π. This is an antipodal transformation, i.e. a reflection through the origin in S 3 . Hence nothing new is obtained physically. The CFS coordinate Tb has the value zero at the beginning and the end of the universe. Again we see that the CFS time coordinate behaves strangely. In the limit that the CFS time approaches infinity, it is reset to come from minus infinity as shown in Figure 21. In order to obtain a correct physical interpretation of this figure, one must take account of the difference between coordinate distances and physical distances in the radial direction, dl = A(R, T )dR. From equations (29), (157), (4) and (161) the scale factor takes the form sgnR . (165) A(T, R) = − b HΛ T
where we have omitted the hat and the tilde on T and R. Looking at Figure 21, we have a coordinate description of particles with constant χ b in the (Tb, R)-system. Tb
6
-
χ = const
38
b R
Figure 21. World lines of reference particles with χ = constant (defining the Hubble flow) in a LIVE dominated universe as observed from the CFS system defined by the transformation (161).
It may be noted that the transformation equation (161) can also be used for universe models with k = 1 containing dust and radiation, but with different coordinate domains. 7. New transformations obtained by composition of generating functions By applying the rule in Appendix B for composing generating functions one may obtain a deeper understanding of the conformal coordinate transformations and find new ones. Let us first consider the transformations to CFS coordinates of type III in universe models with negative spatial curvature. Here the generating function in (111) is obtained as a composition of the generating functions in (99) and (115). Combining the generating functions in (139) and (115) we obtain the generating function f (x) = − cot(x/2) . (166) and the transformation T =
sin η , cos η − cos χ
R=
sin χ , cos χ − cos η
(167)
In the same way as in section 6.4 this only represents a composition of the antipodal transformation with the transformation (139), and therefore nothing new is obtained physically. We have seen in section 3 that it is possible to transform away spatial curvature. In the same way we obtain a more general transformation from spaces with curvature k1 to spaces with curvature k2 if the generating function f satisfies the relation ′
′
f (u)f (v) Sk1
u−v 2
2
= S k2
f (u)−f (v) 2
2
.
(168)
Composition of generating functions can also be used to find coordinates used in Penrose diagrams, as we will investigate in the next section. 7.1. Penrose diagrams Our formalism can be utilized to give simple deductions of the transformations that lead to the Penrose diagrams. Then equation (29) for the CFS scale factor is modified to give a corresponding conformal Einstein space (CES) scale factor, given as A(T, R) =
a(η(T,R)) Sk (χ(T,R)) | sin R |
.
(169)
By means of the composition rule in Appendix B we shall first find a transformation from spaces with negative curvature to spaces with positive curvature, corresponding to 39
Einstein’s static universe. By means of a composition of the generating function in (52) and the inverse of the generating function in (139), we obtain the generating function f (x) = − 1 e−x .
(170)
Ti
This represents a transformation from a space with k = −1 and CFS coordinates of type I to a space with k = 1 and CFS coordinates of type I. Equation (28) then implies that the transformed coordinates T and R are given by tan T =
2Ti eη cosh χ 1−Ti2 e2η
,
tan R =
2Ti eη sinh χ 1+Ti2e2η
.
(171)
The inverse transformation is η
Ti e =
r
cos R−cos T cos R+cos T
,
tanh χ = sin R .
(172)
sin T
Using equation (169) and the transformation (179) we find that the line element (2) with k = −1 as expressed in CES coordinates takes the form ds2 =
a(η(T,R))2 cos2 R−cos2 T
(−dT 2 + dR2 + sin2 R dΩ2 ) =
a(η(T,R))2 cos2 R−cos2 T
ds2E .
(173)
where ds2E is the line element of Einstein’s static universe. In the case of the Milne universe the transformation to the CES coordinates reduces to t sinh χ t cosh χ , tan R = . (174) tan T = 2 2 1−t
1+t
transforming the first quadrant t > 0 and χ > 0 onto the triangle 0 < R < π/2 and R < T < π − R, with inverse transformation r t = cos R−cos T , tanh χ = sin R . (175) cos R+cos T
sin T
The CES form of the line element for the Milne universe is ds2 =
1 (cos R+cos T )2
ds2E .
(176)
Taking a composition of the generating function in (99) and the inverse of the generating function in (139), we obtain the generating function f (x) = 2 arctan(tanh(x/2)) .
(177)
This represents a transformation from a space with k = −1 and CFS coordinates of type II to a space with k = 1 and CFS coordinates of type I. Equation (28) then implies that the transformed coordinates T and R are given by tan T =
sinh η cosh χ
, 40
tan R =
sinh χ cosh η
.
(178)
If one transforms from a universe model with dust and radiation which begins at η = 0, this transformation maps the first quadrant χ > 0, η > 0 onto the triangle 0 < Tb < π/2 b < π/2 − Tb. On the other hand, if one transforms from a universe which is and 0 < R vacuum dominated and infinitely old, the fourth quadrant χ > 0, η < 0 is mapped onto b < π/2 + Tb. The inverse transformation is the triangle −π/2 < Tb < 0 and 0 < R tanh η = sin T
cos R
tanh χ = sin R .
,
(179)
cos T
Using equation (169) and the transformation (179) we find that the line element (2) for a universe model with k = −1 as expressed in CES coordinates takes the form ds2 =
a(η(T,R))2 cos2 T −sin2 R
(−dT 2 + dR2 + sin2 R dΩ2 ) =
a(η(T,R))2 cos2 T −sin2 R
ds2E .
(180)
We shall next find a transformation from flat spaces to spaces with positive curvature, corresponding to Einstein’s static universe. Taking a composition of the generating function f (x) = x for the first type of CFS coordinates in flat space, and the inverse of the generating function in (139), we obtain the generating function f (x) = 2 arctan(x) ,
(181)
which is the inverse of the generating function given in formula (139). Hence the coordinate transformation is the inverse of the one given in equation (140), cot T =
1−(η 2 −χ2 ) 2η
,
cot R =
,
χ=
1+(η 2 −χ2 ) 2χ
.
(182)
with inverse transformation η=
sin T cos T +cos R
sin R cos T +cos R
(183)
Again using (169), this time in combination with the transformation (183), we find the line element for flat space in CES coordinates ds2 =
a(η(T,R))2 (cos T +cos R)2
(−dT 2 + dR2 + sin2 R dΩ2 ) =
a(η(T,R))2 (cos T +cos R)2
ds2E .
(184)
Note that the line element (176) can be immediately obtained by putting a = 1 in equation (184). This is a consequence of the fact that the Milne universe is a coordinate transformation of the interior of the future light cone of the point (0, 0) in the static and spatially flat Minkowski spacetime. As a simple example we shall find the CES coordinates for the Minkowski spacetime with line element ds2 = −dt2 + dr 2 + r 2 dΩ2 . (185)
From equation (169) and the transformation (183) it follows that the CES scale factor for the Minkowski spacetime is [22] A(T, R) =
1 cos T +cos R
41
.
(186)
It may be noted that in the case of universe models with positive spatial curvature, η and χ are CES coordinates. For a deSitter universe with k = 1 one obtains by using equation (157), the CES form of the line element ds2 = correcting an error in reference [23].
1 ds2 b 2 cos2 η E H Λ
,
(187)
8. Some physical properties of the CFS system The parametric time η is widely used in cosmology because many formulae become simpler using this than when expressed in termes of cosmic time t. Important examples are the formulae (9) and (10) for the radius of the particle horizon and the expressions (71), (72) and (149), (150) for the cosmic scale factor in a curved universe with radiation and dust. One may wonder if there are any similar advantages in using the CFS coordinates T and R. According to G.Endean [10-12] there are several. He argues that the picture of the universe obtained with reference to the CFS coordinates is more correct and relevant when interpreting observational data than the usual picture referring to cosmic time t (or the parametric time η) and the standard radial coordinate χ. The criterion for a Big Bang is that A(0, R) = 0. This will be the case if a converges to 0 faster than B when T → 0. Endean claims for example that there was no Big Bang because the recession velocity vanishes at t = 0 (see equations (104) and (119)). The recession velocity is the velocity of a galaxy with χ = constant relative to the CFS system. We prefer the following interpretation : The mentioned expressions tell that the reference particles of the CFS system are initially at rest relative to the galaxes. Then the CFS particles accelerate away from the galaxes, which in no way implies that there was no Big Bang. Endean also claims that in a dust dominated universe with k = 1 there will not be a Big Crunch. The reason for this is the following. At χ = 0 the coordinate transformation (140) gives η T = tan . (188) 2
Furthermore, the cosmic scale factor a = α(1 − cos η) has a maximum value amax = 2α for η = π. This corresponds to a finite cosmic time t1 = πα, but to an infinite CFS time. Hence, in the CFS picture the closed dust dominated universe expands for an infinitely long time, and Big Crunch never happens. However, the CFS clocks are not standard clocks like the cosmic clocks showing t. They are coordinate clocks going at the rate that is adjusted to be in accordance with the first of equations (140). Compared to the standard clocks and clocks showing parametric time, the CFS clocks go increasingly fast. Hence the non-vanishing of a Big Crunch in the CFS picture is a coordinate effect. Further criticism of Endean’s interpretation of the CFS picture has been given by L.Querella [24]. Let us consider a few examples in which the conformal time at χ = 0 can be very simply related to the cosmic time, namely radiation dominated universes with negative,
42
vanishing and positive spatial curvature. With reference to the CFS systems given respectively in the transformations (93), (112) and (140) we find for clocks at χ = 0, T = t .
(189)
p 2βt − kt2 .
(190)
a
In a radiation dominated universe [2,3] a= Hence
T =
r
t 2β−kt
.
(191)
8.1. The CFS Hubble parameters of some universe models The behaviour of the conformal space in different universe models may be investigated by calculating the CFS Hubble parameter defined in equation (48). We first consider negatively curved universe models with respectively dust and radiation as described in CFS systems of type I. Using equation (78) we obtain 2Ti T HR = √ ( T 2 − R2 − Ti )3
(192)
for a dust dominated universe, and by means of equation (79) we get 2Ti2T (T 2 − R2 − Ti2 )2
HR =
(193)
for a radiation dominated universe. As shown in section 4.4 there is continual creation at the boundary of the conformal space given by T 2 − R2 = Ti2 . Hence the CFS Hubble parameter for both of these universe models approaches infinity at the boundary with continual creation. The CFS Hubble parameter for a negatively curved radiation dominated universe with conformal coordinates of type II and III is HR = sgn(T )
b2 )2 −4Tb2 (Tb2 −R b2 ) (1+Tb2−R 2 4β Tb
(194)
Correspondingly for a flat universe one obtains HR =
|T 2 −R2 | (3T 2 +R2 ) βT 2
(195)
For a universe with positive spatial curvature it is HR = sgn(T )
b2 )]2 −4Tb2 (Tb2 −R b2 ) [1−(Tb2−R 4β Tb2
(196)
The Hubble parameter for a LIVE dominated universe model with k = −1 and CFS coordinates of type I is p b0 T HR = Ω0 H (197) Tf
43
In this case the conformal Hubble parameter in independent of the position R. The Hubble parameter for the same universe model but with CFS coordinates of type II and III is b Λ sgn(T ) HR = −H (198)
where we have omitted the hat and the tilde on T . This formula is also valid for a flat LIVE dominated universe. The conformal space therefore expands for T < 0 and contracts for T > 0. This behaviour is a result of two competing motions. The Hubble flow expands exponetially. But as seen in Figure 12, the conformal reference frame contracts relative to the Hubble flow. The sign of conformal Hubble parameter shows that the expansion dominates for T < 0 corresponding to the region −χ < η < 0 in the (η, χ)-plane, while the contaction of the conformal system dominates for T > 0 corresponding to the region 0 < χ < −η. From equations (48) and (159) it follows that the CFS Hubble parameter for a LIVE dominated universe model with positive spatial curvature in CFS coordinates of type I is bΛT . HR = sgn(R) H
(199)
bΛ . HR = sgn(R) H
(200)
The corresponding Hubble parameter in CFS coordinates of type II and III is
8.2. The CFS age of some universe models We shall now calculate the conformal age of some universe models containing radiation. They are of course not realistic universe models, but we shall calculate their conformal age as an illustration. Consider first a negatively curved universe model. The parametric age η0 is determined from the normalization a(η0 ) = 1 in equation (71), which gives sinh η0 = 1/β .
(201)
The cosmic age is given in equation (72) with α = 0. Together with equation (201) this leads to p t0 = 1 + β 2 − β , (202)
where β is given in equation (75) and has the value 9.1 · 10−3 for a radiation density equal to that of the background radiation at the present time. We shall now calculate the present value of the time T0 in a CFS system of type I. It is given by equation (54) which shows that T0 is R-dependent and given by q p (203) T0 = R2 + (1/4) (1 + 1 + β 2 )2 ,
The initial time of the universe model is p Ti = R2 + (1/4) β 2 . 44
(204)
According to equation (1) the unit of time is p l0 /c = (H0 1 − Ωγ0 )−1 = 13.9 · 109 y ,
(205)
where we have used the value of H0 in ref.[25], and that Ωγ0 = 6 · 10−5 . At R = 0 the dimensional initial time is l β (206) (l0 /c) Ti = 0 = 63 · 106 y . 2c
The CFS age of the universe is T0 − Ti , where T0 and Ti are given in equations (203) and (204). Hence the conformal age at R = 0 of this radiation dominated universe model is p l l0 (T0 − Ti ) = 0 {(1 + 1 + β 2 ) − β} = 13.8 · 109 y . (207) c
2c
Using equations (203) and (72) we find that the conformal age and the cosmic age at R = 0 are related by (208) T0 − Ti = 1 (1 + t0 ) . 2
Consider next a negatively curved universe model with CFS coordinates of type II. The CFS age of this universe is q p 2 b2 + β 2 , b (209) T0 = 1 + β − R b < 1 − Tb0 . The conformal age and the cosmic age at R b=0 where 0 < Tb0 < 1 and 0 < R are related by t0 for 0 < t0 < 1 Tb0 = . (210) 1 for t0 > 1 t0
We then consider a flat, radiation dominated universe. The cosmic age of this universe
is
t0 = 1 , 2β
(211)
where β is given in equation (121). As described in a CFS coordinate system given in equation (116), there are some strange coordinate effects. In the CFS coordinate system the universe seems to be divided into two parts. The part U1 outside the future light cone of (η, χ) = (0, 0) appears in the conformal coordinate system with a positive time, coming from Tb = 0 and approaching infinity. The part U2 inside the light cone appears with a b = negative conformal time, coming from minus infinity and converging towards (Tb, R) (0, 0). The conformal age of U1 is p T0 = 1 ( R2 + β 2 − β) . (212) 2
For the other part U2 one can only define the time left until a Big Crunch of the conformal space, which is given by p Tf − T0 = 1 ( R2 + β 2 + β) − R . (213) 2
45
We finally consider a positively curved, radiation dominated universe. Its cosmic age is t0 = β −
p β2 − 1 ,
(214)
where β is given in equation (151). The radiation dominated universe is represented by the region η < π in Figure 14. This region has two parts I and II. In part I the conformal time is positive, and the age is p p T0 = R2 + β 2 − β 2 − 1 . (215)
On the other hand, in part II the conformal time is negative, and the conformal space disappears at Te = 0. In this case the conformal time left until the space disappears is p p (216) − T0 = R2 + β 2 + β 2 − 1 . 8.3. Recession velocity and cosmic redshift in CFS space We have considered several types of conformally flat spacetime coordinate systems. One series has a = b = d = 0 in equation (33). Then the generating function reduces to f (x) = c/Ik (x/2) ,
(217)
and the general expression (46) for the recession velocity takes the form V =
2kRT c2 +k (R2 +T 2 )
.
(218)
The corresponding formula for the Doppler shift factor given in equation (50) is D(T, R) =
1/2
c2 +k (T +R )2 c2 +k (T −R )2
.
(219)
For k = 0 equation (50) gives D(T, R) =
b(T +R−d )−c b(T −R−d )−c
.
(220)
In the case of the Einstein-deSitter universe, i.e. a flat dust dominated universe, the constants in equation (220) have the values b = 2, c = 2 and d = −1, giving D(T, R) =
T0 2T −T0
.
(221)
Using this together with equation (124) we find for the CFS redshift z, 1+z =
2T −T0 T
46
2
.
(222)
The corresponding expression for the redshift i a flat LIVE dominated universe, i.e. the deSitter universe is 1+z = T . (223) 2T −T0
Using equation (51) we find that the cosmic redshift z in a flat universe with f (x) = −1/x is given by A(T0 ,0) , R = T0 − T 1 + z = T +R T −R A(T,R)
where T is the emission time and T0 the time of observation. The redshift z in a universe with k = −1, f (x) = tanh(x/2) r 1+z =
1−(T +R)2 A(T0 ,0) 1−(T −R)2 A(T,R)
.
and in a universe with k = 1, f (x) = tan(x/2) are given by the expression r 1+z =
1+(T +R)2 A(T0 ,0) 1+(T −R)2 A(T,R)
.
(224)
(225)
(226)
9. Conclusion The question ”what happened before the Big Bang?”, which may be asked in the (t, χ)- or the (η, χ)-system, is translated to ”what happened beyond the Big Bang ?” in the (T, R)system. If there was a quantum chaos before the Big Bang in the (η, χ)-system, there was a quantum chaos outside the boundary of the conformal space in the (T, R)-system. In the (η, χ)-system, time and space are created at the Big Bang singularity at t = 0. In the (T, R)-system, new space is also created at later times. Big Bang happens continually at the boundary of space, i.e. on the hyperbola T 2 − R2 = Ti2 in Figure 1. Just as our universe is said not to exist before t = η = 0, our universe did not exist below this hyperbola. Or, alternatively if cosmic time and space was created at the Planck time, conformal space is continually created at the Planck boundary. What then about continual creation of new space? Is this only a coordinate effect? In a way it is. As made clear in the introduction conformal space is a coordinate space. But cosmic space is also a coordinate space. The first one is defined by T = constant and the second one by t = constant. Is the one more fundamental than the other? A difference between the cosmic and the conformal space is that the reference particles of the first one are freely moving, and those of the second ones must be acted upon by non-gravitational forces. The reference particles of the cosmic space constitute an inertial flow, while those of the conformal space do not. In this sense the cosmic space is more fundamental than the conformal space. Continual creation of conformal space, matter and energy is physical and real. But it belongs to a CFS picture of the universe which does not have the same physical significance as the picture of our universe based upon cosmic time and freely moving reference particles, because these particles which define the Hubble flow, are in fact the clusters of galaxes. In a flat universe model the CFS time coordinate in the series with a = b = d = 0 47
in equation (33) is the parametric time η, and the recession velocity vanishes. This does not mean that the universe model is not expanding. The expansion of the universe model is described by the scale factor. A universe model is said to expand if the scale factor associated with the cosmic coordinate system. i.e. the (η, χ)-system, is an increasing function of time. The reference particles with χ = constant define the Hubbble flow. Hence the vanishing of the recession velocity in the CSF system with parametric time, in the cosmic coordinate system means that the recession velocity represents an expansion or a contraction of the CFS system relative to the Hubble flow. Appendix A. Proof that f given in equation (33) satisfies equation (27) We first introduce the function cos x for k = 1 1 for k = 0 Ck (x) = Ik (x)Sk (x) = cosh x for k = −1
(227)
so that Sk′ (x) = Ck (x). This function also satisfies the relation
Furthermore
Sk (u − v) = Sk (u)Ck (v) − Ck (u)Sk (v) .
(228)
h i−2 f ′ (x) = c b Sk x−a + Ck x−a .
(229)
2
2
2
Inserting the function f in equation (27) leads to [f (u)−f (v)]2 4f ′ (u)f ′ (v)
h i i 2 h u−a −1 v−a −1 b + Ik = − b + Ik 2
2
i2 h i2 2 2 h b + Ik v−a Sk u−a Sk v−a b + Ik u−a 2
2
2
2
h i2 2 2 v−a u−a u−a v−a = Ik − Ik Sk Sk . 2
2
2
(230)
2
Using the relation (228) we obtain [f (u)−f (v)]2 4f ′ (u)f ′ (v)
i2 2 h = Sk u−a Ck v−a − Ck u−a Sk v−a = Sk u−v 2
2
2
2
2
(231)
which was to be shown. Appendix B. Composition of generating functions We shall now prove the following rule of composition for generating functions. Let f be a generating function between spaces with curvatures k1 and k2 , and g a generating function between spaces with curvatures k2 and k3 , both satisfying equation (168). Then 48
the composition h of f and g, h(x) = f (g(x)), is a generating function between spaces with curvatures k1 and k3 . The rule is proved by utilizing the chain rule of differentiation. Sk 3
h(u)−h(v) 2
2
= Sk 3
g(f (u))−g(f (v)) 2
2
′
′
f (u)−f (v) 2
= g (f (u)) g (f (v)) Sk2 2 2 = g ′ (f (u)) g ′ (f (v)) f ′ (u) f ′ (v) Sk1 u−v = h′ (u) h′ (v) Sk1 u−v . 2
2
2
(232)
We also have the following rule for the inverse of a generating function. Let f be an invertible generating function between spaces with curvatures k1 and k2 satisfying equation (168). Then the inverse funtion g = f −1 is a generating function between spaces with curvatures k2 and k1 . For if U = f (u) and V = f (v), then
2 f (u)−f (v) 2 1 S = Sk1 u−v = ′1 2 f (u) f ′ (v) k2 2 2 . = g ′ (U) g ′ (V ) Sk2 U −V Sk 1
g(U )−g(V ) 2
2
2
(233)
Appendix C. A general formula for the recession velocity The recession velocity is given in equation (36) which may be written f ′ (f −1 (T +R))−f ′ (f −1 (T −R)) = ′ −1 V = dR ′ −1 dT
χ=constant
f (f
(T +R))+f (f
(T −R))
(234)
Hence we need to calculate the function y = f ′ (f −1 (x)) using the expression (33). This gives f −1 (x) = 2 Ik−1 c − b + a (235) x−d
where
arccot x for k = 1 −1 1/x for k = 0 Ik (x) = arccoth x for k = −1
In order to combine this with formula (229) for f ′ , we √ 1 for x2 +1 Sk (Ik−1 (x)) = 1/x for √1 for x2 −1
(236)
need the following formulae k=1 k=0 k = −1
(237)
and
Ck (Ik−1 (x)) = x Sk (Ik−1 (x)) In the arguments of Sk and Ck of the formula (229) we use that f −1 (x)−a c = Ik−1 −b . 2
x−d
49
(238)
(239)
Combining this with equation (239) we get f ′ (f −1 (x)) = c
2
c x−d
−2
−2 Sk Ik−1 c − b . x−d
(240)
By means of equation (237) we finally obtain
f ′ (f −1 (x)) = 1 [{b(x − d) − c}2 + k(x − d)2 ] 2c
(241)
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