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8/8/2019 Notas Cosmologa

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T O P I C S O F R E L A T I V I S T I C C O S M O L O G Y

N i k o l a i V . M I T S K I E V I C H , G u a d a l a j a r a - N e w D e l h i

1 The general form of FRW metric estab-

lished from the properties of spatial homo-

geneity and isotropy

First we deduce the most general form of the three-dimensional squared in-terval from the postulates of homogeneity and isotropy of the universe takenat fixed cosmological time. The isotropy means, in terms of 3-curvature, thatnot only spherical symmetry is applied, but all three existing in this case di-agonal components of the Ricci tensor have the same values. Moreover, thehomogeneity demands that this unique value should be simply a constantin all three-dimensional space. Thus the squared three-dimensional interval(with the Euclidean signature + + +) has to be

dl2 = e2dr2 + r2(d2 + sin2 d2). (1.1)

From (1.1) immediately follow all three (in this case) connection 1-forms

(1)(2) = e

r(2), (1)(3) = e

r(3), (2)(3) = cot

r(3) (1.2)

as well as the Riemann curvature components

R(1)(2)(1)(2) = R(1)

(3)(1)(3) = e2

r, R(2)(3)(2)(3) =

1

r2

1 e2 (1.3)which yield the Ricci tensor components

R(1)(1) =

2e2

r, R(2)(2) = R(3)(3) =

e2

r 1

r21 e

2 . (1.4)Since these components should be mutually equal and constant,

e2

r=

1

r2

1 e2 = C, (1.5)

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from where e2 = 1 + Cr2. This means that

dl2 =dr2

1 + Cr2+ r2(d2 + sin2 d2). (1.6)

Square root of the first term in this expression can be written as dr1+Cr2

= d,

while r =sinh(

C)

C, so that (1.6) is equivalent to

dl2 = d2 +

sinh

C

C

2

(d2 + sin2 d2). (1.7)

In this expression only three values of C are used: 1, 0, and 1, then giving

r =

= sinh when C = +1,= when C = 0,= sin when C = 1.

(1.8)

They correspond respectively to the open three-dimensional (hyperbolical)space, open plane three-space, and to closed (hyperspherical) three-space. Itis clear that the new variable is dimensionless, thus the interval (1.7) shouldbe multiplied by a factor having dimensionality of squared length. This factorwill be of great importance in cosmology, moreover, it will be not constant,but depend on time thus directly describing the expansion of universe (itsscale factor). This expansion will not contradict to the spatial homogeneityof the universe since the universe will expand as a whole, with the same rateat all its points, naturally taken at one and the same time. This time willbe a privileged one, the time T of the synchronous system of coordinates,alternatively the cosmological time differing from the synchronous one onlyinsignificantly. It is also worth mentioning that in the total ds2 only the partdl2 will contain functions of spatial coordinates, but it will not change its

form under transformations adapted to the spatial homogeneity of dl2, withcoming to a new origin of three spatial coordinates at any new point (the

form-invariance).Thus we now come to the second step, introducing the squared four-

dimensional interval, Einsteins equations for it, and leaving the considerationof perfect fluid to the next section. This will be, in fact, a discussion of theenergy-momentum tensor of matter which fills the universe (supposed to bea perfect fluid as averaged description of its material contents).

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Using the spacetime signature (+,

,

,

), we write down the 4-interval

in the synchronous coordinates (and another, slightly different, system) asds2 = dT2 a2dl2 = a2(dt2 dl2). (1.9)

Here a(T) is the aforementioned scale factor depending on time T (but not onspatial coordinates!); further on the right-hand side of (1.9) the same intervalis written in terms of the so-called cosmological time t, while the scale factoris now supposed to depend on this t: dT = a(t)dt mutually relates thesetwo time variables and the scale factor. The latter will be determined viasolving Einsteins equations when we shall sufficiently know their right-handside, the sources of gravitational field. Prior to considering the descriptionof perfect fluids, we shall deduce now the left-hand side parts of Einsteinsequations for the squared interval (1.9).

The tetrad covector basis is

(0) = adt, (1) = ad,

(2) = aC

sinh(

C)d, (3) = aC

sinh(

C)sin d.

(1.10)

The differentials of these vectors are not too complicated:

d(0) = 0, d(1) = aa2

(0) (1),d(2) = a

a2(0) (2) +

Ca

coth(

C)(1) (2),d(3) = a

a2(0) (3) +

Ca

coth(

C)(1) (3) +Ccot

a sinh(C)

(2) (3).

(1.11)From them the connection 1-forms follow,

(0)(i) =aa2

(i), (1)() = Ca

coth(

C)(),

(2)(3) = Ccot

a sinh(C)

(3)

(1.12)

(here, as usual, the Latin index i takes values from 1 to 3, and additionallywe employed the capital Greek letter as another index to take values 2 and3. Farther calculations yield the following expressions for components of thecurvature:

R(0)(1)(0)(1) = R(0)

(2)(0)(2) = R(0)

(3)(0)(3) =1a2

aa

.,

R(1)(2)(1)(2) = R(1)(3)(1)(3) = R(2)(3)(2)(3) =

aa22 Ca2 . , (1.13)

so that the Ricci tensor components read

R(0)(0) =3

a2

a

a

., R(1)(1) = R(2)(2) = R(3)(3) = 1

a2

a

a

.2

a

a2

2 C

a2

(1.14)

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and the scalar curvature,

R =6

a2

a

a

.+

a

a

2 C

. (1.15)

Hence Einsteins conservative tensor (the left-hand side of Einsteins equa-tions) has components

G(0)(0) = 3a2

aa

2 C ,G(1)(1) = G(2)(2) = G(3)(3) =

1a2

2

aa

.

+

aa

2 C

(1.16)

from which it is obvious that the admissible source of the FRW gravita-tional field on the right-hand side of Einsteins equations can be the energy-momentum tensor of a perfect fluid only (having arbitrary state equation).Thus we now turn to consideration of perfect fluids and their properties.

2 Perfect fluids and reference frames

It is clear that we have here not to delve into the nature of perfect fluids (themain ideas and possible restrictions in the statistical deduction of their prop-erties) though, of course, such considerations might be of interest concerningthe details of this description of the averaged picture of galaxies and radi-ation in the universe at its different evolution stages. In approaches to theFriedmannRobertsonWalker cosmological models dominates the viewpointthat the present state of universe should be well described by a non-coherentdust (perfect fluid without pressure, p = 0), but at much earlier stages therehad to exist more or less large admixture of non-coherent (chaotic) radiation.I do not know if such mixed systems were sufficiently deep investigated, andif it was really proved that they could be well described by the same theoryof perfect fluids, merely with the use of some averaged state equations. Weshall apply here the perfect fluids theory rather formally and taking into ac-

count only simplest equations of state. However one has to keep in mind thatif the universe is really built mainly of exotic materials such as dark energyand dark matter, the situation could occur to be much different from that weintend to consider below following the standard cosmological models. Thereexists also other approach to fluids, not less formal, but much more flexible,in which I am seriously interested [in particular, because I am its author, see

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Mitskievich (1999a, 1999b)]; this is the field-theoretical description of perfect

fluids, but it will not be touched here in any detail since to this end a toolarge additional material should be included in these lectures.

Perfect fluids (also called Pascals fluids) are those which are describedonly by their mass distribution (mass density) and pressure (which is isotro-pic); no viscosity nor thermodynamics are involved. Their theory is extremelysimple, it concentrates around the fluids energy-momentum (perhaps, in thiscase it would be better to say: stress-energy) tensor

T = ( + p)u upg. (2.1)

Here is the mass (energy) density, p is pressure, u, the fluids local four-velocity, and g, the metric tensor. It is clear that we have written thistensor in the abstract completely invariant manner which, however, does notmean that any of the characteristics we used are invariants in all-embracingsense: the four-velocity u is a vector (or covector), and it is only given in theinvariant representation (let us not identify the concepts of a vector and itscomponents, this would be nothing more than to follow a vulgar habit). Themass density and pressure are given here only in the proper sense, i.e., asthe proper mass density and proper pressure, which are nearer to invariantin its ordinary meaning, but should be by no means understood as propertieswhose measured values are independent of the state of motion of the observer

with respect to fluid (or vice versa).In fact, here we, for the first time, encounter a specific problem: how

to separate a general description from its further application to particularcases without destroying their very important unity (mutual succession)? Itis somewhat like the Zeno paradox, in our time existing more psychologicallythan as a real obstacle. The problem consists not in any objective contra-diction, but in a conflict between our more or less vulgar professional habitsand adequate understanding of the concepts we are operating with. The an-swer will automatically come from a deduction and physical interpretation ofone important example: how change the directly observable characteristics

of a perfect fluid with a change of the reference frame in which the fluid isconsidered? To this end we have to discuss not only the subject of perfectfluids, but also that of the reference frames.

The concept of reference frame does not need for its definition neither usenor even introduction of coordinates. When we measure some characteristicsof concrete objects or phenomena occurring in or between them, this can

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be done without coordinates which are completely artificial inventions, while

in such measurements both observer and object (or objects) have to reallyexist. Our old and often misleading habit is to base our definition of referenceframe on coordinates. Moreover, we mix these concepts together, we mistakecoordinates for reference frames and reference frames for coordinates, we treatcomponents of a vector when it is measured, as really observable quantities,and this is completely wrong. Only in very particular cases and very specialcircumstances this may partially work, but not in the curved spacetime, inany case. Well, what is a reference frame? Take the simplest idealization ofan object or an observer, a three-dimensional point. To consider its motionwe have to introduce the fourth dimension, the time, and in spacetime we

now speak about such an object as about a world line. A reference frame ingeneral is an idealization of a swarm, a cloud of observers and their measuringdevices, all being only test objects to create no disturbances in phenomenaunder measurement and scientific investigation in general. The basic ideas inthis formalism of reference frames were independently discovered by J. Ehlersin Germany and A. Zelmanov in the Soviet Union. It is remarkable that theyboth came to these ideas from cosmology and the use there of perfect fluids.

Thus we have to work with a congruence of timelike world lines which isequivalent to a timelike vector field. Since we do not build anything morethan the concept of a reference frame, we take as already existent the metrictensor in order to calculate the square of this vector and finally to normalizeit to unity. Thus we admit a unit vector field (say, ) as a definition ofreference frame.1 A combination of it and the metric tensor gives a newtensor perpendicular to :

b := g (2.2)which simultaneously is the projector on the local subspace (globally, forthe field of, to this field, but now in general not forming a global subspace;such a subspace is formed only in the case of non-rotating congruence) andthe three-dimensional metric tensor (with the signature 0,,,) on thissubspace. 0 in this signature means that b is in fact a three-dimensional

object restricted to the local subspace , though it is formally describedas a four-dimensional one: for example, if we introduce some system of co-ordinates, the components of both and b will be denoted using thefour-dimensional indices, but the determinant of this b will identically vanish

1Since =

= ()X() = etc, the symbolic representation of vectors (and

tensors in general) surely is coordinate- and tetrad-independent, thus invariant.

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(det b

0). A combination of (2.1) and (2.2) gives

T = u upw (2.3)where we have taken w = g u u in analogy to (2.2).

At the same time, the term in (2.2) is another projector, now on (more frequently a simple scalar multiplication by , with a contractionwith the index in whose sense the projection is to be obtained, is used assuch a second projector). This is the projector on the local physical timeof the reference frame described by . Thus g = b + . Let us doa substitution of this expression in ds2 (which can be symbolically writtenwithout using coordinates as ds2 = g(dx, dx) where dx = dx, this time

dx

representing the components of the infinitesimal vector between pointsx and x + dx taken with respect to the vector basis , while the above-written expression with parentheses (dx, dx) means scalar product with therank 2 tensor g in the sense of both its indices; we shall sometimes use suchnotations, also with one of the valences ofg non-occupied). This substitutionyields ds2 = dl2 + dt2 (dl2 := b(dx, dx) and dt := dx, both differentialsnot being total ones, similar to ds; minus in the first expression correspondsto the signature of b). In particular, for propagation of light (ds2 = 0) thisgives

dldt

= v = 1 in every reference frames (including non-inertial ones).Considering the four-velocity u = dx

ds of some object, we can easily find

its three-dimensional velocity in any reference frame as

v =b(dx, )

dt u = dt

ds( + v) (2.4)

where dt is non-total differential of the physical time (taken with respectto the reference frame ) introduced above. In fact, this was a special caseinvolving projection; let us, more generally, consider now an arbitrary four-(co)vector q; its projection onto the monad is a (frame-non-invariant) scalar,

()q := q , (2.5)

thusdtds

=()u , (2.6)

and the projection of q onto the three-space of the reference frame is a (for-mally) four-dimensional (co)vector,

(3)q := b(q, ), (2.7)

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which is by a definition orthogonal to the monad (therefore(3)q is in a certain

sense three-dimensional) and not changes by a repeated b-projection. (Butnote that v in (2.4), also orthogonal to , has other structure.) Hence,

q =()q +

(3)q ; (2.8)

here the factor in the first right-hand side term shows that a more orthodoxform of the projector onto the -congruence is also better logically [see in (2.2) already discussed above].

In (2.1) and (2.3) we had expressions of the energy-momentum tensorof a perfect fluid given in terms of the vector u, the four-velocity of fluid.

This unit vector field is tangent to the congruence of fluids particles worldlines forming a timelike congruence which, naturally, can be considered asa monad. It is clear that this is monad of the frame co-moving with thefluid, thus only in this frame observers measurements will give values of theproper mass density and proper pressure of the fluid. In all other frameseven Pascals fluid pressure must become anisotropic. Let us see this in moredetail. Substitute the expression (2.4) of u in terms of and v in (2.1); yousee that in an arbitrary reference frame the energy-momentum tensor of aperfect fluid takes the form

Tpf =()

u

2

+()

u

2

1p +()u

2

( + p)( v + v + v v)pb. (2.9)Here the scalar coefficient completely written inside the square brackets in

the first term, represents fluids energy (mass) density, the next coefficient()u

2

( +p) (in the second term) expresses coincidence of density of fluids energyflow and of its three-momentum density (when, respectively, multiplied by v and v ); the last term in the same parentheses describes the anisotropicpart of pressure in the direction of fluids motion, while the last term (pb)gives isotropic part of pressure, all this is now done in -reference frame.It is interesting that scalar quantities, the energy density and the pressure,related to the co-moving frame of the fluid, both contribute to the energydensity, its flow density, and anisotropic part of the pressure, taken withrespect to non-co-moving reference frame, while the fluid remains one and the

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same Pascal fluid. By evaluating the involved quantities one has to take into

account the fact that the relativistic factor ()u is always not smaller than unity()u = dt/ds = (1 v2)1/2 1

. Though in cosmological calculations one,

quite obviously, has to use the co-moving frame of fluid modeling contentsof the universe, these considerations of transformation from one referenceframe to another are important in understanding that resulting scalar (i.e.invariant under transformations of coordinates) quantities (mass density andpressure of the fluid) are not invariant under a change of reference frame.

The energy-momentum tensor describes inertial and energetical proper-ties, in our case, of a fluid. The gravitating mass or mass density follows

from participation of this tensor in Einsteins equations, and it is strictlyshown (Mitskievich 2006b) that the latter represents a combination of iner-tial mass and pressure. However this is completely contained in the structureof Einsteins equations, so that the gravitational mass has to be used only inpost-Newtonian approximations of general relativity in celestial mechanics.Thus we have only to use the energy-momentum tensor in Einsteins equa-tions in our cosmological calculations. The only thing which we still need,is to include more one concrete property of the fluid, its equation of state,relating pressure to mass density. Here we shall use the simplest equation ofstate,

p = (2k 1). (2.10)The fluid with k = 1/2 (p = 0) is called incoherent dust, with k = 2/3(thus p = /3, T having zero trace) is incoherent radiation, and k = 1 (p = ,and sound in the fluid propagates with the velocity of light in a vacuum),is stiff matter. Below we shall see that for any value of the constant k it iseasy to find an exact solution of Einsteins equations, even when this k hasno physical meaning.

3 Solution of Einsteins equations for arbi-

trary kLet us assume that the metric (1.9) is written in coordinates corresponding tothe co-moving frame of the matter filling the universe, and that this mattercan be in the large-scale described as a fluid. Then, taking the energy-momentum tensor on the right-hand side of Einsteins equations as (2.1) and

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fluids equation of state as (2.10), we can write these equations as

G(0)(0) = , G(1)(1) = (2k 1). (3.1)The components of conservative tensor G were already given in (1.16). Butfirst we combine these two equations eliminating and thus leaving only oneunknown function, a(t) in G:

a

a

.+ (3k 1)

a

a

2 C

= 0. (3.2)

Of course, we choose a new unknown function, say, X(t) = aa

= (ln a);

then the equation reads X+ (3k1) (X2

C) = 0, i.e.dX

X2C = (3k1)dtwhich is equivalent to 1

2C

dX

XC dX

X+C

= (3k1)dt. Now, ln X+

C

XC

=

2

C(3k 1)t where a new integration constant was included into t as itstranslation. Then

d ln a = Xdt =

Ccoth[

C(3k 1)t]dt =(3k 1)1d ln sinh[C(3k 1)t], (3.3)

and after integration

a(t) =

{sinh[

C(3k

1)t]

}(3k1)1 ,

with one more integration constant (). We choose this constant as =C(3k1)1

, to make the properties of the scale factor a better when C

takes all its three standard values,

a(t) =

sinh[

C(3k 1)t]

C

(3k1)1. (3.4)

Note that for all values of k with positive pressure (and even some with nottoo strong negative one) in the simplest equation of state (2.10) this solution

describes universes having an initial singularity at t = 0.A substitution of (3.4) into the the first of two Einsteins equations in

(3.1) [see also (1.16)] gives time-dependence of the mass density as

=3

C

sinh[

C(3k 1)t]

6k3k1

. (3.5)

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You see that at this initial singularity (t = 0) spacetime still does not exist (all

metric coefficients are equal to zero) while mass density and pressure (if wedo not consider the incoherent dust) have infinite values (at this limit). It isinteresting that the most strong singularity occurs for dust, and the weakest,for stiff matter (the power 6k

3k1 takes in the cases of dust, incoherent radiationand the stiff matter values 6, 4, and 3, respectively), though, of course, thestrongest singularity takes place when k = 1/3, and it occurs not only att = 0, but at all finite times too, thus giving the lower limit for k. Howeverif one would formally consider the values of k below 1/3 (disjoint from thephysically acceptable ones by that perpetual singularity), even non-singularcases of the cosmological bahavour may be found.

References

Mitskievich, N.V. (1999a) Int. J. of Theor. Phys. 38, 997.

Mitskievich, N.V. (1999b) Gen. Relat. and Grav. 31, 713.

Mitskievich, N.V. (2006a) Relativistic Physics in Arbitrary PeferenceFrames (New York: Nova Science Publishers).

Mitskievich, N.V. (2006b) Relativistic generalization of the inertial and

gravitational masses equivalence principle, arXiv: gr-qc/0611017 (Talkat the 11th Marcel Grossmann Meeting, Berlin, July 2006).

Narlikar, J.V. (2002) An Introduction to Cosmology, third edition (Cam-bridge: CUP).

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