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1 Relativity, Special and General

1 Relativity, Special and General

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Relativistic Cosmology



73



In the special theory of relativity Einstein had established that given

an inertial frame of reference, any other frame of reference moving

uniformly and without rotation with respect to the first is also an inertial

frame of reference, and therefore the laws of nature are concordantly

described in the two inertial reference systems (principle of special

relativity). In this theory space and time are no longer absolute but

depend on the direction and magnitude of the translation speeds of the

inertial frame of the observer with respect to the inertial frame where the

observed event takes place. This is true, of course, for speed, comparable

to the speed of propagation of light. Otherwise, i.e. for speeds, v, such

that (v/c)2 << 1, relativistic space and time become indistinguishable

from absolute space and time. On the other hand the velocity of light in

empty space becomes and absolute universal constant, independent of the

relative speed of the reference system of the observer, in agreement with

the Michelson-Morley experiment. So the theory of relativity is more

“absolutist” than generally believed.

But having established, in the special theory of relativity, that

physical laws were the same (invariant) regardless of the speed of the

inertial (i.e. non-accelerated) reference system from which they were

observed, Einstein went a step further, and in his General Theory of

Relativity he established that physical laws were also the same regardless

of the acceleration of the reference frame. He got a clue to this step from

the equivalence of inertial and gravitational mass. An observer

momentaneously moving at constant speed would be unable to attribute a

change in speed either to a sudden acceleration of his motion or to the

sudden appearance of an external gravitational field. Pointing our that

newtonian mechanics assumed a “spatium absolutum” and a “tempus

absolutum” while special relativity assumed a “continuum spatiumtempus absolutum”, he found this still unsatisfactory. This is so because,

on one hand, something (the spatio temporal continuum) which acts by

itself but cannot be acted upon, is conceptually unreasonable in itself,

and, on the other hand, because, for Einstein, the spirit of the principle of

relativity required an extension to non-inertial frames of reference, or

otherwise, such experimental fact as the observational equivalence

between inertial and gravitational mass would not be justifiable. The



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The Intelligible Universe



General Theory of Relativity implies, therefore, that the geometry of

space-time is modified by the presence of massive material bodies. The

gravitational field thus influences the metrics of space-time and even

determines it. Fortunately, first Gauss, introducing arbitrary curvilinear

coordinates adapted to the geometry of any two dimensional surface and,

later, Riemann, generalizing this idea to more than two dimensions, had

prepared the ground for the development of tensorial calculus, which is

the mathematical tool required by the General Theory of Relativity.

As pointed out previously, experimental tests of general relativity

include2 observations on the deflection of light by the sun (and other

astronomical objects), the precession of perihelia of planets, and radar

echo delays. More refined experiments along these lines are under way,

and, for the time being, no conclusive experimental evidence has been

found against its validity. Its beauty and generality speak for themselves,

and it may be said that a general consensus exists among physicists in its

favour. On the other hand, attempts to make it compatible with quantum

theory, which is known to be eminently successful in the atomic and

subatomic realm, have not been too rewarding up to now.

4.2. The Cosmological Dynamic Equations

Einstein’s field equations3 for a spherical, homogeneous and isotropic

system (Robertson-Walker metrics) reduce to



(4.1)

Rɺ 2 =

G ρ R 2 − kc 2

3

where R = R (t) is the scale factor or radius, Rɺ = Rɺ (t ) its time derivative,



G = 6.67 × 10-8 dyn. cm2/g2 is Newton’s gravitational constant,

ρ = ρ (t ) = ρ m (t ) + ρ r (t ) / c 2 the mass density, which involves both

matter and radiation mass, k the spatial curvature, which can in principle

be positive (closed universe) or negative (open universe), and c is the

velocity of light.

It can be noted that for k = 0 (euclidean space) Eq. (4.1) can be

written as



Relativistic Cosmology



 4π



ρ R3 



1 ɺ

3

 = G mM

mR = Gm 

2

R

R



75



(4.2)



which is the classical newtonian equation for the motion of a mass m

moving radially with kinetic energy (1 / 2)mRɺ 2 under the influence of a

central gravitational potential enclosing total mass M and therefore with

potential energy GmM/R, at escape velocity. As the time goes to infinity

(t → ∞) the kinetic and potential energies go simultaneously to zero, because

R → 0, R → ∞. Figure 4.2 depicts the expansion (we know that the

universe is expanding because galaxies are receding from each other

according to the Hubble’s law, Rɺ = HR) of a spherical homogeneous and

isotropic mass distribution in accordance with Eq. (4.1).



Fig. 4.2. General relativity dynamical equation for an expanding spherical, homogeneous,

isotropic mass distribution. The spatial curvature k can be k > 0 (closed universe in which

the expansion phase is necessarily followed by a contraction phase because the expansion

takes place at less than scape velocity), k = 0 (flat euclidean universe expanding for ever

at exactly scape velocity) and k < 0 (open universe, expanding for ever at a velocity

higher than scape velocity).



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The Intelligible Universe



In addition to Eq. (4.1) we can write down the energy conservation

equation

d

( ρ R3 ) = −3 pR3

dR



(4.3)



where p is the pressure due to matter and radiation within the spherical



mass distribution, and the equation of state

p = p( ρ )



(4.4)



which is not known explicitly in general, but it is given by pr = ρr /3 for

radiation pressure only, in particular.

Equations (4.1) (Einstein equation), (4.3) (energy conservation

equation) and (4.4) (equation of state) are the fundamental equations of

dynamical cosmology.

The critical mass density, pc, is defined as the mass density necessary

to make the metrics to the universe flat (k = 0) and therefore, from

Eq. (4.1),



Rɺ 2 =

G ρ R 2 ,i.e. ρ c ≡ 3 Rɺ 2 R 2

3



(



) 8π G



(4.5)



Combining Eqs. (4.1) and (4.5) we get



 R2

−k =  2

c





ρ 

  1 − ρ 

c 





(4.6)



This means that the sign of the space curvature is entirely determined by

the ratio of the actual mass density of the critical mass density in the

universe, i.e.









ρc ) > 1 implies k > 0 (closed universe)



(4.7)



ρc ) ≤ 1 implies k ≤ 0, − k = |k | (open universe)



(4.8)



Present estimates of the mass density, based upon galaxy counts in large

regions surrounding our own galaxy and estimates of the typical mass of



Relativistic Cosmology



77



galaxies deduced from their rotation velocity indicate that (ρ ρ c ) < 1 ,

which supports an open universe. Many theoretical cosmologists,

however, remain unconvinced and hope that some of the various

candidates proposed (from mini black holes to neutrinos) can provide the

extra mass density needed to close the universe. This is the so called

“missing mass problem”.

Assuming that −k = |k| (open universe) we can rewrite Eq. (4.1) as



Rɺ = R



1



2



{(8π 3) G ρ R



3



}



+ k c2 R



1



2



(4.9)



If it is further assumed that after a conveniently unspecified early time

the pressure p becomes negligible, and that according to Eq. (4.3),



ρ R 3 = constant,



(4.10)



the first term within curly brackets in Eq. (4.9) becomes constant also.

Consequently we can define p+ and R3+, whose product is



ρ + R+3 = ρ R 3 = constant,



(4.11)



in such a way that the two terms within curly brackets in Eq. (4.9)

become equal,



(8π 3) G ρ + R+3 = k c2 R+ , i.e, R+ = ( 3 k c2



8π G ρ +



)



1



2



(4.12)



and then proceed with the integration of Eq. (4.9), which reduces to







dt =







R



{(8π G 3)



1



2



ρ + R+3



2



}



+kc R



1



(4.13)

2



The integral in the right hand side can be performed making the change

of variable

x 2 = k c2 R



defining,



(4.14)



78



The Intelligible Universe



a 2 = (8π G/ 3) ρ + R+3 = | k | c 2 R+ ,



(4.15)



and taking into account the result for the indefinite integral







1



x2



{a



2



+x



2



}



1



2



x

dx = a 2 + x 2

2



{



}



2







a2 

ln x − a 2 + x 2

2 



{



}



1



2



 (4.16)





which can be found in tables and can be checked by direct

differentiation.

Using Eqs. (4.14)–(4.16) one gets

1

1

1

1 





2

2







R+  R  2 

R  2

R

R



t = 1 

 1 +

 − ln 

 1 +

  

R

R

R

R

+ 

+ 

 + 

k 2 c  +  



 

R

= 1+ {sinh y cosh y − y}

k 2c



(4.17)



where the definition



( R / R+ )



1



2



≡ sinh y



(4.18)



which implies

R = R+ sinh 2 y



(4.19)



has been used. Eqs. (4.17) and (4.19) give the time t and the scale factor

R in terms of y in parametric form, and are equivalent to the well known

Friedmann5 solutions for an open universe. For a closed universe (k > 0)

the solutions are very similar and are given in terms of the trigonometric

functions siny, cosy instead of the hyperbolic functions sinhy, coshy.

The basic parameters to describe the time evolution of the universe

are the Hubble parameter (H), the density parameter (Ω) and the

deceleration parameter (q), defined by

H ≡ Rɺ R



(4.20)



Relativistic Cosmology



79



Fig. 4.3. E.P. Hubble.



Fig. 4.4. M.L. Humason.



Ω ≡ ρ / ρC = 1 −



k c2

R2



(see Eq. (4.6))



ɺɺ Rɺ 2

q ≡ − RR



(4.21)

(4.22)



All these parameters are given in terms of

Rɺ = ( dR dy ) ( dt dy ) ,



and,



(4.23)



80



The Intelligible Universe



R = ( dRɺ dy ) ( dt dy )



(4.24)



which can be easily calculated from Eqs. (4.17) and (4.19), resulting in

Rɺ = |k |



1



2



c



cosh y

sinh y



(4.25)



and

1



2

ɺɺ = − 1 |k | c 1

R

2 R+ sinh 4 y





(4.26)



••



Substituting now these values for R and R in Eqs. (4.10)–(4.24) one

finally gets



(



)(



H = |k |h c R+ cosh y sinh 3 y



)



(4.27)



Ω = 1 cosh 2 y



(4.28)



q = 1 2 cosh 2 y



(4.29)



We can see that for R « R+, i.e. y « 1, one has

 1



H ( y → 0) ≈  | k | 2 c R+  y −3 → ∞







(4.30)



Ω ( y → 0) ≈ 1



(4.31)



q( y → 0) ≈ 1/ 2



(4.32)



 1



H + = (0.9507)  | k | 2 c R+ 







(4.33)



Ω+ = 0.4199



(4.34)



q+ = 0.2099



(4.35)



Relativistic Cosmology



81



ɺɺR Rɺ 2 upon the parameter y = sinh−1(R/R+)1/2

Fig. 4.5. Dependence of Ω ≡ ρ/ρc and q ≡ − R

which is related to the degree of expansion of the universe.



And, finally, for R » R+, i.e. y » 1,

1

H ( y → ∞) ≈  | k | 2 c Rt  4e − 2 y → 0







(4.36)



Ω ( y → ∞) ≈ 4e−2 y → 0



(4.37)



q( y → 0) ≈ 2e −2 y → 0



(4.38)



The dependence of Ω and q on y = sinh-1 [(R/R+)]1/2 is depicted in

Fig. 4.5 where it can be seen that both decrease monotonously towards

zero for k < 0 as the expansion proceeds. When the expansion is in the

very early stages (R « R+) the value of Ω = ρ/ρc becomes almost the

same for k < 0, k = 0, and k > 0, making it difficult to distinguish

weather the universe is open, flat or closed.



82



The Intelligible Universe



It is of interest to set upper and lower limits to the present density

(Ω) = (ρ0/ρc0) and present age (τ0 = t(y0)/H0-1) of the universe based

upon Eqs. (4.28) and (4.17) and upon the available information on

the minimum mass density estimated from the mass in galaxies,6

the minimum age of the universe estimated from the age of the oldest

stars in the Milky Way,7 and present reasonable values of Hubble’s

constant,8 whose observational value is known within a factor of two

only, i.e. it lays within 0.7 H0 and 1.4 H0, being H0, its most probable

value.

Assuming Ω0 = 0.025, H0 = 2.1 × 10-18 s-1, one can make use of

Eq. (4.28) to get the corresponding value of the parameter y0 through

y = cosh −1 (1/ Ω0 )



1



2



= 2.53



(4.39)



and then get the corresponding age of the universe by Eq. (4.17)



R+



t=

k

=



1



2



c



( sinh y0 cosh y0 − y0 )



cosh y0 sinh3 y0

( sinh y0 cosh y0 − y0 )

H0



(4.40)



= 4.57 × 1017 s = 1.45 × 1010 years

which is very close to the estimated age of the oldest stars in the Milky

Way (t ≈ 1.5 × 1010 years).

Figure 4.6 gives τ0 ≈ t0 /H0-1 as a function of Ω0 in the interval 10-2 <

Ω0 < 1 for three different values of the Hubble constant (0.7 HQ, H0 and

1.4 H0) spanning the uncertainly range for this fundamental cosmological

parameter. It can be seen that, while Ω0 ≈ 0.025 is favoured, values as

large as Ω0 ≈ 0.7 are not completely ruled out by other observational

constraints.



Relativistic Cosmology



83



Constraints

(a) Minimum mass in galaxies

(b) Maximum Hubble’s constant

(c) Age of oldest stars in Milky way

Fig. 4.6. Limits on the mass density and the age of the universe compatible

with cosmological equations (Eqs. (4.26) and (4.17)) for an open (k < 0) universe (H0 =

2.1 × 10-18 s-1).



4.3. The Matter Dominated and the Radiation Dominated Eras

We know that at present times in the expansion of the universe the mass

density associated with matter (ρm) is much larger than the mass density

associated with background radiation (ρr /c2). The former, which is

concentrated in stars and cosmic dust within galaxies, can be estimated as



ρ m 0 ≈ 1.97 × 10 −31 g/cm3



(4.41)



corresponding to Ω0 = ρm0/ρc0 = 0.025, where ρc0 = 3H2/8πG, with H0, taken

as H0 = 2.1 × 10-18 s-1. The mass density associated with background



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