1 Relativity, Special and General
Tải bản đầy đủ - 0trang
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
74
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
8π
(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).
76
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),
8π
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.
(ρ
(ρ
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