DRAFT OF A CHAPTER
PHYSICS BACKGROUND FOR BIG BANG
Charles P. Poole, Jr.
CONTENTS
1. Introduction
2. Forces in Nature
3. Distance Parameters
4. Time Parameters
5. Dimensionless Physical Constants
6. Electromagnetic Spectrum
7. Special Relativity
8. Orthogonality and Subspaces
9. The Existence of God
` 10. Ockham's Razor
1. INTRODUCTION
In
the Big Bang chapter we recount the standard scientific model for describing
the creation of the Universe, and its development to its present status at the
beginning of the third millenium of human-history
time. The Anthropic Principle chapter adds an
approach which facilitates the formulation in the Weltanschuung
chapter of an overall world view of creation that takes into account both
scientific and theological perspectives.
To accomplish these objectives it will be helpful to provide in the
present chapter some technical scientific and other details which would
interrupt the flow of the argument if they appeared in the other chapters.
These details, as well as this entire chapter, can be skipped by readers with
very little interest in these matters.
2.
FORCES IN NATURE
We
know from modern theories of physics that there are four fundamental forces in nature,
namely the gravitational, weak, electromagnetic, and strong forces with the
approximate relative coupling strengths "G = 10-39, "w = 10-6, "E = " =1/137, and "S = 1. These forces have associated characteristic
energies, characteristic masses, characteristic distances, and characteristic
times. They also have different length
dependencies, so the assignment of their relative coupling strengths is a
little subtle.
Two
of these force laws are long range in nature, exerting their influence at astronomical
distances r. They are of the inverse
square type, and for two masses m1 and m2, and two
electrical charges Q1 and Q2 we have the respective
gravitational FG and electrostatic FE forces:
FG = G m1 m2 /r2
FE
= k Q1 Q2
/r2 (1)
where the gravitational and
electrostatic universal constants, G and k respectively, have the values listed in the table. Since they both depend inversely on the
distance, we can assume that the ratio of the coupling strengths "E / "G for the electromagnetic to
the gravitational interactions involving two protons is the ratio of their
respective forces FE / FG from Eq.
(1)
"E / "G = FE / FG
(2)
where the charge Q on the proton
is +e, the same magnitude as the negative
charge -e on the electron , and the
value of e given in Table 1.
The
strong and weak forces are short range in nature, so they have no effect beyond
their range of influence. The strong force is an attractive force between
nucleons, where a nucleon is either a proton or a neutron, and it operates at
about the distance of one femptometer or 10-15
meters. At this distance the
electromagnetic force is 1/137 as strong.
The weak force is only operative at much shorter distances, below 10-17
or 10-18 meters. The weak
force is involved in some nuclear decay schemes, the strong force is what holds
nucleons together in an atomic nucleus, the electromagnetic force attracts
electrons to their nucleus in an atom and holds the atom together, and the
gravitational force dominates at macroscopic or much larger distances,
including astronomical distances.
3. DISTANCE PARAMETERS
There
are several important distance parameters in physics. The smallest such characteristic length is
the Planck length RP
which depends on the universal gravitational constant G as follows:
RP = ( SG/c3)½ =
1.6161x10-35 meters, (3)
where the reduced Planck
constant S = h/ 2B is defined in terms of Planck’s constant h, and c
is the velocity of light in vacuo. We have seen that the range of the weak
interaction is between 10-17 or 10-18 meters, and the
range of the strong interaction is 10-15 meters. The classical radius of an electron re
is the ratio of the electromagnetic interaction constant ke2 to the
rest energy of the electron mec2, that is:
re =
ke2 / mec2 = 2.8179x10-15
meters. (4)
A length parameter associated with the scattering of
electrons is the reduced
8C = S/mec = 3.8774x10-11 meters (5)
The radius of a hydrogen atom, called the Bohr
radius aO, is given by:
aO =
S2/
meke2 = 5.2918x10-11 meters (6)
These are related through the expression:
re = " 8C = "2 aO (7)
where " - 1/137 is the dimensionless
fine structure constant which is defined in the next section. Finally the longest distance parameter, the
Hubble radius rH = c/HO, is
given by
rH = c/HO = " re (rec2/G
me) = 8.61x1025 meters =
9.1x109 light years (8)
which is comparable to the size
of the known universe. Comparing Eqs. (3) and (8) we obtain
RP2 rH = re3 , (9)
a rather curious
relation.
4. TIME PARAMETERS
The
shortest fundamental time is the Planck time JP given by:
JP = RP/c = 5.391x10-44
sec (10)
The three strongest forces have characteristic times
associated with the average lifetime of an elementary particle that decays by
that particular interaction. The strong
interaction characteristic time JS = dP /c is the time that it takes a particle moving
at the speed of light c to travel the diameter dP
=
2rP of a proton
JS = (10-15
meters/ 3x108 meters per sec) "S2 - 10-23 sec (11)
where the dimensionless strong
coupling strength "S = 1 is
inserted to clarify the next two formulae.
The typical electromagnetic interaction decay time JEM depends on the
electromagnetic coupling strength " = 1/ 137 as follows:
JEM = ("S /"EM)2 JS - 10-19 sec (12)
In like
manner for weak interactions we have:
JW = ("S /"W)2 JS - 10-11 sec (13)
The (Bohr) period JB = h/EB associated with a hydrogen atom ground state
energy EB is:
JB =2aOh/ke2 = 3.04x10-16 sec
The Hubble Time JH = 1/HO,
which is comparable with the age of the universe, is obtained from a variant of
Eq. (8)
JH = 1/HO = " re (rec/G me) = 2.87x1017 sec =
9.1x109 years (14)
From Eqs. (10) and (14) we obtain
the result:
JP2 JH = (re/c)3 (15)
5. DIMENSIONLESS PHYSICAL CONSTANTS
Perhaps
the best known dimensionless physical constant in physics is the fine structure
constant ", which was mentioned above as the coupling strength
of the electromagnetic interaction. A
photon is considered as the particle which mediates the electromagnetic
interaction, and " is the ratio of the electromagnetic force parameter
ke2 to the photon parameter Sc:
" = ke2 / Sc = 7.2974x1–3 - 1/137 (16)
This an example of a dimensionless
universal number which is in the vicinity of unity. Another important dimensionless ratio is
proton-electron mass ratio:
mp /me
- 1837 (17)
A much larger dimensionless number is obtained from
the ratio of the electrostatic force ke2/r2 to the
gravitational force Gme2/r2
between two electrons:
(ke2 /Gme2)
= (re c2 /Gme)
= 4.17x1042 (18)
a result that can be useful
for evaluating Eqs. (3) and (8). Barrow and Tipler
mention that many decades ago Arthur Eddington
estimated the number of particles N in the universe from the formula N
= 2257 " = 3.17x1079 - 1079 , which is
called the Eddington Number. Another Eddington
expression gives a similar answer
N = (me /mP)2
(re c2 /Gme)2 = 5.15x1078 (19)
Hartmann and Impey mention
that there are about 5x1010 galaxies and 1020 stars in
the Universe.
6. ELECTROMAGNETIC SPECTRUM
Visible
light constitutes one octave of frequencies in the middle of the
electromagnetic spectrum. The lowest frequencies are called audio (100 to
100,000 Hz), then come radio frequencies in the amplitude modulation (AM, 530
to 1710 kHz) and the frequency modulation (FM, 88 to 108 MHz) bands. Television channels broadcast in the very
high frequency (VHF, 30 to 300 MHz) and the ultrahigh frequency (UHF, 300 to
3,000 MHz) bands, and radar operates at microwave frequencies (1 to 100
GHz). The optical range includes
infrared light (1012 to 4x1014 Hz), visible light (4x1014
to 7.5x1014 Hz), and ultraviolet light (7.5x1014 to 3x1015
Hz). Much more penetrating higher
frequency radiation is found in the x-ray (1016 to 1019
Hz), gamma ray ((-ray, 1019 to 1022 Hz), and
cosmic ray (beyond 1022 Hz) ranges.
Each
radio station adds music and speech to its fundamental or carrier broadcasting
frequency by modulating or superimposing audio range frequencies on the
carrier. Each television station has a
VHF or UHF carrier frequency which it modulates with an audio side band for sound
and a higher frequency video side band with the picture information. Some materials are transparent, but most
materials are opaque to optical range frequencies. X-rays are more penetrating, being attenuated
in matter, whereas gamma rays are only slightly attenuated. Neutrinos, which are produced in nuclear
reactions, are so penetrating that they are observed in the deepest mines in
the earth where the gamma rays incident from the sun and outer space have been
attenuated to zero. At any particular locality
on the surface of the earth many of these frequencies are present
simultaneously, that is they are passing through that location at the speed of
light. They do so without interfering
with each other. Their presence is
easily detected with the aid of a suitable detector, such as a radio receiver,
a TV antenna, or a Geiger counter.
7. SPECIAL RELATIVITY
The
special theory of relativity, introduced by Albert Einstein a hundred years ago
(1905), postulates a four-dimensional world, a world with three space
dimensions, and one time dimension. It also postulates that the speed of light
c in a vacuum is the same in all frames of reference, about three centimeters
per microsecond (3x108 m/sec).
The three space coordinates are denoted by x, y and z, and the time
coordinate is denoted by ct to give it the dimensions of distance like the
other three coordinates. It is well
known that the distance r between the point x1, y1, z1
denoted by r1 and the point x2, y2, z2
denoted by r2 is given by the expression:
r = [(x1
- x2)2
+ (y1 - y2)2 + (z1 - z2)2]1/2.
(20)
Relativity theory predicts that the space-time
separation R between an event with the coordinates x1, y1,
z1 , ct1 and another event with
the coordinates x2, y2, z2 , ct2 is
given by the expression:
R = [(x1
- x2)2
+ (y1 - y2)2 + (z1 - z2)2
- c2(t1 - t2)2 ]1/2 , (21)
where the time term is subtracted
from the space term instead of being added to it. To take this into account mathematically, the
time coordinate can be selected as ict instead of ct,
where i is the square root of minus one (-1)1/2. Since ict contains the quantity i =
(-1)1/2 it is called an imaginary number, while ordinary numbers like x, y and z are
called real numbers. Equation (21) can
also be written with a plus sign:
R = [(x1
- x2)2
+ (y1 - y2)2 + (z1 - z2)2
+ (ict1 - ict2)2 ]1/2 (22)
since i2 = -1. If the magnitude of the space part (x1-
x2)2
+ (y1 - y2)2 + (z1 - z2)2
is larger than the time part c2(t1-
t2)2 then the two events are called space-like. If, on the other hand, the magnitude if the
space part (x1- x2)2 + (y1 - y2)2 + (z1 - z2)2
exceeds that of the time part c2(t1
- t2)2 then the two events are called time-like.
There is an important
physical consequence of this peculiar introduction of the quantity i into the time coordinate.
If a Lorentz transformation is employed to
transform the point of observation to a new primed coordinate system that is
moving at a velocity v relative to the initial point of observation then the
space-time separation R’ of these same two events with the new coordinates x1’, y1’, z1‘,
ict1‘ and x2’, y2’, z2‘, ict2‘
will change to the value:
R’ = [(x1‘- x2’)2 + (y1‘ - y2’)2 + (z1‘ - z2’)2
- c2(t1‘ - t2’)2 ]1/2 (23)
If R was originally space-like then R’ will also be
space-like, and if R was originally time-like then R’ will also be time
like. In other words the space-like and
time-like properties do not change during a Lorentz
transformation to a moving frame of reference, always remembering that the
velocity of the transformation v cannot exceed the speed of light c. For the space like case it is always possible
to find a Lorentz transformation so that the two
events are simultaneous in the new primed frame of reference, that is t1‘ = t2’, and as a result from Eq. (23) we see that R’ becomes a real number with the
value R’ = [(x1‘- x2’)2 + (y1‘ - y2’)2 + (z1‘ - z2’)2]1/2. In like manner for the time-like case it is always
possible to find a transformation which places the two events at the same
location in space, that is x1’ = x2’, y1‘ = y2’,
and z1’ = z2‘, and
from Eq. (23) we see that this makes R’ an imaginary
number: R’ = ic(t1‘
- t2’).
An
example of a pair of time-like events is the takeoff of an airplane from
8.ORTHOGONALITY
Three
dimensional space is most often described in terms of the cartesian coordinates x, y, z, where these three
directions are perpendicular to or orthogonal to each other. On can also
define two dimensional subspaces
such as the x, y plane for which the coordinate z = 0. There is an infinite number of parallel x, y planes for different values of
z, such as the x, y plane for the value z = +1 meter. The z direction is
perpendicular to all x, y planes. One
can move along the z direction to change from a position on the lower z = 0
plane to one on the upper z = +1
plane. To accomplish this one leaves
the two dimensional subspace by projecting out into a new third dimension,
moving along this third dimension, and then arriving at the other two
dimensional subspace. A two dimensional
observer confined to a two dimensional subspace has no awareness of the
presence of a third orthogonal direction.
There
are another three dimensional orthogonal coordinate systems besides the
Cartesian one that we just described.
The most important one is called spherical polar coordinates, and its
makes use of two angular variables 1 and N plus one distance
coordinate r = (x2 + y2 + z2)1/2, where r is
the distance from the origin of the coordinate system. In this coordinate system a sphere with a
particular radius rO is a two dimensional
surface, with the variable
1 related to
the latitude, and the variable N corresponding to the longitude. This two
dimensional surface has the interesting property that an observer moving along
a great circle, that is along a circular path with the radius rO, will return to his starting position after traveling
the distance 2BrO.
Because of this property a sphere is called a closed surface. In contrast to this the x, y plane discussed
above is an infinite two dimensional subspace, because motion along a straight
line in any direction can continue forever without reaching an edge, or
retracing its steps.
A two
dimensional model of an expanding universe is a sphere of radius r = rOt/tO, where t is the time, and tO is called the doubling time. If the sphere is covered with dots
representing galaxies with the average separation dO
at the time t = tO, then the average
separation of the galaxies will be 2dO at the later time t = 2tO,
and the density of the galaxies (number per cubic meter) will be only 1/4 of
its earlier value.
9. THE EXISTENCE OF GOD
Thomas
Aquinas, in his Summa Theologica I, Question 2,
Article 3, presents five proofs for the existence of God: (a) argument from motion: everything moved has a
mover, and the prime mover is God; (b) argument from causality: every effect
has an efficient cause, and the first efficient cause is God; ( c) argument
from possibility and necessity: some beings are possible, some are necessary,
God is the being having of itself its own necessity; (d) argument from gradations
of perfection (goodness, truth, nobility, etc,) in beings, the maximum of a
perfection is its cause, and God is the cause of all perfections; (e) argument
from design: things such as natural bodies act toward an end, directed by a
being with knowledge and intelligence; an intelligent being called God exists
who directs all things toward their ends.
Other proofs of God’s existence have also been advanced. Stephen Barr (2003, p.
263) argues that every contingent being must have a cause which cannot run to
an infinite regress, so there must be a first cause itself uncaused, a cause
which is a necessary (not contingent) being which we call God. The cosmological argument (Polkinghorne 1946, p. 79) asserts
that the existence of the world requires an explanation which could only
involve its creation by a God who is being itself. The ontological argument of Anselm
(c.1033-1109) asserts that God is the being than which no greater can be
imagined, then it asserts that God requires the
property of existence to be such a being, hence God exists. There is also a universal consent of mankind
argument.
A
number of authors have commented on these arguments. Frederick Coppleston
(Vol. 2, 1948) says that “the five proofs” are “an explication of the words of
the Book of Wisdom (Chap. 13) and of
Thomas
Aquinas (c.1222-1274) ( Summa Theologica
I, Question 2, Article 1) asserted the proposition that “God exists, of itself,
is self-evident” before he presented his five proofs. The first Vatican Council (1870, Sess. 3, Chap. 2; see Vatican II, Dei Verbum,
Chap. 1) proclaimed that God can be known through the certain light of natural
reason, and the Fourth Lateran Council (1215, Const. 1) proclaimed that God, in
His omnipotent power, created (condidit) from the
beginning of time and out of nothing (simul ab initio temporis,
utramque de nihilo),
creatures that are spiritual as well as corporeal, angelic clearly and earthly,
and then humans constituted of spirit and body (ex spiritu
et corpore constitutam).
My original intention was to
formulate and present arguments for the existence of God that I considered
cogent from the viewpoint of a scientist.
Then I found that modern books of this type written by scientists do not
attempt to do this, and some implicitly minimize their importance. When I thought about it further I decided not
to embark on this task.
10. OCKHAM’S RAZOR
William
of Ockham was a Franciscan theologian who belonged to
a general movement which championed the simplification of philosophical
terminology. He is famous for what is
called his razor: “Entia non sunt
multiplicanda sine ratione,” or entities should
not be multiplied without reason.
Frederick Copleston (1953, Chap. 3) calls this the principle of economy, and it could also be referred
to as the principle of parsimony. Ockham’s contemporary fellow Franciscan Petrus
Aureoli made use of the closely related adage: pluralitas non est
ponenda sine necessitate (multiplicity should not be
posited without necessity). These
principles have influenced physicists to have a preference for simpler theories
instead of more
complex ones.
Table 1. Values of several physical
constants.
Name Symbol
Value
Boltzmann
constant kB 1.3807x10-23
J/K
Electron charge (negative)
e 1.6022x10-19
C
Electron mass me 9.1094x10-31
Kg
Electrostatic force constant k 8.9875x109 m3
Kg /s2 C2
Fine structure constant " 7.2974x10-3 - 1/137
Gravitational constant G 6.6726x10-11 m3/Kg
s2
Planck constant h 6.6261x10-34
J s
Proton charge (positive) +e 1.6022x10-19
C
Proton mass mp 1.8835x10-27
Kg
Reduced Planck constant S
1.0546x10-34 J s
Speed of light c 2.9979x108
m/s