In the Beginning

The Expanding Universe

Using the Doppler shift to measure the speeds of distant stars, Astronomer

Edwin Hubble discovered in 1929 that all but a few of the closest galaxies

are moving away from us, and that the speed at which each is moving away

is approximately proportional to its distance from us. The

proportionality constant is called the Hubble constant, so that the speed

with which a galaxy recedes from us is approximately given by H times its

current distance from us. The best current estimate of the Hubble constant

is about H = 5*10^-11/year. In other words, a galaxy which is currently R

miles away from us is now receding from us at the rate of about H*R miles/year.

At this rate, every 1/H years it recedes HR(1/H) = R miles, and so if we

assume that it has always been receding from us at this same velocity, it is

easy to see that 1/H years ago it must have been 0 miles from us! Since the

same thing can be said for any distant galaxy, we calculate that all the

galaxies would have been very closely clustered together some 1/H = 20 billion

years ago, when the current "expansion" must have begun. Actually, the

gravitational attraction between stars and galaxies must be gradually slowing

down this expansion, so that the expansion rate in the past must have been

even higher. This means that 20 billion years is an upper limit to the time

which has passed since the beginning of the expansion of the universe. In

fact, if the only significant force between astronomical bodies, the force of

gravity, works against expansion, what force sent the galaxies hurdling away

from each other through space 20 billion years ago? It must have required

quite an explosion, quite a "big bang", to overcome the force of gravity and

send all this cosmic "debris" flying out through space.

In order to use the apparent expansion of the universe to interpret either the

past or the future, we must make some assumptions. The basic assumption of

cosmology is the "cosmological principle", which states that the universe

appears essentially the same (i.e., modulo minor local variations) to observers

at any point in it. This means that matter must be distributed more or less

evenly throughout the universe, so that we may treat the matter in it as if it

were a gas of more or less constant density. Of course there are local

variations: an observer in the middle of the Milky Way galaxy will see a lot

more stars in his sky than an observer halfway between galaxies, but we are

talking about the larger view, where galaxies can be considered the "molecules"

of the gas!

Notice that the cosmological principle also implies that the universe cannot

have "boundaries" because then it would appear different to an observer near

the edge than to one far from the boundary (a boundary would be rather

problematic in any case!). Surprisingly, this does not necessarily mean that

the universe has to be infinite in volume, with an infinite quantity of matter

distributed throughout. According to Einstein's general theory of relativity,

the universe could be "finite but unbounded." A "finite but unbounded"

universe is a concept which can only be grasped intuitively by saying that the

universe would appear to us like the surface of a sphere would appear to a

creature who can only imagine two dimensions. The surface of a sphere has a

finite area but it has no boundaries, and it looks the same to all its two

dimensional inhabitants, wherever they live on it. Our universe is three

dimensional but, according to Einstein, we can think of it as being embedded

in a higher dimensional space.

According to the general theory of relativity, space is warped, or curved, in

the vicinity of matter, and if the density of matter (which we are still

assuming to be approximately the same throughout the universe) is greater than

some critical density, the curvature of space will be large enough to ensure

that the universe is closed, comparable to the surface of a sphere. If the

density is less than this critical density, we can compare our universe to a

curved but open surface in 3D space, such as a hyperbolic paraboloid.

Is the cosmological principle justified by the evidence? Although the part of

the universe that we can see does appear very homogeneous, the justification

for the cosmological principle is really as much philosophical as

observational, since we can presumably see only a small portion of the whole

universe. However, it is hard to see how we could assume anything else--if

the cosmological assumption is not valid, there is not much hope for modeling

the expansion of the universe.

A Model for the Expanding Universe

Now we are ready to talk about the future of the universe. Will the universe

continue to expand forever, or will the attractive force of gravity slow the

expansion to zero, then cause the universe to begin contracting? To answer

this question would seem to require the more modern ideas about gravity given

by Einstein's general theory of relativity (which are beyond the scope of this

discussion), but in fact it is possible to pose this question in a

mathematically correct way using the classical (Newtonian) theory of gravity.

If I dig a hole to the center of the Earth and excavate a small chamber there,

I will be able to float about weightlessly in my chamber, because the

gravitational attraction of the Earth on my body is equally strong in all

directions. What if I only tunnel halfway to the center? As you might expect,

I will weigh less than I did on the surface, but I will not be completely

weightless; there is still a net force toward the center of the Earth. It can

be shown by summing up the gravitational forces exerted on my body by all the

molecules in the Earth (thanks to integral calculus, this is not as difficult

as it sounds), that the net force exerted by that portion of the Earth which is

closer to the surface than I am is zero, so that only the portion closer to the

center than I contributes to my weight. If I could hollow out the inner core

of the Earth, the entire hollow core would be a giant weightless chamber and I

could float about in it, because at any point within the hollow core the tug of

gravity would be the same in every direction. Thus the Earth's gravitational

attraction on my body can be calculated by throwing away the outer shell and

pretending that I am standing on the surface of a smaller planet, half the

diameter of the Earth.

Now let us replace the Earth in our story by the entire universe, and let us

take our planet to be the center of the universe. (Copernicus showed us that

the Earth is not the center of the universe, but the cosmological principle

and the theory of relativity tell us that it is as good a center as any!) Then

consider a certain galaxy (A) whose distance from the Earth is given as a

function of time by R(t), and let us calculate the "weight" of that

galaxy--that is, the gravitational force with which the rest of the matter in

the universe pulls it toward the center (us). By the same reasoning as used

above, we conclude that we can ignore all matter further away from the center

than A, and calculate the force pulling A toward the center as the force of

gravity between A and a sphere of matter whose center is the Earth and whose

radius is R. According to classical gravitational theory, this force is

GMm/R(t)², where G is the universal gravitation constant, m is the

mass of A, and M is the mass of the above described sphere, on whose surface

A rests. M is just the density r(t) times the volume of this sphere,

(4p/3)R(t)³. As the universe expands or contracts, this sphere

will expand or contract proportionally, so the quantity of matter in the sphere

will remain constant. Then we may use the current (t=0) values for r(t)

and R(t), and write M = (4p/3)r₀R₀³.

This gives, for the gravitational force on A:

By Newton's second law, then, the acceleration (R") of A is equal to this

force divided by the mass (m) of A:

where the negative sign is used because the acceleration is negative, that is,

gravity decelerates (slows down) the expansion.

The initial conditions for this differential equation are obtained by noting

that at t=0 (now) we have R(0) = R₀ and R'(0) = H R₀, since the rate at

which any galaxy is receding from us is supposed to be approximately H (the

current Hubble constant) times its distance from us.

If r(t) is defined to be R(t)/R₀, r can be interpreted as the size of the

universe normalized to make the current size equal to 1. Then the differential

equation and initial conditions simplify to:

r" = -(4p/3)G r₀/r² [1.1]

r(0) = 1

r'(0) = H

Now there is an objection which the reader may raise to the way in which

(1.1) was derived. It may be argued that there is no net gravitational force

on either the Earth or A, because in either case the pull of the rest of the

matter in the universe is equally strong in all directions--either can be

considered the center of the universe. The answer is that the ultimate

justification for (1.1) comes from the general theory of relativity. However,

if we look at any sphere of small (cosmologically speaking!) radius, the

general theory of relativity allows us to ignore the gravitational effects of

material outside that sphere, and to use classical gravitational theory inside

the sphere. And we will get equation (1.1) using classical ideas if we take

the universe to be a sphere of radius M, where M is arbitrary, with center at

any particular point--Earth, galaxy A, or a neutral third party.

Using techniques found in any elementary differential equations text we find

that (1.1) implies:

[1.2]

where C is found, by applying the initial conditions, to be

Now if C is positive, that is, if r₀ < 3H²/(8p G), then it is clear

from (1.2) that the universe will continue expanding forever, since r' will

always be positive. On the other hand, if r₀ > 3H²/(8p G), C will be

negative and there will be a value of r which will make r'=0, so that when the

normalized size of the universe reaches that value of r, it will stop

expanding and begin to contract. This contraction would presumably continue

until the universe ends in a "big squeeze".

Thus there is a critical value of r, namely r_c=3H²/(8pG), such

that if the current density is larger than this value the gravitational

attraction between galaxies will be strong enough to eventually stop the

expansion of the universe and cause it to contract; otherwise it will continue

expanding forever. It turns out that r_c is also exactly the critical

density which will "close" space. In other words, if r₀ is larger than

r_c not only can we conclude that the expansion of the universe will

eventually halt, but also that the universe is finite but unbounded (closed).

Thus a closed universe will eventually contract, while an open universe will

continue expanding forever.

Is r₀ larger than the critical value? The best current estimates say no,

so that the universe is infinite in size, and will continue expanding forever.

But it is very difficult to estimate the density of matter in the universe, and

the current estimates of r₀ are close enough to r_c to leave considerable doubt

as to the size and future of the universe, and our estimates keep increasing.

I believe the universe is finite, because, in my opinion, "infinity" is a concept

which exists only in pure mathematics, I don't believe there could really be an

infinite amount of anything. But I could be wrong, it is a surprising universe

we live in.

The Big Bang

If we give a rock an initial upward velocity which equals or exceeds the

Earth's escape velocity, it will continue upward forever and never return to

Earth. If we give it an upward velocity less than this escape velocity, the

attraction of the Earth's gravity will cause the rock to slow down and finally

stop and begin falling back down. But in any case, if we see a rock flying

upward through the air, even if we are unable to calculate whether its future

holds a return to Earth or an eternal flight through outer space, we can

confidently "predict" its past.

In a similar manner, whatever the future of the universe may be, it is clear

that at some time in the past r(t) was very small, and the matter in the

universe was very tightly packed. If we solve (1.2) for r(t) (this is not easy

to do, but it can be done using standard differential equation solution

techniques) we find that, no matter what value we assign to r₀, at a

particular negative (past) value of t, the normalized size of the universe

was r=0, and the velocity of expansion was r'=∞. That r'=∞ when

r=0 can be seen directly from (1.2), and it reflects the fact that the

gravitational attraction between bodies becomes infinite as their separation

approaches zero, and so an infinitely large initial velocity would be required

to overcome the infinitely strong attraction of gravity associated with the

state r=0.

This is suggestive of a very "big bang". The value of t for which r=0

depends on our estimate of r₀, but we have already seen that it cannot

have been more than 20 billion years ago. For example, if r₀ = r_c,

we can compute that the big bang must have occurred about (2/3)H^-1 = 13.3

billion years ago.

The justification from classical physics for the simple equation (1.1) would

seem to break down when r is close to 0, because other forces are no longer

negligible then, but the real justification for (1.1) is not classical physics

but the general theory of relativity. A model of the universe which is

based on the general theory of relativity still predicts a singularity,

with r=0 and r'=∞ in the finite past. The currently accepted model,

called the "standard model", still says that the universe arose from nothing,

with a "big bang."

For a while scientists were divided between the big bang theory and the "steady

state" theory of the universe. The steady state theory holds that the average

density of the universe is maintained at a constant level as the universe

expands, by the creation (somehow) of new matter. However, there are several

theoretical and experimental reasons why the steady state model has now been

rejected in favor of the big bang theory. In particular, the 1965 discovery by

a pair of radio astronomers of a background of microwave radiation permeating

the universe was spectacular confirmation of one of the predictions of the big

bang model. Proponents of the big bang theory had calculated that at a time

shortly after its beginning, when the temperature of the universe was about

3,000^o Kelvin, the universe must have been filled with high energy, that is,

short wavelength, photons. They calculated that the entire universe at

that point would function as a "blackbody", and that the photon wavelengths

would be distributed in the manner characteristic of blackbody spectra. As

the universe expanded, this radiation cooled and by now, they calculated, it

should have an equivalent temperature of about 3^o Kelvin (very cold!). With

the drop in temperature, the photon wavelength distribution would shift upward

and would now have a peak in the microwave range, at about 0.1cm.

It was this 3^o K remnant microwave radiation, emanating not from any

particular astronomical object but from the entire sky, that Bell Telephone

Laboratory radio astronomers Arno Penzias and Robert Wilson discovered in

1965. Not only was the peak close to the expected value, but the form of the

observed wavelength distribution curve conformed closely to the predicted

shape. Robert Jastrow, founder and director of NASA's Goddard Institute for

Space Studies, and professor at Columbia University, describes the discovery

of the background microwave radiation in layman's language [Jastrow 1978]:

"The two physicists were puzzled by their discovery. They were not thinking

about the origin of the universe, and they did not realize that they had

stumbled upon the answer to one of the cosmic mysteries. Scientists who

believed in the theory of the big bang had long asserted that the universe

must have resembled a white-hot fireball in the first moments after the big

bang occurred. Gradually, as the universe expanded and cooled, the fireball

would have become less brilliant, but its radiation would never have

disappeared entirely. It was the diffuse glow of this ancient radiation,

dating back to the birth of the universe, that Penzias and Wilson apparently

discovered.

No explanation other than the big bang has been found for the fireball

radiation. The clincher, which has convinced almost the last doubting Thomas,

is that the radiation discovered by Penzias and Wilson has exactly the pattern

of wavelengths expected for the light and heat produced in a great explosion.

Supporters of the steady state theory have tried desperately to find an

alternative explanation, but they have failed. At the present time, the big

bang theory has no competitors."

We suggested earlier that our 3D universe, if finite, may appear to us like the

surface of a sphere would appear to a 2D creature who cannot even comprehend

the concept of a third dimension. The inflation of a sphere or balloon is

perhaps a better analogy than an explosion to illustrate the expanding universe.

If air is pumped into a balloon, it will expand in such a way that every point

in this 2D universe (the balloon surface) recedes from every other point.

This analogy also helps us understand how the attraction of gravity could

cause the expansion to slow, or to reverse itself, even though the pull of

gravity on any star is the same in every direction. The word "explosion"

implies that a volume of empty space is already there, and an explosion at

one point in that volume sends debris flying out in all directions through

the already-existing space. But our expanding universe is more like the

surface of a sphere whose radius has expanded from r=0 to its current size.

There was no universe, not even an empty one, before the big bang, and it is

the entire universe--empty space and all--which is expanding. r=0 does not

mean a very small, dense, universe, it means nothing existed: not only no

matter or energy, but no space or time either!

The Finite Age of the Universe

Although the discovery of the background microwave radiation permeating the

universe is the most important reason that the big bang theory is the

"standard" model today, there are other reasons to believe the universe had a

beginning, and most are consistent with the time frame of 20 billion years

estimated from the expansion rate of the universe. The ages of various

celestial objects can been estimated, and all are found to be less than 20

billion years old. For example, radioactive dating techniques can be used to

compute the age of a meteorite; the fraction of a radioactive isotope remaining

tells us how many half-lives have passed since the isotope was created.

The fraction of the matter in the universe represented by hydrogen is

continually decreasing, as hydrogen atoms in the stars fuse to make helium and

other heavier elements. If the universe were infinitely old, all the hydrogen

would presumably have been consumed long ago.

A similar argument is based on the second law of thermodynamics, which states

that the "entropy" (disorder) in the universe continually increases. Every

time hot and cold water mix to make lukewarm water, or two gases mix, or

mechanical energy is converted by friction into heat energy, the specific

entropy (entropy per unit volume) of the universe increases irreversibly--the

universe "winds down" a little. But, again, if the universe were infinitely

old, the continual increase in randomness predicted by the second law of

thermodynamics would ensure that all that would have been left by now would be

a homogeneous, "wound down" universe.

Russian astrophysicists Y.B. Zel'Dovich and I.D. Novikov [Zel'Dovich and Novikov,

1983] argue that even if we conjecture that the universe goes through cycles of expansion

and contraction (and they see no evidence of any repulsive force which could turn

contraction back into expansion) the second law of thermodynamics still

guarantees that the age of the universe is finite. They write: "It follows

from this that the universe has lived through only a finite number of cycles in

the past and has a finite time of existence because in each cycle the entropy

increases by a finite amount. For an infinite number of cycles, therefore, the

specific entropy would be infinite; but this is not the case."

It is conceivable, though it seems extremely unlikely, that evidence will

someday be uncovered which forces us back to a steady-state or oscillating

universe theory. But it is inconceivable that natural processes will be

discovered which can reverse the normal flow of entropy, and cause disorder to

reorganize itself into order. Thus Nature's irreversible tendency toward

disorder will not allow us to avoid the problem of a true beginning of time, of

a "moment with no moment preceding it" (Arthur Eddington).

Philosophical Implications

In the introduction to his book "The First Three Minutes" [Weinberg 1977] Steven

Weinberg wrote:

"How then did we come to the 'standard model'? And how has it supplanted

other theories, like the steady state model? It is a tribute to the essential

objectivity of modern astrophysics that this consensus has been brought about,

not by shifts in philosophical preference or by the influence of astrophysical

mandarins, but by the pressure of empirical data."

To say that rejection of the steady state model in favor of the big bang theory

was not due to shifts in philosophical preference is an understatement, because

many scientists would agree with Weinberg that the steady state model is

"philosophically far more attractive." Einstein introduced an arbitrary

additional term into his equations of general relativity in an attempt (which

he later regretted) to avoid the expanding universe solution. Robert Jastrow

[Jastrow 1978] writes that:

"Some prominent scientists began to feel the same irritation over the

expanding universe that Einstein had expressed earlier. Eddington wrote in

1931, 'I have no ax to grind in this discussion, but the notion of a

beginning is repugnant to me. The expanding universe is preposterous. . .

incredible, it leaves me cold.' The German chemist Walter Nernst wrote 'To

deny the infinite duration of time would be to betray the very foundation of

science.'"

The reason that many scientists were reluctant to accept the big bang is

obvious: it points out the incompleteness of science. If the goal of science

is, as Joseph Le Conte [Le Conte 1888] put it, to explain how "each state or

condition grew naturally out of the immediately preceding", then this pursuit

meets a dead end in the big bang, for the chain of causality must end with the

beginning of time. The implications of the discovery that the entire universe

--matter, energy, space and time--had a true beginning are enormous, and do

not yet seem to have "sunk in" to our scientific consciousness; many scientists

still gloss over the big bang as if it were just another explosion.

Scientists still tend to think of religions as systems of beliefs

which have no root in science, and of atheism as the absence of any such

unprovable beliefs. The truth is that now all theories of origins, theistic

or atheistic, involve speculation as to the nature of the supernatural forces

which created our universe out of nothingness, because there were no "natural"

causes before Nature came into existence. The question is only, was it an

intelligent or an unintelligent supernatural force that created time, space,

matter and energy out of nothingness?

Some religious people do not like the big bang theory because they believe it

is an attempt to explain scientifically how our universe came into existence.

But while the big bang theory attempts to explain what happened from the

early stages onward, it does not attempt to explain how or why our universe

came into being from nothingness--how could a scientific theory ever do that?

It only states that, according to the evidence, that is exactly what happened.

British physicist Edmund Wittaker [quoted in Jastrow 1978] stated what other scientists

had to be thinking: "What came before the beginning? There is no ground for

supposing that matter and energy existed before and were suddenly galvanized

into action. For what could distinguish that moment from all other moments

in eternity? It is simpler to postulate creation ex nihilio--Divine will

constituting Nature from nothingness."

The Anthropic Principle

The development of models of the early universe involves primarily theoretical

calculations, intended to reconstruct what must have happened in the

aftermath of the big bang. These computations, and many others made by

physicists, show that we are the beneficiaries of some very lucky coincidences.

In an interview published in [Varghese 1984] Robert Jastrow discusses what he calls

"the most theistic result ever to come out of science":

"According to the picture of the evolution of the universe developed by the

astronomer and his fellow scientists, the smallest change in any of the

circumstances of the natural world, such as the relative strengths of the

forces of Nature, or the properties of the elementary particles, would have

led to a universe in which there could be no life and no man."

As an example, Jastrow cites the forces binding the nuclei of atoms together.

If the nuclear force were increased in strength by a small amount, he says, this

attraction would have been sufficient to cause all hydrogen nuclei (protons) to

fuse together into helium during the early stages of the universe, and there

would be no hydrogen left to fuel the stars. On the other hand, if the nuclear

force were slightly decreased in strength, the attraction would have been

insufficient to drive the nuclear fusion reactions which created elements

heavier than helium (such as carbon and oxygen), and it is impossible to

imagine how any complex life forms could be constructed out of hydrogen and

helium alone.

Jastrow continues:

"It is possible to make the same argument about changes in the strengths of

the electromagnetic force, the force of gravity, or any other constants of the

material universe, and so come to the conclusion that in a slightly changed

universe there could be no life, and no man. Thus according to the physicist

and the astronomer, it appears that the universe was constructed within very

narrow limits, in such a way that man could dwell in it. This result is

called the anthropic principle.

Some scientists suggest, in an effort to avoid a theistic or teleological

implication in their findings, that there must be an infinite number of

universes, representing all possible combinations of basic forces and

conditions, and that our universe is one of an infinitely small fraction, in

this great plenitude of universes, in which life exists."

Now the Darwinist would argue that a different universe, which might be hostile

to life as we know it, would only have resulted in life forms which are

adapted to different conditions. For example, if the Earth had been a bit

further from the sun than it is, we might have evolved thicker skin to adjust

to the cold, and if it were a little closer, we might have developed cooling

fins. However, we are not talking about conditions which are hostile to life

as we know it on Earth, but rather conditions so hostile that any imaginable

form of life would be impossible. A.J. Leggett [Leggett 1987] lists several ways in

which the development of life depends sensitively on the values of the

universal constants, and says,

"The list could be multiplied endlessly, and it is easy to draw the conclusion

that for any kind of conscious beings to exist at all, the basic constants of

Nature have to be exactly what they are, or at least extremely close to it.

The anthropic principle then turns this statement around and says, in effect,

that the reason the fundamental constants have the values they do is because

otherwise we would not be here to wonder about them."

Physicist Steven Hawking discusses some of these fundamental constants of

Nature and says [Hawking 1988], "The remarkable fact is that the values of these numbers

seem to have been very finely adjusted to make possible the development of

life."

Edward Harrison [Harrison 1981], mentions some other bad things which would happen if

certain constants were tampered with:

"We first notice that alterations in the known values of c [speed of light],

h [Planck's constant], and e [electronic charge] cause huge changes

in the structure of atoms and atomic nuclei. Even when the changes are

only slight, most atomic nuclei are unstable and cannot exist. . .We also

find that slight changes in the values of c, G [gravitational constant], h,

e, and the masses of subatomic particles cause huge changes in the

structure and evolution of stars. The majority of universes will actually

not contain any stars at all, and in the few that do, the stars either are

nonluminous or are so luminous that their lifetimes are too short for

biological evolution. . .Our universe is therefore finely tuned, and we would

not exist if the constants of nature had different values."

But we have to ask ourselves not only, why do the gravitational, nuclear and

electromagnetic forces have the strengths that they have, and why do electrons,

protons and neutrons have the masses and charges they do, but why are there

particles at all, and why are there forces between them? We need to wonder not

only why the speed of light is 299,792 km/sec, but why are there photons? We

should not only wonder why Planck's constant, which appears in the Schrodinger

equations, has such a lucky value, but why are the motions of all particles

governed by these partial differential equations?! One of the most amazing

things about our universe is the beautiful way in which mathematical equations

can be used to elegantly model physical processes. In the case of macroscopic

processes, such as diffusion or fluid flow, we can derive the equations from

more basic processes, so that in these cases we feel we "understand" why the

mathematics fits the physics so nicely. But when we get down to the most

fundamental particles and forces, we find they still obey elegant mathematical

equations, and we have absolutely no idea why. There is no conceivable reason why

the effect that the fundamental forces have on the fundamental particles should

be given by the (complex-valued!!) solution to a wave or eigenvalue partial

differential equation, except that it results in elements and chemical compounds

with extremely rich and useful chemical properties, and gives partial differential

equation software developers like me some very interesting applications to

solve.

Are we to assume that in all these other universes there is still space and

time, gravity and electromagnetic forces, electrons, protons, neutrons and

photons, and Schrodinger equations, but their forces, masses and charges have

different values, generated by some random number generator?

Paul Davies, in "Other Worlds" [Davies 1980] appeals to the anthropic principle no

fewer than 10 times to explain benevolent features of our universe. Citing

the calculations of various physicists and astronomers, he shows that if the

matter in the universe were distributed a little more--or less--uniformly;

if the material density of the universe were a little higher--or lower; if

either the strong or the weak nuclear force were a little stronger--or weaker;

then the resulting universe would have been very hostile toward the conception

and development of any form of life.

Davies estimates the odds against one of these coincidences to be (very

roughly, of course!) 10^{1000000000000000000000000000000} to 1. And he adds

that "there are probably many more features of the world that are vital to the

existence of life and which contribute to the general impression of the

improbability of the observed world."

Although Davies recognizes that some may see design in the fortuitous features

of our universe, he prefers the multiple universes theory. "If we believe that

there are countless other universes, either in space or time, or in superspace,

there is no longer anything astounding about the enormous degree of cosmic

organization that we observe. We have selected it by our very existence. The

world is just an accident that was bound to happen sooner or later," he says.

Davies compares the anthropic principle's explanation of why the laws,

particles and forces of physics are so friendly toward life to the traditional

scientific explanation of why conditions on Earth are so ideally suited for

life: "The many universes theory does provide an explanation for why many

things around us are the way they are. Just as we can explain why we are

living on a planet near a stable star by pointing out that only in such

locations can life form, so we can perhaps explain many of the more general

features of the universe by this anthropic selection process."

As Michael Behe points out in The Edge of Evolution [Behe

2007], however, anthropic selection only claims to explain why we

live in a universe which can support life, it does not explain why

we live in such a "lush" universe, where the fundamental laws

of physics not only make life possible, they make it interesting.

For example, some of the heavier chemical elements (for example,

iron, copper or uranium), which are probably not vital for life

itself, have played a critical role in the progress of science and

technology, and the existence and useful chemical properties of

these elements can also be traced to lucky features of our physical

laws.

A related argument is now being made regarding the "privi-

leged" position of our planet. It is well-documented that the con-

ditions on Earth are very "fine-tuned" for the development of life:

our planet is the right size, with the right kind of atmosphere, it

circles the right kind of star, it is the right distance from this star,

to name only the most obvious. Of course, there are many stars,

so it has always been argued that there were bound to be some

planets in this huge universe with the conditions needed for life.

(In fact, there is now some doubt about this, as the known list of

privileges enjoyed by our planet continues to increase.) But now

some scientists, such as astronomer Guillermo Gonzalez [Gonzalez

and Richards 2004], argue that our planet enjoys other "privileges"

which are rare in the universe, which have nothing to do with sur-

vival, but seem to give us an ideal platform from which to view

the universe.

According to the picture drawn by the popular media, primitive

man attributed many phenomena in Nature to design, but science

has progressively removed the need for the design hypothesis from

these phenomena one by one, and now a group of religious fanatics

is trying to make a last stand in biological origins, where things

are most difficult to explain. The true picture is very different;

in fact, we are discovering that primitive man was not wrong in

attributing many natural phenomena to design, the design just

2.2. APPENDIX: THE INVERSE SQUARE LAW OF GRAVITY 21

dates back much farther than he imagined, to the origin of the

universe. And of course all of the arguments in this chapter take for

granted that once the right conditions to support life are present,

life can spontaneously develop, an assumption for which there is

absolutely no supporting evidence. As Richard Dawkins famously

admitted in the movie Expelled, no one really has any idea how

life could have originated.

It is difficult to argue with those who appeal to "anthropic selection" to

explain improbable circumstances; about all you can say is that there is a

simpler explanation. But other universes are by definition beyond

observation, so that the anthropic principle is untestable, and therefore

unscientific. It is interesting to see how those who for many years have

criticized the creationists for inventing a force external to our universe

to account for the appearance of man are now reduced to inventing other

universes to explain our existence.

Fred Heeren [Heeren 1995] illustrates the silliness of the idea that, given enough

universes, everything will eventually happen. If there are an infinite

number of universes, he says, one of them would be just like ours except

that in that one Elvis Presley kicked his drug habit, got involved in

Tennessee politics, and became president of the United States. It seems

much simpler to believe that there is only one universe, and it appears to

be cleverly designed because it is cleverly designed.

Appendix: The Inverse Square Law of Gravity

In our universe, the force of gravity between two bodies of masses

M and m, at a distance r apart, is given by F = −GMm/r², where

G is the gravitational constant. We have already mentioned that

if we play with the constant G and make it a tiny bit larger or

smaller (by just 0.1%), terrible things would happen that make it

impossible for our universe to support life, but it requires some

advanced physics to show this. However, it requires only a little

physics, and a little calculus, to see what would happen if we play

with the r² term in the denominator, so let's do this. The results

are not nearly as striking, but this is one of the few examples of

"fine-tuning" that can be understood without a lot of advanced

physics.

So let us replace our inverse square law of gravity by F =

−GMm/rⁿ, where n may be something other than 2, and let us

look at the orbit of the Earth around the sun. If the position of our

sun, of mass M, is taken to be fixed at the origin, and the position

of Earth, of mass m, is given by (x(t), y(t), z(t)), Newton's second

law says

m(x′′, y′′, z′′) = −GMm/rⁿ (x/r, y/r, z/r)

that is, the mass of the Earth times its acceleration vector is equal

to the force of gravity on the Earth, which is a vector of magnitude

GMm/rⁿ in the direction of the unit vector −(x/r, y/r , z/r), ie, toward

the sun.

Since orbits will remain in the plane they start in, and we can

take the z axis to be normal to this plane, we can express the

Earth's position using polar coordinates as x(t) = r sin(θ), y(t) =

r sin(θ), z(t) = 0. Now, after a bit of work, the above differential

equations can be written in polar coordinates as:

r′′ − r(θ′)² = −GM/rⁿ

2r′ θ′ + r θ ′′ = 0

The second equation, after multiplying through by r, is equiva-

lent to (r² θ′)′ = 0, which means r² θ′ = c, a constant. Substituting

θ′ = c/r² into the first equation, we get a differential equation for

r(t):

r′′ = c²/r³ − GM/rⁿ (2.1)

From equation (2.1) we can see why orbits can be stable in an

inverse-square force field: if n = 2, then when r gets too small,

the positive term (due to the centrifugal force) dominates, and the

radial acceleration is positive, which tends to increase r. When r

gets too large, the negative term (due to gravity) dominates, and

the radial acceleration is back toward the sun. But what if we

increase n, to 3? Now, r′′ = (c² − GM)/r³ and if c² − GM is

positive the acceleration will always be positive, and the Earth

would spiral away out of the solar system; if c² − GM is negative,

the Earth would spiral into the sun. Neither outcome would be

very healthy for life on Earth! If n is even larger than 3, the

negative term in (2.1) dominates when r is small, and the positive

term dominates when r is large, so that all orbits of all planets are

again unstable.

Orbits can still be stable if we decrease n, to 1. But now the po-

tential energy for an object of mass m, associated with the Earth's

gravitational field, would be p(r) = GMm ln(r), where M is the

mass of the Earth. Note that the potential energy at r = ∞ would

be infinite (in an inverse square field it is finite). This means there

would be no theoretical limit to the energy with which a meteor

or other object of a given mass could hit the Earth. Exploration

of deep space would also be difficult, obviously.

Now one could argue that it is only natural that in our three-

dimensional universe, gravity would obey an inverse-square law. In

an N-dimensional universe, the energy from a source (e.g., the sun)

is spread out, at a distance r, over an "area" of size proportional

to 1/r^N−1, so its intensity is proportional to 1/r^N−1. Thus it

seems reasonable that the effect of the sun's gravity would also

die out at this rate as we move away from it. But in that case,

we can say we are lucky we live in a three-dimensional universe,

because if N were 4 or more, gravity would obey an inverse cube

(or worse) law, and since orbits would still be planar (a planet

would remain in the 2D plane spanned by its initial position and

velocity vectors), the above polar coordinate analysis is still valid

and shows that all orbits would be unstable in universes of more

than three dimensions. And who wants to live in a 1D or 2D

universe, where all we could see would be points or lines!