Entropy and the Universe

Contents

General Introduction

Introduction to Research and Analysis

Introducing Entropy

  1. What is Entropy?
  2. Specific Entropy
  3. Entropy and Gravity
  4. Conclusions about Entropy

Evidence about the Universe

  1. Evidence for Models of the Universe
  2. What is the Red-shift?
  3. The First Law and the Red-shift

The Big Bang Theory

  1. What is the Big Bang?
  2. The Nature of the Big Bang
  3. The Entropy of a Closed Universe
  4. The Entropy of Open and Flat Universes

Variations of the Big Bang Theory

  1. What is an Oscillating Universe?
  2. The Entropy of an Oscillating Universe
  3. The Cosmological Constant
  4. What is Inflation?
  5. The Entropy of Inflation

Other theories of the Universe

  1. The Steady State
  2. Religeous Theories
  3. The Positive Alternative

Conclusion and Evaluation

Bibliography

General Introduction

"Publish and be damned!" A. Wellesley

When I was seventeen, I had to do a project on entropy as part of my A-level physics course. This report was the result, and through dint of getting carried away with the research and getting about three times as much information as I could possibly use, gave me top marks.

After that, it got pushed away and forgotten, until several years later I rediscovered it when looking through old files on my computer. Despite the extra distance that time gives, it still seemed pretty correct and interesting, so I decided that it would be a good thing to put it on my website so that other people could read it as well.

I make no pretense that this report is groundbreaking, novel or indeed accurate; it merely summarises the position as I saw, from the information I could gather at that time. Nonetheless, I hope you enjoy it.

Introduction to Research and Analysis

"A beginning, a muddle, and an end." D. Larkin

This investigation deals with the relationship between entropy and the universe as a whole. In particular, emphasis is given to what entropy can tell us about the nature of the universe and to the role it plays in the models of the universe used by cosmologists.

For reasons of clarity, I split the project into five sections - 'Introducing Entropy', 'Evidence about the Universe', 'The Big Bang Theory', 'Variations of the Big Bang Theory', and 'Other Theories of the Universe'.

As no publications dealt directly with my topic, I used parts of many books. Reference to which books is providing which information on the same page would take up too much space within the text. Therefore I decided to include a page in the bibliography, linking the pages with a code for the books used. When I have included my own views or ideas in the project, I have tried to make that clear in the text. These ideas are also uncoded in the pages relating ideas and resource books.

Where books contradict each other I have, in order to save space, taken the view of the majority of the books, unless I believed there was a good reason to include the other view. For the sake of brevity, I have also missed out books which did not help me in my project, although I consulted them when looking for information.

I have avoided using excessive diagrams in this project, as when diagrams were included in my reference books, they rarely made things any clearer and usually confused the issue. In particular, the books usually only managed to give good illustrations of analogies with the universe, not of models of the universe itself - if it is hard to visualise the beginning of everything, it is even harder to give a good impression of it in a two dimensional picture. Instead, I have tried to keep my writing as concise as possible.

I hope you enjoy and learn as much from reading the project as I did researching it.

What is Entropy?

"If things look bad, I always think: 'It could be worse.' And sure enough, it gets worse." R. Asprin

The actions of anything, from the flight of an insect to the movements of the largest galaxies, are caused by energy changes. If we are to comprehend the universe, it is vital to understand energy and the rules governing it. In particular, it is vital to understand the laws of thermodynamics.

The first of these laws tells us there is a fixed amount of energy in the universe. Energy can be changed from one form to another, but never created or destroyed. Energy which appears to have disappeared has in fact been converted into a form we cannot detect - for example, sound energy seems to be lost, but really turns into minute quantities of heat.

Although energy cannot be destroyed, it is of little use to anyone if it cannot make things happen. Unfortunately, the second law of thermodynamics tells us all energy changes decrease the amount of useful energy in the universe.

Consider a box of small magnets. If the small magnets are lined up in the same direction, as a group they can attract other metal objects. If they are not lined up in the same direction, individual magnets cancel each other's effect and cannot do useful work. The same is true of energy - it is useful when it is ordered, but when it is disordered, its effects cancel each other out. For example, although the twelve men below all have the same strength, the 'ordered' six can push a lorry (useful work) and the other six cannot.

Six men pushing one lorry forwards and six other men pushing in opposite directions failing to move another

Entropy is a measure of the lack of order in the energy. There is no definite value of entropy for a given system (as there is for, say, mass), as entropy is a purely statistical measure. When there is zero entropy, all the energy can be used. As the entropy increases, available energy decreases until, with maximum entropy, no useful energy is available.

Continuing with the concept of the magnets, imagine you must move them to a different box in order to use them. As you are moving them, you may put some into the new box the wrong way round - the useful energy will then have decreased. Of course, the slower and more carefully you make the exchange, the fewer mistakes you will make. The same is true of energy - the entropy in the system always increases, unless the rate of change is infinitesimally small.

But why doesn't entropy (the disorder) decrease? What prevents only those magnets facing the wrong way being turned round? This could happen in two ways -

  1. The first possibility is that someone decides to increase the order in the system. However, as anyone tries to order the system, that person is doing work - and so the system's entropy decrease would be balanced by a hefty increase in that person's entropy. Thus entropy would increase on the whole.
  2. Otherwise it could happen by chance. This is very unlikely because, for both magnets and energy, there are a lot more ways in which things can be disordered than ways in which they can be ordered. This means it is practically impossible an ordered arrangement will appear by accident and practically certain any ordered arrangement will become less ordered. With one hundred magnets, it is more likely that you win the national lottery jackpot four times in a row than that they all point the same way by chance. With the many millions of atoms in any system, for all intents and purposes, entropy will never decrease.

All systems, therefore, tend towards a state with maximum entropy. In most cases heat is the energy form with most entropy, so all energy tends to become heat. As a heat difference has some order (the heat flows in one direction, which can be used to do work), any heat differences will decrease. Thus an object with maximum entropy is completely homogenous (same throughout) in terms of temperature and has no energy but heat.

The rest of this project grapples with the remaining conundrums of entropy - namely, why was the universe so ordered in the first place, how much (or how little) order is left in the universe, and where does the universe go on from here?

Specific Entropy

"There is a measure in all things made." R. Kipling

So far I have discussed the fact that, although the energy in the universe remains constant, less and less of it can be used to do work, as entropy increases. Though the entropy of a small system is easy to calculate (compare the 'useful' energy to the heat energy within it), measuring entropy on the scale of the universe presents many problems -

  1. We cannot physically measure the 'useful' energy or the heat energy even in nearby stars, let alone the rest of the universe.
  2. Although we can roughly infer the heat energy of a region of space, due to its radiation, we cannot measure the level of heat accurately.
  3. Even if we could measure a region's entropy accurately, we couldn't scale it up to the size of the universe. We know neither if that region of space reflects the universe's entropy as a whole, nor the size of the universe to scale it up to.

The specific entropy of the universe helps solve these problems. Before I define it, I will explain how the idea was developed, and in so doing, why it makes sense.

All hot objects tend to give out electromagnetic radiation (light) - the hotter the object is, the more photons (units of radiation) are given off. As heat increases anywhere, so the number of photons will increase. Therefore photon numbers are a good indicator of the amount of entropy. However, counting them all is impossible. Instead we can count their numbers in a given volume of space. As photons travel very fast and move through a vacuum, the number of photons is likely to be fairly uniform throughout space.

This idea runs into problems when we realise that in most models of the universe space is not static but expanding. Even if the number of photons in the universe stays the same, their numbers in a fixed volume would vary - we need to count the numbers of photons relative to something which remains constant. The proton (a fundamental particle found in the centre of atoms) was chosen for this task. The photon to proton ratio is called the specific entropy and is used to approximate the true entropy of the universe.

The specific entropy is an important tool in understanding the universe, because it gives a consistant, accurately defined and easily measurable idea of the universe's entropy.

Entropy of Gravity

"Make no mistake - gravity really gets me down." Anon.

Looking up to the heavens, one can see millions upon millions of stars, every one of which is seemingly refuting the second law of thermodynamnics. The universe should be on the way to become a homogenous mass of matter with a uniform temperature, but instead stars form, which are so hot and massive that they are seen millions of light-years away.

However, none of the books I have read give a good explanation of this problem. They either ignore the problem or toy (briefly) with the idea that the universe 'prefers' to be gravitationally dimpled. To clarify, gravitationally dimpled means space-time is distorted a lot. The laws of relativity state that mass distorts space-time, the larger the mass, the greater the distortion. Therefore, to dimple the universe matter must move together - the explanation is no more than the original problem formulated in a different way.

Two balls dimpling space time varying amounts

In resolving this problem, I was initially side-tracked into wondering how gravitational dimpling could effect entropy in any case and came to the conclusion it could not. By generalising the problem, though, I worked out a good reason why gravity works.

If gravity contradicts entropy, then surely all other forces should also do so. Electromagnetism (the force responsible for the effects of magnets and electricity) is probably the best known of these other forces, so let's consider this. When an electromagnetic force is used (say a magnet picks up a paper clip), it does so by exchanging a force particle. In this case, that particle is a photon.

Now let's think back to the specific entropy - the ratio of photons to protons. When electromagnetism is used, photons are produced, so the specific entropy must rise. Yet what is true of one force should be true of them all - there should be an entropy rise due to an increase in any force carrying particles. Specific entropy therefore should be the ratio of force carrying particles to protons - the photon to proton ratio was an over-simplification. As any force acts, the specific entropy rises, and so does the universe's entropy.

Yet there are problems with this view of gravity, indeed with the entire question I was trying to answer. The problem lies in accepting the books' claim that star formation causes an entropy contradiction, because matter is getting more organised. This is clearly irrelevant, because the laws of thermodynamics deal purely with energy change. The second law definitely allows star formation - gravitational energy becomes energy with greater entropy (usually heat) as the star forms and the excess heat is radiated away as light to decrease the heat difference created.

The only reason for my argument was to reconcile star formation with entropy. Even if the argument was correct, it was clearly superfluous. However, I have included it as it is elegant and could have been relevant. Moreover, it is important to show how reasonable sounding arguments can arise from mistaken assumptions, which is always worth keeping in mind when considering the many conflicting views of the universe's nature.

The arrangement of matter has really no bearing on the energy changes and thus entropy. Even where matter seems to exert a force to create energy changes, it is not caused by the matter, but by the energy in the matter. Pressure, for example, can be used to do work, but is caused by the kinetic energy of the particles, not the particles themselves.

On the same line, I would like to correct one illustrative example which is given in most books about thermodynamics. They claim entropy rises when two gases are mixed. In the light of what has been said, this is incorrect. Although Boltzmann did his initial experiments on gases and extrapolated the principles he learnt from them when formulating the second law, mixing gases is not an example of the second law itself. Rather, it shows the statistical processes involved in governing the second law, and the books should claim only this.

The insights gained in both gravity and entropy are clearly important. In particular, giving gravity a good theoretical basis is vital, as gravity is the only force strong enough to act between stars and so hold the universe together.

Conclusions about entropy

"The answer to life, the universe and everything is ... 42." D. Adams

We have discussed what specific entropy is and how gravitational entropy works, but we have not explained what entropy actually tells us about the universe we live in.

The conventional view is that we live in a low entropy universe. This idea is supported by the fact that our planet hosts complex life, the stars are pouring out energy, in fact all the visible universe is fantastically intricate. As entropy increases, more and more energy will become unavailable heat, stars will burn up and be swallowed up by black holes; the universe as we know it will die a heat death.

Specific entropy, on the other hand, gives a vastly different view. Over its life span, the sun will produce an increase of about 106 photons for each proton within it. The universe at present has 1010 photons per proton. This is ten thousand times as much specific entropy as the entire sun has or will ever produce. When we compare the amount of entropy that is produced now to the vast quantities which were produced in the past, we must conclude (in all but the smallest details) that the 'heat death' has already occurred.

By counting the abundance of different elements in our sun, scientists believe there can only have been (at most) two or three stars before the current set of stars. This indicates that the entropy increase today is only marginal, compared to what it was in the past, and that entropy was increased by a different mechanism then compared to now.

However, the question remains - whatever state of entropy we have at present, the entropy of a black hole is much greater, so why isn't there a black hole here? Although they take time to form, there have been at least 15 million years available (current estimate according to the big bang theory, though estimates vary from a thousand years to infinitely long). The best explanation is the anthropic principle. If there was a black hole here, no humans would be there to question its absence - so there can't be a black hole.

The last question could only be answered within the framework of a universe model (in this case the big bang theory). Few further conclusions can be drawn from entropy alone without having an idea of how the universe has evolved. For this reason, we will now look at how entropy works in the context of different models of the universe.

Evidence for Models of the Universe

"No evidence can be read less than three ways." A. C. Doyle

A good model of the universe has to explain a number of physical effects. I do not want to dwell on these, yet I will briefly mention all the main effects and how the big bang theory explains them, as this is the most accepted theory.

The red-shift is the most important physical evidence, and is discussed in greater depth further on. In essence, it shows us that all the stars in the sky are moving away from us. This implies that all stars are receding from each other, so the universe was denser in the past than it is now. This laid the groundwork for models showing an expanding universe.

The infra-red background radiation is uniform heat radiation found everywhere in space. The big bang theory states it is the light from the big bang red-shifted to a fantastic extent.

The helium abundance has its origin in nuclear synthesis (formation of the elements in the stars). Stars create all elements naturally during their lifetime, in the proportions that are close to those existing in the universe (which in turn suggests they formed these elements), but there is too large a proportion of helium, lithium and other light elements. The big bang theory claims these were formed in the heat of the big bang.

Additionally, a good model of the universe should have several particular features. It should ideally only use existing and experimentally proven physical laws (there is no point in creating new laws that may or may not exist). The big bang theory is in accordance with this, in that it logically follows on from Einstein's theory of relativity and a small number of assumptions. Secondly, the theory behind a good model should not contravene any accepted fundamental physical laws (a model, for example, which demands something moving faster than light-speed would be suspect). Finally, it should be capable of making predictions which can then be tested (it is all very well to theorise about the moon being made of green cheese, but if the theory cannot be proven or disproven, it is useless).

Although I so far have only mentioned how the big bang model would explain these effects, many comprehensive models of the universe explain most if not all of them equally well. However, as the most important factor in understanding the universe is without doubt the red-shift, I feel I should spend a little time exploring it further.

What is the Red-shift?

"Nor dim nor red, like god's own head, the glorious sun uprist." S. T. Coleridge

The red-shift is an effect which all theories of the universe's evolution must account for. I have therefore decided to devote a little space to investigate it.

Imagine you are in a boat on a lake. Every five seconds you splash the water to produce a ripple. An observer at the lake's edge would notice a ripple coming from you every five seconds. However, if you are moving, the observer would not necessarily see a ripple every five seconds. If you are moving towards the observer, each new ripple would have to travel a shorter distance than the one preceding it. As this will take less time, the frequency of the ripples' arrival would increase. If you are moving away, each ripple must travel further than the ripple preceding it. This takes longer and the frequency decreases.

A picture of eyes observing 'ripples' of light

Imagine now that the boat is a star, the lake space, and the observer earth. Instead of the person regularly creating ripples, imagine the hydrogen in the star giving off particular frequencies of light - its emission spectrum. If the star is at rest relative to earth, that is what we would expect to see from hydrogen. If, however, the star is moving, the frequency of the light we observe is different. If the star comes towards earth, the frequency becomes higher, but it decreases if the star is moving away. As blue has a high and red a low frequency, these changes are called blue-shifts and red- shifts respectively. As the emission spectrum of hydrogen is the same throughout the universe, an observer on earth can work out the speed and direction the stars are travelling relative to us. Scientists have found that all the galaxies in the universe are rushing away from each other - the universe is expanding.

Yet the red-shift violates the first law of thermodynamics. We must resolve this problem.

The First Law and the Red-shift

"Make the green one red." W. Shakespeare

The red-shift of light seemingly contradicts one of the most fundamental laws of science - the first law of thermodynamics. The energy of light is dependant on its frequency - the higher the frequency, the higher the energy. This means energy is created when light is blue-shifted, and destroyed when light is red-shifted. I decided to try to find a plausible way this could happen, and came up with the following theory...

Photons have a mass and travel at the same speed relative to all objects. This means, when colliding with an object, they always give it the same momentum increase, whatever its speed. The kinetic energy gained, however is not constant. Objects moving quickly away from the photon source gain more kinetic energy, objects moving towards the source loose kinetic energy and the change in kinetic energy is hardly noticeable to objects remaining still. My hypothesis is that the kinetic energy gained or lost corresponds to that lost or gained in the red- or blue-shift of that photon.

Working through the problem mathematically, first we must define terms;

General

c = speed of light
h = Planck's constant
R(v) = red-shift for a given velocity

Photon

F = frequency
E = energy
P = momentum

Object hit

e = energy
p = momentum
v = velocity

Modifiers

means before red-shift
means after red-shift

normal typeface means before collision
bold typeface means after collision

Then we must decide what formulae we need to use;

F = F x R(v) (1)
Kinetic Energy = mv(2)
p = mv = mv (3)
E = hF (4)
p = E/c (5)
R(v) = (1- v/c) / (1-v/c) (6)

Thus we get the following.

p = hF/c (7) (from 4, 5)
E = hF (8)
P = hF/c (9) (from 1, 4, 7)
E = hF = (hF)(R(v)) (10)
P = hF/c = (hF/c)(R(v)) (11)
e = mv (12)
p = mv (13) (from 2, 3)
e = 0 (14)
p = 0 (15)
p = mv + (hF/c) (16)
p = (hF/c)(R(v)) (17) (from 3, 8, 9, 13, 14)
v = v + (hF/c)(1/m) (18)
v = (hF/c)(R(v))(1/m) (19) (from 3, 16, 17)
e = m (v + (hF/c)(1/m)) (20)
e = m ((hF/c)(R(v))(1/m)) (21) (from 2, 18, 19)
e - e = m (v + (hF/c)(1/m)) - mv = vhF/c + (hF/2mc) (22) (from 12, 14, 20, 21)
e - e = (hF/c)(R(v))(1/2m) - 0 = R(v)hF/2mc (23)

Combining (22) and (23) shows us the difference in kinetic energy viewed with respect to the object, and the object emitting the photon. This is;

(e - e) - (e - e) = vhF/c + (hF/2mc) - R(v)hF/2mc (24)

and the energy lost or gained by the photon in the red-shift is;

E - E = hF - hF R(v) (25)

These should be the same, so combining (22) and (23);

vhF/c + (hF/2mc) - R(v)hF/2mc = hF - hF R(v)
v/c + (hF/2mc) - R(v)hF/2mc = 1 - R(v) (remove hF)
1 - v/c + (hF/2mc)(1 - R(v)) = R(v) (rearranging)

As h/c 0 and (1 - R(v)) 0 at low speeds, we can assume (hF/2mc)(1 - R(v)) 0. So;

1 - v/c R(v); therefore; 1 - v/c (1- v/c) / (1-v/c)

As this is evidently true, the energy change in the red-shift equals the change in kinetic energy relative to the photon source. This vindicates my hypothesis that the two are the same, although more work is required to prove the matter beyond all reasonable doubt.

What is the Big Bang?

"Down with the Big Bang." Editorial title, Nature, 1989

At present, the big bang and its countless variations are the dominant theories in cosmology. For this reason, when examining entropy and the universe, the big bang theory is clearly the point to start. The 'standard model' is the simplest version.

According to the standard model, the universe was born in a massive explosion about 15 thousand million years ago. Time, space and matter were all created in an infinitely hot and dense fireball. Cosmologists still do not know what happened at the actual moment of the big bang. The closer to the big bang they investigate, the hotter and denser the matter is and the more tentative our knowledge of the particle physics involved at that stage becomes. What happens after the big bang is easier to work out.

As the universe expanded, the matter within it became cooler and less dense. The temperature dropped from 1011 K at the time of one hundredth of a second after the big bang to 1010 K one second later and then to 109 K three minutes after the big bang. About 700,000 years later, temperatures had dropped enough to let stable atoms form. Fluctuations from place to place in the density or expansion rate of the smooth and featureless universe created slight matter imbalances, which gave rise to the first stars.

Although the universe has been expanding ever since the big bang, it will not necessarily always continue expanding, because gravity is pulling everything back together. The fate of the universe depends on how strong this retarding force is. There are three possibilities.

  1. The universe may expand forever, because the gravitational force is not strong enough to halt the expansion. This is called an open universe and is judged the most likely possibility by observational evidence.
  2. The universe may be balanced exactly between the two forces - the universe may just stop expanding in (infinite) time, but never contract. This is a flat universe.
  3. Gravity may be strong enough to reverse the expansion and the universe will collapse back to a singularity. This is called a closed universe.

First, I will consider the nature of a big bang universe, then think about entropy and its effect in open, flat or closed universes.

The Nature of a Big Bang Universe

"Nature is usually wrong." J. Whistler

The nature of a big bang universe is a description of how its space-time is warped overall. It depends on what 'shape' the universe is (open, flat or closed) and tells us whether the universe is finite of infinite.

If the universe is closed, everything must collapse back into a singularity. As this includes light, no light can escape the universe's gravitational pull. As space is defined as somewhere which has the potential for matter or energy to be within it, this means the universe must have a limited volume - it must be spatially finite. As space-time has no boundaries, this means the universe is curved in on itself, like the surface of a sphere.

If the universe is flat, then space-time is also said to be flat, and the universe is infinite in size. To my mind this raises a contradiction. The larger an object is, the less dense it needs to be to have the same gravitational attraction as a smaller object. An infinitely large object would need an infinitesimally low density to have enough gravitational attraction to stop light escaping. As the observable universe has a density larger than zero and the universe as a whole is homogenous (a key assumption in the big bang theory), an infinite universe would immediately become a black hole. As a black hole is technically the same as a finite universe, even if an infinite universe existed in the past, it now would be finite.

If the universe is open, even greater questions arise, as space-time is said to be warped 'like a saddle'. Apart from suffering from the same contradiction as the flat universe, there is no explanation why it should be shaped in this way, or what shapes it. In any case, as I see it, there is no reason why an open universe should not be spatially finite, but still expand forever.

Pictures of spherical, planar and hyperbolic geometries

Although in my opinion the idea of an infinite universe makes little sense, I will include it when considering entropy in a big bang universe, in case my argument is flawed.

The Entropy of a Closed Universe

"It is not the end... but it is, perhaps, the beginning of the end" W. Churchill

If the universe contracts, no amount of technology will avert the impending doom, because, although life may continue until about a billion years before the end, all matter will then vapourise. Fortunately for civilisation the universe is unlikely to contract; quite apart from the fact that astronomers can only find 1% of the matter required to make the universe a 'closed' one, the laws of thermodynamics give a number of reason why a closed universe cannot exist.

The first problem arises when asking why the big bang happened at all, indeed why the universe is expanding. It has been shown that it is favourable in terms of entropy that matter falls together to form stars and galaxies. So why did the largest concentration of matter ever decrease its entropy massively by exploding outwards? Furthermore, how did it do it? Not even light can escape a singularity, and even if all the particles, by some stroke of luck, escaped, then gravity would pull them back into a singularity instantly.

The second contradiction arises when considering what happens between the big bang and the big crunch (the singularity the universe will become). As both singularities have the same entropy, any decrease in entropy (like the sun giving off light) must be matched by an increase in entropy at some other time - a clear violation of the second law. There have been a few attempts to get round this problem.

  1. As the universe contracts, the second law of thermodynamics reverses, so then entropy always decreases. With a little thought, this is clearly wrong. As the law of thermodynamics reverses, it no longer becomes favourable for objects to fall together, so the law of gravity reverses. Thus the universe no longer recontracts. The only way to resolve this paradox is to say that entropy does not reverse.
  2. The second singularity is more disordered than the first, so there is no entropy contradiction. Unfortunately, a property of black holes is that all black holes of a given mass, rotation etc. are indistinguishable. So to be more disordered, the second singularity needs to have gained mass or kinetic energy, all of which is forbidden by the first law of thermodynamics.

Luckily for the big bang theory generally, the open universe model has far fewer problems.

The Entropy of Open and Flat Universes

"This is the way the world ends - not with a bang but a whimper." T. S. Eliot

The models of the open and the flat universe (which is a special case of an open universe) seem most likely to be correct, both in terms of observation and thermodynamics. I will first consider implications for the future and then examine the problems with the theory.

As the universe ages, all stars will use up their nuclear fuel and die, left as white dwarfs or black holes. The black holes will coalesce (join), becoming larger and eventually dominating the universe. It may be hundreds of billions of years, though, before life becomes impossible. We might have to move to a different star or galaxy, as ours would lose available energy, but we could survive. Problems with the theory start to emerge, though, when looking back into the past.

If the universe is finite, an open universe has the same problem of escaping from a singularity in the big bang as a closed universe. If the universe is infinite, however, there are even more problems, as all of space would not have fitted into a singularity. In other words, space would already have been in existence when the big bang happened, and the big bang caused all the particles to move away from each other.

However, the big bang is supposed to account for the entire creation of the universe. Yet a fledgling universe must exist first to be expanded - so where does this universe come from? Furthermore, for the big bang to occur simultaneously in all of the (infinite) universe, a signal must be transmitted instantaneously, violating the laws of physics governing the speed of information flow.

On the whole, the theory of the big bang creating an open universe is a successful one. It even explains one of the conundrums of thermodynamics - why the universe is ordered enough to degenerate into disorder. The universe was highly ordered at the beginning, because of the large amounts of gravitational energy it had following the initial explosion. Since then, this energy has been converted into higher entropy forms of energy, and entropy has consequentially increased.

In the next section, I will explore the variations of the big bang theory, to see whether they explain entropy better than the standard model.

What is an Oscillating Universe?

"For dust thou art, and unto dust shall return." Bible (Genesis)

The model of an oscillating universe is an extension of the closed big bang theory. Instead of the universe simply ending at the big crunch, it bounces back into a new big bang - our universe is just one in a infinite line of universes. There is no real way to verify this theory - as far as humans are concerned we can detect only this universe. However, the theory solves some of the problems plaguing the closed big bang model.

Firstly, the problems of what caused the big bang and how matter escaped from the singularity are resolved. The previous big crunch sparked the big bang off and matter had no problem in escaping the singularity because there was no singularity to escape - the bounce mechanism headed the matter off before it could form one. In addition, there is no problem about how matter was created; it has always existed, as has energy.

The oscillating universe theory also solves the problem of the universe not being able to become more complex from singularity to singularity. As no singularities form, the entropy can increase and be passed on to the next universe.

The main problem is therefore to find a plausable bounce mechanism. Ideas are endless, but theoretical work carried out in Cambridge has proved that none of the current models are correct. However, until we have a working knowledge of all the fundamnetal laws, the fact that such a mechanism exists cannot be completely ruled out.

Ideas for what the bounce mechanism could be include: The idea that gravity becomes repulsive at high densities or when quantum effects are taken into account. That singularities twist themselves out of space-time to start a new big bang. That a false vacuum repells matter which forms a big bang (see inflation). That the energy released when matter annihilates with anti-matter causes the big bang. That all the galaxies just miss each other and fly back into space. And so on; the list is very long.

Next, I will investigate how the entropy of an oscillating universe works and how this has to take account of the fact that the universe's volume is always changing.

The Entropy of an Oscillating Universe

"Beginning is often the end." T. S. Eliot

Any finite universe (as Newton pointed out) is unstable and must continue expanding or contracting. However, thermodynamic equilibrium cannot be established as long as the universe keeps oscillating so, by definition, it always increases in entropy.

As the entropy increases with every oscillation, the specific entropy must increase with it. As light has a stronger gravitational attraction than the equivalent energy tied up in matter, the gravitational attraction increases as entropy increases. This means the universe is pulled together harder and faster than before. The bounce mechanism conserves this force, so the next big bang is a little larger, thus the next cycle is bigger and longer.

As entropy rises infinitely, this implies one of two things. Either, there is a maximum entropy, coupled with a rate of entropy increase which is always changing as to never quite reaching the maximum value, or the maximum entropy does not remain constant: (as entropy increases, so does the maximum entropy, the level which the entropy of a given universe is tending towards.)

However, there are problems with this oscillating universe model. Firstly, if the specific entropy is always increasing, and infinite cycles have already gone by, where are all the photons? There should be an infinite amount of them. Similarly, if black holes form, they would not be destroyed, so after infinite cycles, we should easily be able detect them - which we cannot - which provides further evidence that this is not the correct model.

And when all has been said and done, to break down the elements formed in previous oscillations to hydrogen, the temperature during the bounce must reach 10 million degrees Kelvin. A temperature this extreme would destroy all information in the universe, so the same universe hardly endures, nor does one cycle really follow another - if nothing can pass from one universe to the next, what difference does it make to us if they exist or not?

In the next section I will turn my attention to the current view of how the universe was created; to Einstein's cosmological constant and to inflation.

The Cosmological Constant

"When I have an idea, I stop to savour it, for it is bound to be wrong." G. Wald

When Einstein had completed his theory of relativity, the dominant view of the world was that the universe is static and unchanging. To reconcile his equations with this view, he added the 'cosmological constant' to his equations - a move he was later to regard as "my greatest blunder."

The cosmological constant is the counterpart of gravity. It exerts a repulsive effect on distant stars proportional to the square of their distance. Einstein was convinced that this would counterbalance gravity, allowing the universe to remain unchanging. Unfortunately, there were two fundamental flaws. The first was mathematical; Einstein had devided both sides of his equation by a constant which could possibly be zero - if it turned out to be zero, the model would automatically be invalidated. Secondly the universe was highly unstable. If it expanded or contracted slightly, one of the forces would be greater than the other and force the universe away from equilibrium, either to a singularity or to infinity.

Furthermore, the cosmological constant comes into conflict with observational evidence of the universe - no red-shifts were predicted, there is no explanation for the background radiation. In terms of thermodynamics, too, the model was lacking. If the universe remained static infinitely, then it should have reached thermodynamic equilibrium by now - there would be no available energy and we would not exist.

However, Einstein's speculations convinced the public, because the idea was fundamentally attractive - a spherical space-time, static and eternally unchanging. Scientists, however, spotted the flaws in Einstein's argument, detailed above. In particular, Lamaitre showed that the universe model was wrong and that the universe was either expanding or contracting. As a response to Lamaitre's work, Einstein rejected the cosmological constant in his equations.

Ironically, Lamaitre and other cosmologists did not drop the cosmological constant so readily. It was convenient, and they could use it to slow down or speed up the universe's evolution as they pleased. However, the cosmological constant only really seized the limelight with the invention of inflation, as vacuum energy in inflation is formally equivalent to the repulsion force. We will investigate the inflation idea next.

What is Inflation?

"A bigger bang for your buck." Anon.

In 1978, a modification of the big bang model called 'inflation' was proposed, which solved some of its problems. I will not mention these problems, because I want to concentrate on the entropy of the models.

According to the theory of inflation, shortly after the big bang the universe went through a period in which it expanded massively, after which it returned to a more leisurely pace of expansion. The rapid expansion epoch took a portion of space the size of a grapefruit and expanded it to a size bigger than the observable universe, after which the energy powering the inflation was lost as a big release of particles with mass. Calculations show the mass released is exactly that required for a closed universe model.

This theory appears to have a few contradictions. In particular, to expand this fast, objects must have been moving faster than the speed of light. This objection is resolved in that, although objects in space cannot travel faster than the speed of light, space itself can expand this fast, carrying the objects with it.

Secondly, there is no proven mechanism for creating this mass expansion. Einstein's corrective force was used initially, but this would still be expanding the universe now. A concept called false vacuum has been dreamt up to explain the effect. In essence, when the big bang took place, there was only one type of superforce. As the universe grew, this split into the four forces we have today - gravity, electromagnetism and the strong and weak nuclear forces. The energy released during this split drove inflation. Unfortunately, until we can perform experiments at 1028K there is no way in which to prove the theory as either correct or incorrect.

Finally, there is no observational evidence for inflation. One of the few claims it makes - that the universe is a closed one - is flatly contradicted by the evidence. Additionally, the problems that it solves can usually be solved in a much simpler way than by inventing a brand new theory of how the universe began.

However, it is still important to see how inflation fares when facing up to the laws of thermodynamics.

The Entropy of Inflation

"Mighty things from small beginnings grow" J. Dryden

The entropy of the inflation model suffers from many of the problems that beset the model of the closed big bang. Although it manages to resolve some of these problems it creates others as well.

The first effect it manages to explain is how the universe overcame its gravitational attraction. When the universe had moved away from the singularity sufficiently, inflation took over, stretching space and as such separating all the matter by a sufficient distance to prevent the universe falling back into a singularity immediately.

It has less luck solving the problem of the universe not being able to increase in entropy from big bang to big crunch. There are two possibilities.

  1. Energy is returned to the false vacuum at the big crunch. This clearly would not solve the problem at all, as there could be no entropy change from one singularity to the other - the contradiction still stands.
  2. Energy is not returned to the false vacuum. This poses a contradiction in terms of the first law of thermodynamics, as energy should not be able to be created out of nothing. It also asks questions of the big bang theory, namely where the energy came from in the first place, why the false vacuum existed at the time of the big bang, and what the difference between the two singularities, is to allow a false vacuum to form in one and not the other.

If the problems with the second option can be solved, then inflation works very neatly indeed, in terms of thermodynamics. Inflation created the highly ordered initial universe, by spreading matter out so far apart that non-uniformities became uniform. The energy of the negative gravity and positive matter cancel, so matter was formed without violating the first law, and the universe will become a true singularity in the big crunch, when it will have maximum entropy. In effect, inflation created and wound up the universe.

But there are more models of the universe than just variations of the big bang. I will investigate these now.

Steady State Theory

"More steady than an ebbing sea..." J. Ford

In 1948, some cosmologists who were unhappy with the big bang proposed a radical new model of the universe, based on the idea that the universe was not only homogenous in space (a fundamental concept in most models of the universe) but in time. This means, as the universe expands and galaxies separate, new galaxies form to fill the gaps created.

This theory appealed to many, as it had none of the problems associated with the big bang - what preceded it, what caused it, and so forth. Furthermore, it explained most of the effects that the big bang did - the red-shift is explained by the expansion, the background radiation is the light from former stars red-shifted. Furthermore, the theory requires only a minor modification in the laws of relativity, and the rate of matter creation is too low to be observable in any case.

Observational evidence proved to be the undoing of the steady state theory. The number of quasars increases when looking back into the past, violating the perfect cosmological principle. Furthermore, as time is infinite, another intelligent species should already have colonised the observable universe, which is obviously not the case.

In terms of thermodynamics, the steady state theory resolves some problems and generates others. It allows the universe to have existed for an infinite time, because as entropy increases, more matter appears which increases the order again - the new matter is an inexhaustible supply of negative entropy. In the same way, this balancing of entropy with new matter could be the very reason for space's expansion.

However, the steady state theory breaks both the first and second law of thermodynamics with the matter increases, although novel ways around the problem have been suggested - for example, the universe could be like a massive black hole, which absorbs extra matter when it expands. Even so, though, the matter increase is, to my mind, too small to balance the entropy increases caused by, for example, the stars burning.

The steady state theory flourished in the sixties, but today, few people believe it. In contrast, the theory I will outline next has been believed for millenniums - that god created the universe.

Entropy and God

"God is really only another artist." P. Picasso

If god is omnipotent, would he obey his own laws? Considering the laws of entropy, it seems he probably does not. I decided to take an irreligious look at religion.

An overriding feature of many creation stories is that chaos is overcome by god(s), which heralds the beginning of the universe. In other words, the gods' actions formed the universe, by decreasing its entropy enough to free up energy. Greek legends speak of Cronos (the father of the gods) overcoming chaos to found the universe, while the bible says that 'the earth was without form and void' before god created it. Other cultures limit their gods to creating the universe in a low entropy state. Madagascan legend tells us that Zanahary made earth but left it empty (Ratovoantany created everything on it), and Zulu myths say that Unkulunkulu evolved alone in emptiness before creating men from grass.

A further important feature of creation is the separation of land from sea - another entropy contradiction. The bible says that god parted them after he had separated heaven from earth, and New Zealand was allegedly pulled out of the ocean floor by Maui with a magic fishhook. In other religions, the gods emerged from the waters instead. The Egyptian god Re brought forth the first pair of gods as he emerged from the waters. In Babylon, pairs of gods rose to the surface as the waters of Abzu and Tiamat met.

Most, but not all gods, conserve entropy in that they allow only one universe, which degenerates from a perfect beginning (symbolised in the bible by the eviction of Adam and Eve from paradise). This is not true of Hinduism or Buddhism, in which creation ebbs and flows on a vast scale. In Hinduism, this is symbolised by the god Brahma sleeping and waking - each universe is a single dream.

The question most creation stories leave unanswered is the question of what will happen at the end of the universe, preferring to let it continue infinitely or indefinitely. Even when a definite end is mentioned, as in the bible, there is no definite date for the end - perhaps this is god's way of telling us that we have nothing to fear in the future.

Thanks to entropy, there is agreement that the universe must end. Some disagree, so I will look at the views of those who believe we have little to worry about concerning entropy.

The Positive View

"Always look at the bight side of life." M. Python

So far I have said that due to the constantly increasing entropy the universe as we know it is doomed. This is not the entire story - many scientists believe entropy can be overcome.

Firstly, many believe that entropy has been overgeneralised in applying it to the universe as a whole. On the basis of a few experiments done with modest containers of gases, people have leap-frogged into believing the universe is dying. For one thing gases are already highly disordered - on mixing solids, there is no entropy increase. And if oil and water are mixed, they separate causing an entropy decrease - so we shouldn't let these experiments determine all our assumptions. In addition, entropy ignores forces. If you have a plasma, the ions in it will form filaments due to magnetic attractions - again entropy has dropped.

Secondly, if entropy exists, it may exist in more than one form - disorder of matter, entropy in gravity, specific entropy (electromagnetic) and so on. Although entropy in all these forms tends to increase, we could use one form to decrease the entropy of another, and reverse the process to decrease the overall entropy, as shown below - thus the overall entropy problem is not insurmountable.

A picture using one type of entropy to decrease another type of entropy

An extension of this approach is the idea that different factors take over at different points of the universe's evolution - where one entropy cycle ends, another begins. Tracing the evolution of the universe so far, one can imagine it starting in a plasma like state with only Maxwell's laws of electromagnetism applying. The plasma formed filaments, which in time evolved to the filaments of stars seen in the universe today. These filaments created matter imbalances which allowed gravity to take over. The gravity compressed matter into stars, allowing nuclear forces to take over, and so on. When one entropy cycle ends, another more efficient one will take the universe further - the maximum entropy always increases.

A further approach is to go back to the magnet shuffling example. When will the magnets be completely disordered? After transferring them three times? Thirteen times? Thirty or three hundred times? With such a number of atoms as the universe has, the answer to when they would all be disordered is probably never - there is always somewhere that can get less ordered, so entropy always rises. This means that there is always some energy which can be harnessed and used to do work.

A similar approach considers the always changing size of the universe. If the universe ever reached thermodynamic equilibrium, the increase in the universe's size would disturb the equilibrium, and so entropy rises would continue.

A different approach tackles entropy from a new direction. It claims that although order is decreasing, the complexity (or organisation) of the remaining order is increasing. Compare, for example, a block of marble with a statue - the block of marble is bigger and has more order, but the statue is more complex. The continuation of life, from this point of view, does not depend on minimising entropy (although this comes into it) but maximising the complexity gained from it.

A quite novel approach is to disprove entropy altogether. If black holes form, they must increase entropy by doing so. Quantum physics demands that they release radiation. Losing radiation, they lose mass and therefore energy. This means that eventually black holes are destroyed, so entropy must have reversed. If the laws of thermodynamics are true, then this theory states that either quantum physics (the most reliable theory anyone has ever discovered) is false or black holes (and thus all variations of the big bang theory) don't exist. As the big bang model was built on the assumptions of entropy, and most scientists would back quantum physics rather than entropy, this theory would conclude that, one way or another, the laws of thermodynamics are wrong.

Finally, even if the universe should be destroyed, there would have to be a mechanism of sorts to destroy it. Since everything which exists, exists in this universe, the destruction mechanism must also be within the universe. So the mechanism would either destroy itself in the process, in which case the destruction would stop, or not destroy itself, in which case it would be left in the universe. Either way, there will always remain something of the universe.

Conclusion and Evaluation

"A beginning, a muddle, and an end." D. Larkin

Looking at the project, most conclusions are drawn in the relevant sections. However, one unmentioned conclusion is outstanding - what the most likely model of the universe is.

According to my investigation the most likely overall theory seems to be the big bang theory - all the other theories contain several entropy contradictions and need extra effects (like the creation of matter or a false vacuum) which have not been observed. In particular, the open universe model contains least entropy contradictions, and agrees with observational evidence - the only problem it needs to solve is how the gravitational attraction of the big bang singularity could have been overcome.

Whatever the model of the universe, the universe itself is already some way into the heat death. However, entropy is currently rising so slowly that mankind will probably never live long enough to suffer the loss of all available energy into heat.

Evaluating my completed project, on the whole I am pleased with what I have achieved. All the points I wanted to raise have been discussed, and I believe I have given a comprehensive view of how entropy considerations effect the models of the universe.

What I am less pleased with is the length of the essay. When starting the research and analysis project, I had a difficult time finding any information which was of use, as there were no books which dealt directly with the problem. In order to make sure I had enough resources, I borrowed most books with any information on the topic from the local and university's library system. By the time it came to write up the project, I found myself with too much material, and with the problem that I could not shorten the project without sacrificing important information or clarity. When I spoke to my teachers about the problem it was decided that clarity of the information mattered most.

For another research and analysis project I would choose a much more restricted topic (the universe is a little too large to cope with) and carefully define how much information is required. However, I am glad I chose the topic I did, because I was interested in learning about the beginning of the universe and how entropy effects it.

Bibliography

"You cannot open a book without learning something." Chinese Proverb

Which books information came from

Introducing Entropy

  1. What is Entropy? 5, 6, 9, 14, 16, 28, 30
  2. Specific Entropy 5, 11, 14, 16, 28
  3. Entropy and Gravity 10, 14, 17, 24, 28
  4. Conclusions about Entropy 12, 14, 16, 28

Evidence about the Universe

  1. Evidence for Models of the Universe 2, 3, 5, 8, 10, 12, 16, 20, 23, 24, 27, 29, 32
  2. What is the Red-shift? 2, 3, 5, 8, 10, 12, 16, 20, 23, 24, 27, 29
  3. The First Law and the Red-shift 13, 23, 24, 27

The Big Bang Theory

  1. What is the Big Bang? 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 16, 17, 18, 20, 21, 23, 24, 27, 29, 31, 32
  2. The Nature of the Big Bang 1, 2, 3, 8, 10, 11, 16, 17, 23, 25, 27, 32
  3. The Entropy of a Closed Universe 3, 6, 12, 16, 17, 27, 29, 31, Internet pages
  4. The Entropy of Open and Flat Universes 3, 6, 12, 16, 27, 29, 31

Variations of the Big Bang Theory

  1. What is an Oscillating Universe? 2, 3, 5, 10, 11, 12, 16, 17, 23, 24, 29, 32
  2. The Entropy of an Oscillating Universe 2, 3, 11, 12, 16, 17
  3. The Cosmological Constant 2, 3, 5, 16, 17, 23
  4. What is Inflation? 2, 3, 5, 10, 15, 16, 17, 23, 27, 29, 32
  5. The Entropy of Inflation 3, 5, 10, 16, 17

Other theories of the Universe

  1. The Steady State 2, 3, 6, 12, 16, 17, 18, 20, 23
  2. Religeous Theories 3, 4, 6, 8, 16, 17, 22
  3. The Positive Alternative 1, 10, 16, 17, 18, 23, 24, 26, 29

Books

  1. A Brief History of Time
    SW Hawking
  2. Afterglow of Creation
    Marcus Chown Published 1996, by University Science Books
  3. Ancient Light
    Alan Lightman
    Published 1991, by Harvard University Press
  4. Astronomy and the Bible
    Donald B Deyoung
    Published 1989, by Baker Book House Company
  5. Before the Beginning
    Martin Rees
  6. Black Holes - The End of the Universe?
    John Taylor
    Published 1973, by Souvenir Press Ltd.
  7. Black Holes and Quasars and other Mysteries
    Stan Joinler
  8. Cosmic Horizons
    Wagoner and Goldsmith
    Published 1982, by Stanford Alumini Association
  9. Dreams of a Final Theory
    Stephen Weinberg
  10. Einstein's Greatest Blunder
    Donald Goldsmith
    Published 1995, by Harvard University Press
  11. General Relativity - an Einstein Centenery Survey
    Edited by SW Hawking
  12. In Search of the Big Bang
    John Gribbin
    Published 1986, by Heinemann Books Ltd.
  13. Relativity for the Layman
    James Coleman
    Published 1954, by Penguin Books Ltd.
  14. Relativity, Thermodynamics and Cosmology
    Tolman
  15. Stars and Galaxies
    Robin Kerrod
    Published 1990, by Wayland (publishers) Ltd.
  16. The Accidental Universe
    PCW Davis
    Published 1982, by Cambridge University Press
  17. The Big Bang Never Happened
    Eric Lerner
    Published 1992, by Simon and Schuster Ltd.
  18. The Cosmic Blueprint
    Paul Davies
    Published 1987, by Heinemann Books Ltd.
  19. The Cosmic Onion
    Frank Close
    Published 1984, by Heinemann Books Ltd.
  20. The Exploding Universe
    Nigel Henbest
    Published 1979, by Marshall Caverdish Books Ltd.
  21. The First Three Minutes
    Stephen Weinberg
    Published 1977, by Andre Deutsch Ltd.
  22. The Illustrated Encylopedia of Myths and Legends
    Arthur Cotterell
    Published 1989, by Guild Publishing
  23. The Last Three Minutes
    Paul Davies
    Published 1994, by The Guernsey Press Company
  24. The Left Hand of Creation
    JD Barrow and J Silk
    Published 1984, by Basic Books Ltd.
  25. The Nature of Space and Time
    S Hawking and R Penrose
    Published 1996, by Princeton University Press.
  26. The New Physics
    Edited by Paul Davies
    Published 1989, by Cambridge University Press
  27. The Omega Point
    John Gribbin
    Published 1987, by Heinemann Books Ltd.
  28. The Refridgerator and the Universe
    Martin and Inge Goldstein
    Published 1995, by Harvard University Press.
  29. The Runaway Universe
    Paul Davies
    Published 1978, by Biddles Ltd.
  30. The Second Law
    Henry Bent
    Published 1965, by Oxford University Press
  31. The Universe for Beginners
    F Pirani and C Roche
    Published 1993, by Icon Books Ltd.
  32. Three Big Bangs
    Dauber Muller

The remaining sources were used mainly for a general overview, or to answer specific questions, so I have not allocated reference numbers to them.

Computer related

Others

References sources printed above in a bold typeface are particularly recommended for further reading on this subject.


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