The mystery of time May 17, 2011
Time is of the utmost importance in the biblical worldview, if only because the God of the Bible inhabits eternity (Isaiah 57:15) and thus exists outside of time and space (though remaining active in time and space). The subject underlies many topical issues, ranging from the origin of the universe to ‘open theism’ and more. I am therefore taking the unusual step of reproducing here as an article the whole chapter from “Who made God?” that deals with the subject (at some length and in some depth). Please note that this material is copyright but short extracts may be reproduced without further permission as long as the source is clearly acknowledged.
8. Steam engine to the stars
‘Time is the longest distance between two places’.
Tennessee Williams, The glass menagerie (1945).
‘Most of the methods for measuring time have, I believe, been the contrivance of monks and religious recluses who, finding time hangs heavy on their hands, were at some pains to see how they got rid of it’.
William Hazlitt, ‘On a sundial’ (Sketches and essays, 1839).
Let me introduce you to Wells, Who and Hawking. Don’t worry, they are not my solicitors but merely a few of the people who have messed with the mystery of time — and speculated how to get rid of it. In 1888, novelist H. G. Wells published The chronic argonauts, a short story featuring time-travellers who visit the past and the future using a time-machine. Wells followed this in 1895 with his better known story The time machine. Then there’s Doctor Who — British TV’s intrepid, perennial and accident-prone ‘time-lord’ — whose temperamental Tardis (a space-ship-cum-time-machine) lands him in all kinds of uncomfortable times, places and situations. He should need no further introduction. Finally, as we have already seen, Stephen Hawking covered the serious stuff by writing A brief history of time: from the big bang to black holes.
When I talk about ‘getting rid of time’ I don’t quite mean what Hazlitt meant by it, but am referring to any attempt to escape the restrictions that time imposes on us. In real life that’s impossible. In fiction it’s easy; just use a time-machine. In science it’s somewhere in between — rather more difficult than for Wells and Who but not conceptually impossible, as Professor Paul Davies has argued in his entertaining but exceedingly far-fetched book How to build a time machine. As we shall see, those who seek to escape the implications of big bang cosmology are actually trying to cast off the shackles of time. And who can blame them? (no pun intended). As the Persian poet Omar Khayyám reminds us, time is the ultimate taskmaster.
‘The moving finger writes and having writ
Moves on; nor all thy piety nor wit
Shall lure it back to cancel half a line,
Nor all thy tears wash out a word of it’.
Nicely put and all too true; but what if we could somehow get rid of time? According to our biblical hypothesis, of course, God already has, for having invented time he cannot be subject to it. He was already there before time began and time is just as much part of the created material order as are space, matter and energy. As we have seen, this accords entirely with general relativity and basic big bang cosmology. [Note: elsewhere in the book I explain that although I accept the big bang theory as a legitimate evidence-based scientific model, I nevertheless insist on a fully historical interpretation of Genesis Ch.1]. God himself, the creator, stands outside the created order and thus outside of time. Psalm 90 declares: ‘Before the mountains were brought forth, or ever you had formed the earth and the world, even from everlasting to everlasting, you are God. You turn man to destruction and say, “Return, O children of men”. For a thousand years in your sight are like yesterday when it is past, and like a watch in the night’.
Look at it this way. God is time’s cartographer — time for God is spread out like a map. Just as we might study a route map showing villages, townships and cities, together with their environments and the roads that link them, so God surveys all history at a glance — encompassing everything that is to us past, present and future. As Isaiah 46:9-10 puts it: ‘I am God, and there is no other; I am God, and there is none like me, declaring the end from the beginning, and from ancient times things that are not yet done, saying, “My counsel shall stand, and I will do all my pleasure”’.
This implies, of course, the intriguing concept that all time still exists. In the three dimensions of space, I can travel from London to Manchester and onwards to Glasgow. In terms of my experience, once I reach Manchester, London lies in the past and Glasgow in the future. But this doesn’t mean that London has stopped existing or that Glasgow is still a green-field site. So with time. The fact that we are confined to ‘now’ and can visit neither yesterday nor tomorrow, doesn’t mean that yesterday has ceased to exist or tomorrow doesn’t yet exist. It is, in fact, one of the inevitable conclusions of relativity theory that the whole of space-time must have a real and continuing existence — regardless of our perception of time as being divided into past, present and future. If you doubt my word, Brian Greene sets out detailed arguments to prove this and concludes: ‘Just as we envision all of space as really being out there, as really existing, we should also envision all of time as really being out there, as really existing too’. The biblical idea that God surveys all time is therefore predictive of what has only recently become apparent to science.
Isaiah 57:15 provides a further example of the Bible’s teaching on this subject, naming God as ‘the high and lofty one who inhabits eternity, whose name is holy’ (emphasis added). The Hebrew word translated ‘holy’ means ‘separate’ and speaks of the ‘otherness’ of God, both in a moral sense and in respect of his eternal self-existent and non-material nature. As Frederic Amiel put it, ‘Time and space are fragments of the infinite for the use of finite creatures’. That’s us, not God. So let’s take a look at time — first of all as science sees it.
The arrow of time
Einstein’s special and general theories of relativity, developed in 1905 and 1915 respectively, caused a seismic shift in science’s perception of the universe. Until then, space and time were considered to be the ‘given’ and unchanging stage on which the drama of existence was played out. In Newtonian physics, everything happened in time and space but nothing ever happened to time and space. With Einstein all that changed.
Even before Einstein, physics already viewed time as a ‘fourth dimension’ and had come to terms with the idea of a four-dimensional ‘space-time continuum’. For example, the equations derived by James Clerk Maxwell in 1876 — which describe how electromagnetic waves like light, heat and radio waves propagate through space — showed that time behaves like a fourth spatial dimension divided by the speed of light. But even before that, in an 1848 essay on cosmology (appropriately titled Eureka), the American poet and writer Edgar Allan Poe had concluded, after 90 pages of philosophical reasoning, that ‘space and duration are one’. This was apparently the first time anyone had suggested in print that space and time were linked. Following Poe’s pronouncement, H. G. Wells later wrote: ‘There is no difference between time and any of the three dimensions of space except that our consciousness moves along it’ — adding, ‘Scientific people … know very well that time is only a kind of space’.
So what difference did Einstein make? He showed that the space-time continuum was not the rigid and unchanging ‘stage’ (or frame of reference) for everything else, as had previously been supposed, but was itself an actor in the cosmic drama. It was part of the fabric of the universe, capable of change and distortion. Thus gravity ceased to be some mysterious action-at-a-distance and became instead the result of massive objects causing curvature in space-time. In a relativistic universe, the three dimensions of space and the one dimension of time are all equally flexible. However, the equivalence of space and time is only partial. We can move freely backwards and forwards in any of the three dimensions of space, but we can only go forwards in the ‘pseudo-space’ we call ‘time’. For convenience we can (and shall) talk about time itself being unidirectional, or time ‘flowing’ in one direction, but strictly speaking it is we who are moving in one direction through the pre-existing landscape of time. Except in science fiction, the past is forbidden territory. Conversely, you can remember the past but not the future. This unique unidirectional nature of time is called ‘the arrow of time’, a term coined in 1927 by British astronomer Sir Arthur Eddington.
Interestingly, almost all the equations and theories of science are symmetrical with respect to time — that is, they hold true whether events are running forwards or backwards in time. For all they care, the clock can go ‘tick-tock’ or ‘tock-tick’, it makes no difference. They contain nothing to indicate what we all know to be true, namely, that we can only move in one direction through time. But Eddington identified one glaring exception — a single physical law that is not indifferent to the direction of time. The law in question is the ‘second law of thermodynamics’, a long-established principle that describes and governs growth in randomness. The law states that while the randomness (technically, the ‘entropy’) of an isolated system may increase with the passage of time or remain unchanged, it can never decrease. Eddington concluded that as far as science is concerned, the arrow of time is a property of entropy alone. So, what’s entropy all about?
By steam engine to the stars
Per ardua ad astra. Let’s take a journey to the stars by steam engine. I know it sounds like a Wallace and Gromit project, but in fact our companions on this trip will be a child of the French revolution, a Pomeranian ambulance driver, a British Baron and the son of a Viennese tax inspector. More respectfully, they were all leaders in the development of the science of thermodynamics — Sadi Carnot, a French military engineer whose father served on the French revolutionary council; Rudolph Clausius, a German scientist born in Pomerania who organised an ambulance service during the Franco-Prussian war; Lord Kelvin who, as William Thomson, was appointed to Glasgow’s Chair of Natural Philosophy when he was just 22; and Ludwig Boltzmann who, in 1869, became Professor of Mathematical Physics at the University of Graz at the similarly early age of 25.
Carnot wanted to improve the efficiency of the steam engine and wondered what factors might be involved. To simplify the problem, he imagined an idealised ‘heat engine’ in which heat flowed from a source at high temperature (like the fire of a steam engine) to a sink or reservoir at a lower temperature (like the steam exhausted from the engine) to produce mechanical work (as the steam moves the engine’s piston and turns a shaft). Carnot pictured an ideal engine with no friction or external interference in which the whole process could be reversed by driving the shaft backwards and pumping heat back from the cool reservoir to the hot source. Such a ‘reversible’ engine could be taken through a complete cycle and brought back to its original state without losing any heat energy. Carnot realised that no heat engine could be more efficient than a reversible one, since if it were more efficient it would deliver perpetual motion (it could drive the shaft without expending energy).
Carnot also showed that the efficiency of a reversible heat engine didn’t depend on the details but only on the temperature difference between the heat source and the heat sink. E. T. Jaynes comments: ‘Carnot’s reasoning is outstandingly beautiful because it deduces so much from so little — and with such a sweeping generality that rises above all tedious details — but at the same time with such a compelling logical force. In this respect I think that Carnot’s principle ranks with Einstein’s principle of relativity’.
Next, enter William Thomson (later Lord Kelvin), who along with James Joule established that heat and mechanical work are equivalent — each is a form of energy that can be converted into the other. The original idea and measurements are credited to Joule (and independently to Julius von Mayer) but Thomson was the one who analysed, explained and published the results. The Kelvin or ‘absolute’ temperature scale used by scientists today was a direct result of this work.
Another James — not Joule but Bond — famously required his cocktails to be ‘shaken, not stirred’. When you agitate a cocktail shaker you are performing mechanical work on the contents and as a result the liquid warms up (as a thermometer would show). If the shaker were perfectly insulated so that no heat could escape, the heat generated inside the shaker would equal the mechanical energy supplied, and no energy would be lost. (Mind you, James, the same would be true if the drink were stirred rather than shaken. In fact Joule first demonstrated the point by using a falling weight to drive a stirrer which, in turn, raised the temperature of a water-bath).
But it was Clausius who came up with the perfect mix. Taking one part of Carnot’s heat engine and adding two parts of Kelvin-Joule’s mechanical equivalence of heat, he produced a cocktail that could well be called ‘steam engine to the stars’. What did he discover? That if any isolated system goes from one equilibrium state to another, the net heat flow (called Q) divided by the temperature (T) at which it occurs can only increase or remain constant — it cannot decrease. Clausius called this quantity (Q/T) the ‘entropy’, so whatever happens to such a system either leaves the entropy unchanged (only true for a perfectly reversible system) or causes it to increase. Furthermore, Clausius had the audacity to point out that this applied not only to idealised steam engines but to the ultimate ‘isolated system’ — the universe itself.
(Note that even in a ‘closed’ system, where heat but not mass is allowed to pass in or out of the system, the principle of the second law of thermodynamics still applies).
Bicycle pumps and rubber bands
Other scientists (notably J. Willard Gibbs and Ludwig Boltzmann) carried the subject further. They showed that entropy has a statistical character — the more ways there are to arrange the components of a system (e.g. a collection of atoms, molecules or soup-bowl fragments, see chapter 1) the higher is its entropy. If you compress a gas (as when pumping up a bicycle tyre) its temperature increases but its entropy decreases because there are fewer places for the air molecules to go in the smaller volume available. The lesson? You have to work hard to decrease entropy.
A rubber band is another example of a closed system where energy must be injected to reduce the entropy. In an unstretched band, each of the long segmented rubber molecules is coiled up randomly into a ball. Since there are a very large number of ways in which its many segments can be arranged in such a coil, the entropy of an unstretched molecule is high. But when you stretch the rubber band the molecules uncoil and elongate, lining up roughly parallel to one another. Since there are only a small number of ways to achieve this highly ordered molecular arrangement, the entropy of the stretched band is correspondingly low.
To go from the unstretched state (high entropy) to the stretched state (low entropy) we have to apply a stretching force and perform work on the rubber band — an energy input is needed to reduce the entropy. Notice that this energy input is not random. We can supply energy randomly by zapping the rubber in a microwave oven, but that won’t make it stretch. To reduce the entropy the energy supplied must be specifically controlled and directed. When the tension is released, the molecules revert to their normal form of random coils and, in the process, the rubber band can give back energy to its surroundings — for example, by driving the propeller of a model aeroplane.
The chief thing I want you to notice here is that rubber bands can’t stretch themselves — it takes effort to put the rubber into a state of low entropy (a stretched condition having a high degree of order with all the molecules lined up). Equally, it takes effort to work the bicycle pump that compresses air and reduces its entropy — the gas will not spontaneously compress itself. So in an isolated system like the universe, low overall entropy can no more ‘just happen’ than can a rubber band stretch itself or a gas compress itself. On the other hand, a state of low entropy (high order) can spontaneously ‘unwind’ to one of high entropy, releasing energy in the process — just as a stretched rubber band can turn a model propeller or a gas spontaneously expand (let go of the pump handle at the end of a compression stroke and the handle will spring back as the gas expands).
Clausius’ cocktail hour
In the days when the law required British pubs to close at a fixed hour of the night, the landlord would cry out in a loud voice, ‘time gentlemen please’ to advise his customers that the statutory moment had arrived. The Clausius cocktail hour also has profound implications for time but on a much grander scale. Put briefly, the only way science can explain the phenomenon of time, with its one-directional ‘arrow’, is by assuming that the universe began in a highly ordered (low entropy) state. Physicist Brian Greene puts it thus: ‘Conditions at the birth of the universe are critical to directing time’s arrow. The future is indeed the direction of increasing entropy. The arrow of time — the fact that things start like this and finish like that but never start like that and end like this — began its flight in the highly ordered, low-entropy state of the universe at its inception’.
So, according to science, we experience time in one direction because the universe can only run downhill from high order to low order — the entropy or randomness can increase but never decrease. Scientifically speaking, then, time is simply the measure (and experience) of change, as the universe is transformed progressively from an initial highly ordered state of low entropy to some final condition of maximum entropy. In this final condition the universe would be in a ‘reversible’ state — entropy would stop increasing and time would cease to flow. The question then arises: what or who provided the energy input that established this low-entropy condition in the early universe?
Basic big bang cosmology can say only that the universe was created that way (in a low entropy state). However, the latest idea is that immediately after the universe originated, it underwent an ‘inflationary’ stage in which the force of gravity was actually negative, thus pushing apart space and its contents (whatever they were at that point, which is by no means agreed) at a fantastic speed. This inflation, it is argued, smoothed out any irregularities in the chaotic pre-inflationary universe and produced an expanded universe having low entropy. This scenario involves assumptions and speculations which we can’t go into here, but it does agree with our conclusion that energy must have been input to create a low-entropy universe. In the inflationary scenario, of course, it was the negative gravitational field (called the ‘inflaton field’) that donated this energy.
But where did the inflaton field get its energy? From ‘a statistical fluctuation from primordial chaos’ we are told — from an initial nugget of chaos that weighed a mere 20 lbs and was just 10-26 cm in diameter (that’s about one billionth of one billionth of one billionth of a centimetre). Twenty pounds doesn’t sound heavy, but a one-pound jam-jar full of this chaos stuff would weigh a billion billion billion billion billion billion billion billion billion metric tons — and that’s not counting the weight of the jar itself.
But whether all this turns out to be true or not, the fact remains that to get the whole process of time and ‘creation’ going, somewhere back there, someone or something somehow supplied energy that wasn’t already present in some prior shape or form. It’s either God or turtles, so make your choice.
Full marks for the hypothesis of God
It seems to me that whichever scientific theory you adopt, it accords completely with the predictions of the biblical hypothesis of God. Let’s take stock.
Firstly, the scientific scenarios outlined above validate the biblical concept of eternity. ‘Before’ the big bang origin, time could not exist. Why not? Because until the universe was there in some shape or form it could not be in any entropy state, low or otherwise.
Secondly, when the universe originated, it must either have been created ab initio in a low entropy state or subsequently put into such a state. Either way, as rubber bands and bicycle pumps teach us, it takes effort to create low entropy — this isn’t something that can just happen spontaneously. It follows that an ‘energetic’ act of creation must have occurred at some stage before time could begin to run.
Thirdly, as entropy began to increase (producing the phenomenon of time) the universe began to change and evolve. Modern cosmology seeks to chart and describe this development, and may or may not have got it right. But the Bible clearly teaches a progressive development of the universe. Not only does Genesis record a progressive creation (including the unfolding of the ‘six days of creation’) but other Bible passages clearly describe a universe that changes and ‘runs down’ with the passage of time. For example:
‘Of old you laid the foundation of the earth, and the heavens are the work of your hands.
They will perish but you will endure; Yes, they will all grow old like a garment; like a cloak you will change them [or fold them up] and they will be changed. But you are the same and your years will have no end’ (emphasis added).
Finally, according to the Bible, there was not only a beginning of time but there will be an end. St Peter writes in the New Testament: ‘do not forget this one thing, that one day is with the Lord as a thousand years, and a thousand years as one day. The Lord is not slack concerning his promise, as some count slackness, but is longsuffering towards us, not willing that any should perish, but that all should come to repentance. But the day of the Lord will come as a thief in the night, in which the heavens will pass away with a great noise and the elements will melt with fervent heat; both the earth and the works that are in it will be burned up’.
None of this ‘proves’ the existence of God. What it does do, however, is confirm that the biblical hypothesis of God makes correct predictions — predictions that are only now being confirmed by cosmological research, thousands of years after the hypothesis was first advanced. We are beginning to see, therefore, that our hypothesis survives the acid test — it has predictive power.
Getting rid of time
Needless to say, many have tried (and are still trying) to get rid of the theological implications of all this. So let’s see how some scientists try to get rid of time and, specifically, the idea that it actually started. There are several current ideas including the evergreen speculation that the universe has gone through a succession of big bangs, expansions, contractions and ‘big crunches’ — the cycle repeating ad infinitum. But where, in this scenario, does the energy come from to create the low starting entropy necessary to explain the arrow of time? You might think that the low entropy that started the present cycle could be supplied by the energy of the collapsing cosmos as it shrank to a point at the end of the previous cycle. But this doesn’t work. If the net energy of the universe is zero (as Hawking and others maintain, the ‘positive’ energy of matter and radiation being exactly balanced by the ‘negative’ energy of gravitational attraction) — if, in other words, the universe is the ultimate ‘free lunch’ — then any ‘big crunch’ would cause the positive and negative energies to annihilate each other leaving no energy to recreate a low entropy state. The free lunch would vanish leaving the cupboard completely bare of energy. But if there was no energy left over to put the newly crunched universe into a low entropy state — so that the flow of time could start all over again — how did this low entropy come about?
One answer is that time doesn’t have to restart following a big crunch. It could be argued that time continues to flow smoothly throughout successive expansions, contractions and ‘bounces’ (or punches, crunches and free lunches, if you prefer) — with entropy continuing to increase all the time. Each new big bang would simply begin with the entropy left over from the previous cosmic cycle without any resetting of the clock. This would certainly satisfy the second law of thermodynamics but it fails to explain the existence of time. Why? Because if there was no beginning, but only an eternal sequence of bangs and crunches, then entropy would have been increasing for ever. But sooner or later the entropy of the universe must reaches a maximum, which will happen once complete disorder is established. If entropy had been increasing for an infinite time, then maximum entropy would have been achieved and time would have ceased to flow. When nothing changes, time stands still.
There are other more sophisticated ways of getting rid of time. Dr Stenger, for example, seems somewhat ambivalent on the origin of the universe. On p.133 of God the failed hypothesis he talks about the ‘nothing-to-something transition’ as if it really happened, but just a few pages earlier he suggests that perhaps there was, after all, ‘something’ rather than ‘nothing’ before the big bang. He writes on p.126: ‘In The comprehensible cosmos I presented a specific scenario for the purely natural origin of the universe … based on the “no boundary model” of James Hartle and Stephen Hawking. In that model the universe had no beginning or end in space or time. In the scenario I presented, our universe is described as having “tunnelled” through the chaos … from a prior universe that existed for all previous time’.
The idea that our own universe is just one of a bunch is quite fashionable among those who speculate about time. Read any popular book on cosmology or time and you’ll find yourself tripping over universes by the dozen. It is almost taken for granted that we exist not so much in a universe as in a ‘multiverse’. So, what do we know about these other universes or, in Stenger’s case, this eternally prior universe? Precisely and absolutely nothing — which is a good deal less than we know about God. Yet Dr Stenger has the chutzpah to conclude: ‘I do not dispute that the exact nature of the origin of the universe remains a gap in scientific knowledge, but I deny that we are bereft of any conceivable way to account for that origin scientifically’ (my italics). The fact is that neither Dr Stenger nor anyone else has a clue — exact or inexact — about how the universe might have originated by material causes. This must be true because if the origin of the universe did have a material cause it wasn’t the origin of the universe at all, but merely a later stage in its development. Watch out for those turtles! Furthermore, to invoke an invisible, inaccessible, eternal and totally unknowable prior universe as the material cause of the one we know, can hardly be dignified as a ‘scientific’ account of origins. Science fiction and pop-science can get away with such speculations but real science demands a little more evidence.
Finally, Stenger’s claim to have found a completely material explanation for the origin of the universe invokes not only alternative universes but also the idea that our own universe is finite but ‘unbounded in time’, having neither beginning nor end. The idea (the Hartle-Hawking theory) is borrowed from Stephen Hawking, who pictures such a universe as a sphere — like the earth marked with lines of latitude and longitude. The size of the universe at any time is represented by the length of a given line of latitude. Thus, for example, the maximum size of the universe is related to the area encompassed by the equator. (Of course, in this picture we have to drop two of the dimensions of space, so that the one-dimensional line of latitude actually represent the three-dimensional volume of the universe).
What about the lines of longitude that run from pole to pole? These represent trajectories in time — but (wait for it) not in ‘real’ time! What Dr Stenger omits to mention is that you only get a ‘universe [that] had no beginning or end in space or time’ if you replace time by imaginary time — a perfectly legitimate mathematical concept introduced to help solve the equations underlying the model. Hawking emphasises that ‘we may regard our use of imaginary time … as merely a mathematical device (or trick) to calculate answers about real space-time’. ‘Trick’ is Hawking’s word, not mine.
This ‘imaginary time’ model suffers no singularities — the laws of science never break down nor do any physical quantities rise to infinity, even at the poles where the size of the universe shrinks to zero. Just as you can walk through the north pole without anything dreadful happening to you, so the universe could pass back and forth through zero size without any breakdown of the physics that describe it. This is what Hawking means by a finite but unbounded universe. Such a universe could indeed undergo successive collapses and bounces — but only in imaginary time, never in real time, as Hawking is at pains to point out. He writes, ‘Only if we could picture the universe in terms of imaginary time would there be no singularities. … When we go back to the real time in which we live, however, there will still be singularities’ — of which of course an ex nihilo creation of space and time would be a perfect example. And that is bad news for anyone wanting to get rid of time.
Much more could be said about attempts to banish the beginning of time but space forbids. So who should be allowed to have the last word on this subject? Might I suggest we defer to
Sir Roger Penrose, the esteemed Oxford mathematician who worked with Stephen Hawking on the theory of black holes and other cosmological problems. Writing in New Scientist on 27 April 1996, Gabrielle Walker cites him thus: ‘The standard big bang model is agreed, says Oxford mathematician Roger Penrose, and everything else is “embellishments and flights of fancy”.’
This should alert us to the danger of being seduced by speculation masquerading as science. Whatever flights of fancy we may undertake, travelling in imagination to other universes in cosmic bubbles or time machines, real science offers no prospect of escaping the inescapable fact that time really did begin!
 H. G. Wells The chronic argonauts (1888; Kessinger Publishing 2004).
 H. G. Wells, The time machine (1895; New American Library, 2002).
 Stephen Hawking, A brief history of time (Bantam Press, 1988).
 Paul Davies, How to build a time machine (Penguin Books, 2002).
 Omar Khayyám, in Francis Turner Palgrave, The golden treasury (OUP, 1861).
 Bible references; Romans 1:20; Ephesians 1:4; 2 Timothy 1:9.
 Psalm 90:2-4
 Brian Greene, The fabric of the cosmos (Alfred A. Knopf, New York, 2004) pp. 132-139.
 Frederic Amiel, Journal, 16 Nov 1864.
 Edgar Allan Poe, Eureka (Kessinger Publishing 2007).
 H. G. Wells, The time machine (1895).
 Sir Arthur Eddington, The Nature of the Physical World (1928).
 E. T. Jaynes, http://bayes.wustl.edu/etj/articles/ccarnot.pdf
 Yunus A. Cengel and Michael A. Boles, Thermodynamics: an engineering approach (6th Edition;
McGraw-Hill, 2008) Ch. 6, ‘The second law of thermodynamics’, p.283.
 Brian Greene, as reference 8, p.175.
 Brian Greene, as reference 8, p.319.
 Psalm 102:25-27; emphasis added.
 2 Peter 3:9-10.
 See Jay M. Pasachoff and Alex Filippenko, The Cosmos: Astronomy in the new Millennium (Brooks/Cole, the Wadsworth Group, 2001).
 For those who really want to know, imaginary time is real time multiplied by the square root of -1 (minus one), which is denoted by the symbol i. The quantity i cannot be a real number because if you square any real number, either positive or negative, you always get a positive number, never a negative one. That is why the square root of a negative number is called an ‘imaginary’ number.
 Stephen Hawking, as reference 3, p.135.
 Stephen Hawking, as reference 3, pp. 138-139.