If there is one obvious landmark in the history of cosmology which my thesis‘ introduction absolutely couldn’t ignore, it is the Big Bang theory. The realisation that the Universe is in constant expansion may have challenged a certain view of the Cosmos as a static stage for physical phenomena – but the acceptance of a paradigm where the Universe had an actual beginning absolutely revolutionised the way we see our place in cosmic history. Sure, the idea that there was a moment (or process) of Creation is at least as ancient as recorded history – but the Big Bang was the first time such an idea could be framed in scientific terms. Thanks to the analytical power of General Relativity, the Big Bang became a fully-fledged cosmological model which could be adjusted to changes in knowledge about the contents of the Universe, scoured for internal inconsistencies, and from which testable predictions could be extracted. Perhaps most importantly, the Big Bang definitely transformed physics as a whole from a field obsessed with immutable laws into one with a very ingrained notion of evolution in the natural order of things1.
Naturally, I felt compelled not only to find the original reference for the first proposal of the Big Bang but also to read it. I felt reassured that this would be an easy task, seeing as it is a really short paper – the whole thing taking up less than half a page. As it turns out, I was betrayed by the paper’s only reference, in its very first sentence:
“Sir Arthur Eddington1 states that, philosophically, the notion of a beginning of the present order of Nature is repugnant to him.“
GEORGES LEMAÎTRE, THE BEGINNING OF THE WORLD FROM THE POINT OF VIEW OF QUANTUM THEORY, NATURE 127:706, MAY 1931
This first sentence intrigued me. Although everything that followed was understandable and self-contained, this sentence made it sound like I was looking at one short isolated reply taken from a broader conversation. As it turns out, I was right – and trying to grasp the substance and breadth of this conversation ended up taking me down a delightful rabbit hole. I was forced to reassess what I thought I knew about the original argument for the Big Bang, and even about who was its original author.
Before I read Lemaître’s paper I expected to find a simple argument of the sort we are taught at University cosmology courses: take the Einstein (or Friedmann) equations for a Universe populated with a simple-but-sensible-as-a-large-scale-approximation ideal gas, see how they predict an expanding Universe, then observe that the solution you find shows the Universe as being infinitely small and infinitely dense at some point in the distant (but not infinitely distant) past. Instead, I found a much richer (and less straightforward) discussion focused on much more basic concepts.
Context: Lemaître and Eddington
Has we have previously seen, Eddington and Lemaître were often good allies. When Lemaître rediscovered the theoretical arguments for an expanding Universe, Eddington championed Lemaître’s proposal in the face of staunch opposition by Einstein himself (who had previously condemned Friedmann’s original discovery to obscurity).
In fact, the two giants had a close scientific relationship dating back to 1923, when Lemaître worked as a research assistant under Eddington at Cambridge. One may hypothesize the two men may have seen each other as a sort of kindred spirits: two scientists of strong religious convictions2 in a community which increasingly regarded such views with suspicion. At least in the article that Lemaître cites above, Eddington certainly comes across as an enthusiastic supporter of Lemaître when he says “We have recently learnt, mainly through the work of Prof. Lemaître, that (…) space is expanding” (which seems normal until you realise he’s completely dismissing Hubble’s contribution).
The issue of the beginning of time, however, saw them take different sides in a scientific and epistemological dispute. Reading these papers, one gets the impression that they must have both been surprised by how this discussion laid bare such fundamental differences between the way both men looked at science.
Eddington and the end (and beginning) of the world
Trying to get a good grasp of what exactly Lemaître was referring to in that first sentence, I dived into the Eddington article he was citing (published just two months before Lemaître’s paper). I must say it is one of the scientific papers I’ve enjoyed the most, and the one which breaks the most “rules”: its prose is anything but dry (it reads like a literary piece), and it touches upon a great number of important topics with only a draft of a complete argument in each (which, from my perspective as a modern reader, leads to glaring mistakes, but of the kind that are both instructive and revealing of how a great astronomer thought about physics as a “big picture”).
In what follows I try to present as simplified and coherent a version of Eddington’s position as possible. In great part due to the wide range of topics he invokes in the course of this article, in practice this means I am not faithful to the order of his exposition – but still (I think) convey essentially the same arguments.
The arrow of time
Eddington begins by noting that, from the point of view of general relativity, it seems like time is just another dimension, like the three spatial dimensions in which we routinely move, which just happens to have a different sort of geometry. In other words: sure, spacetime may have a non-intuitive geometry, but there seems to be no fundamental difference between the physical nature (or mathematical description) of time and that of space. In that case, Eddington reasons, it seems logical to reject the notion that time is special in the sense that we’re only allowed to move “forward” in it. Instead, he argues that it seems logical to posit that time is not special at all but our minds trick us into assigning a direction to time which makes life feel like some absolute flow from past to future – but this is ultimately an illusion inside our heads rather than a fact about reality. But how do our brains pick a direction to define as the “right” one?
(At this point, it is worth going on a long tangent to flesh out exactly what Eddington is trying to say about the nature of time. Nevertheless, if you’re familiar with the second law of thermodynamics, please feel free to skip all the way to the next subsection – I promise a very quick recap of where this tangent leads.)
The Universe according to Eddington is a huge four-dimensional landscape of which we are all part. In a sense, our past and future both exist “at the same time”, merely corresponding to different positions in the dimension we call time – in the same way our heads and feet both exist at the same time, in different points of some vertical dimension. If you want, just like you can speak of someone’s height or width, you could also speak of someone’s “length” along the time dimension: this would just correspond to someone’s lifespan! After all, in this picture humans are not what we’d call human-shaped. They’re more like four-dimensional tubes whose spatial sections look human-shaped – humans are to “human shapes” as pyramids are to triangles!
If you were to look at a human from outside this four-dimensional Universe, and to cut out only one thin slice along its spatial dimensions, you’d essentially be left holding that human at a particular point in its life. If you were then able to peer into the configuration of this human’s brain and to translate the information in its visual cortex into a coherent image (and, if you were such a powerful higher-dimensional being, why wouldn’t you?), then you’d be able to look at the world as seen by that human in the moment from which you removed it.
Now suppose you thinly sliced a human in this manner all along its entire time-length, and obtained an image from its visual cortex for each one; so you basically end up with a silent film of this human’s life from a first-person perspective. Now further suppose that you physically printed those images and clumsily dropped them on the floor and they ended up all mixed up. Now ask yourself: is there any way I can put these images back in their right order?
As it turns out, it’s not hard for you (being a powerful higher-dimensional being and all) to figure out which frames should go next to each other. All you need to do is make sure the assembled film shows images in a continuous succession (i.e, so it looks like an actual film rather than just a random succession of unrelated frames). But how can you know the direction in which to assemble them – which is to say, if you put a film together in this way and then play it, how can you tell if it is playing backwards? And if you can do this, doesn’t that amount to saying that there is an objective distinction between past and future after all?
If you (being a powerful higher-dimensional being and all) happen to be sufficiently unfamiliar with the way humans work, this is not that easy a task. You’re not allowed to resort to simple facts of life which you (in real life) know because you have lived a lifetime as a human: you can’t use your knowledge that humans generally walk forwards, or that clock pointers move clockwise, etc. “Hold on!”, you might say. “Why can’t I just look for some point in the film when some object is falling and use my knowledge that gravity pulls objects downwards?” Excellent question!
Suppose there is a point in the film when this particular human was playing basketball. You see the ball in the human’s hand, then you see the ball leaving the hand in a downward trajectory, hit the floor, then bounce back to the hand. If you played these frames in the reverse order, you’d see… exactly the same sequence of events (assuming a perfectly elastic ball)! “Well yeah”, you may interject, “but what if you see something dropped which isn’t perfectly elastic? What about something like a pen?” Then, as it happens, you would not see the same sequence of events if you played the film backwards, but you’d still be seeing two versions of history which are perfectly compatible with the laws of physics. “How come, if gravity doesn’t make balls shoot upwards?” That is why it’s an excellent question!
Consider in more detail what happens when this human drops a pen. The pen leaves the human’s hand, then the pen falls in a downward trajectory with an acceleration of 9.8 m/s2 (neglecting air resistance), then the pen hits the floor… and then what? “Then nothing? Then the pen stops because there’s nowhere for it to go?” Well, that would violate the law of conservation of energy, wouldn’t it? What happens to all that kinetic energy the pen acquired during its accelerated fall? “Well, I guess it gets transferred to the atoms in the floor?” Indeed, that is the right sequence of events: the pen leaves the human’s hand, then the pen falls in a downward trajectory with an acceleration of 9.8 m/s2, then the pen hits the floor and its kinetic energy is dispersed through a myriad of tiny atoms: first at the point of contact, then spreading all across the floor. In fact, presumably the floor would be ever so slightly hotter due to this energy transfer (and so would the pen, since it is also made up of atoms). In physics parlance, the pen’s kinetic energy is converted into heat. The point is not that gravity doesn’t pull pens upward. After all, unlikely as it may be, it is technically not impossible for the atoms in the floor (and in the pen) to have trajectories so that a part of their energy is converted into kinetic energy pushing the pen upwards! Suppose you watched exactly that in your backwards film: the pen is pushed upwards by a transfer of heat into kinetic energy (due to all the atoms in this picture just happening to have the right trajectories to make this happen), then the pen decelerates at a rate of 9.8m/s2 (as its kinetic energy is converted into gravitational potential energy), then it is caught by the human’s hand.
The initial condition giving rise to this bizarre occurrence may seem freakishly unlikely, but it is not inconceivable. In other words, unlikely as this situation may be, it does not break any fundamental law of physics. In fact, no matter what happens in this human’s life, there is nothing that may show up in a backwards-playing film of its life which actually breaks any fundamental law of physics – because fundamental laws of physics (at least those known to Eddington in 1931) are symmetric with respect to past-future inversions of this kind! In fact, a regular human’s life-film played in reverse would be full of just such odd situations which should be extremely unlikely yet are in accordance with the fundamental laws of physics: from billiards balls in chaotic positions happening to conspire to collide in just the right angles to end up in a near-perfect triangle configuration, to shards of glass jolting from the floor and assembling into a pristine-looking cup, to the atoms in a soup spontaneously rearranging themselves into a mixture of water and an assortment of vegetables!
And that is the real “arrow” that allows physicists (and our brains – and powerful higher-dimensional beings) to decide there is an objective difference between the future and the past! Even if the laws of physics treat past and future the same way as left and right, there is a right order to assemble the frames of a human’s life: because the alternative will always show a film where countless improbable (but not impossible!) events happen all the time! In the language of thermodynamics, the future is the direction in time where entropy is increasing!
(Entropy, in this context, is essentially a measure of “disorder”: how much you don’t know about the microscopic state of the Universe if all you know are macroscopic quantities. Due to non-obvious mathematical reasons, saying that entropy always increases is equivalent to saying that heat can never be converted into work without energy being lost forever – an example of which would be the heat from the floor being converted into work that makes the pen shoot upwards.)
“Hold on one fleeting minute!”, you interject. “Isn’t that basically cheating? After all, even if it’s tremendously unlikely, it is still possible for a powerful higher-dimensional being to assemble a human’s life-film in the wrong order by following that criterion! In fact, since this requirement that entropy increase is not a fundamental law of physics but rather one which will surely be verified almost but never exactly 100% of the time, if this higher-dimensional being repeats this procedure with an arbitrarily large number of humans it is sure to get it wrong at least once!” As it happens, Eddington gives a very detailed and very subtle answer to that exact complaint.
“I have sometimes been taken to task for not sufficiently emphasising in my discussion of these problems that the results about entropy are a matter of probability, not of certainty. I said above that if we observe a system at two instants, the instant corresponding to the greater entropy will be the later. Strictly speaking, I ought to have said that for a smallish system the chances are, say, 1020 to 1, that it is the later. Some critics seem to have been shocked at my lax morality in making such a statement, when I was well aware of the 1 in 1020 chance of its being wrong. Let me make a confession. I have in the past twenty-five years written a good many papers and books, broadcasting a large number of statements about the physical world. I fear that for not many of these statements is the risk of error so small as 1 in 1020. Except in the domain of pure mathematics, the trustworthiness of my conclusions is usually to be rated at nearer 10 to 1 than 1020 to 1; even that may be unduly boastful. I do not think it would be for the benefit of the world that no statement should be allowed to be made if there were a 1 in 1020 chance of its being untrue; conversation would languish somewhat. The only persons entitled to open their mouths would presumably be the pure mathematicians.”
Arthur Eddington, The End of the World: from the Standpoint of Mathematical Physics, NATURE 127:427-453, MARCH 1931
In fact, Eddington goes further than that. To cut a long argument short, Eddington contends that the very foundation of our exploration of the laws of nature rests on the assumption that extremely unlikely events are impossible for all intents and purposes. Therefore, even so-called fundamental laws of physics rank beneath the requirement that entropy must never decrease macroscopically – because they are derived from observations with far weaker statistical strength. In a hypothetical situation where Eddington witnesses a violation of this law of entropy with his very own eyes3, he claims he will rationally sooner renege on the idea of there existing laws of Nature (and admit that “the apparent uniformities so far observed are merely coincidences”) than concede “that the event is entirely in accordance with the laws of Nature, but that an improbable coincidence has happened (…); because [his] whole reason for accepting the laws of Nature rests on the assumption that improbable coincidences do not happen – at least, that they do not happen in [his] experience”.
Interestingly, at some point in the course of unpacking this view, Eddington spares a thought for “Dr. P. A. M. Dirac[‘s] wave equation”. If the future is kind to me, I shall look into this particular equation in some detail at some point in another post. For now, all you need to know about it is that it describes the dynamics of (particles like) electrons in a way that entails the first theoretical prediction of the existence of positrons – particles which can be equivalently defined as “opposite-charge twins” of electrons or as electrons moving “backwards” in time. Eddington comes across as both respectful towards Dirac and charitably dismissing this prediction (and, it seems to me, not fully grasping the mathematics behind Dirac’s calculations). The problem, he argues, is that the production of these particles in Dirac’s theory had only ever been shown to take place (with low probability) in mathematical problems involving very few particles. Individual electrons, he proposed, can be thought of as computing the arrow of time, as dictated by entropy in their local vicinity, in real time. According to him, in problems with so few particles this entropy is allowed to have much larger fluctuations than in any realistic scenario, causing Dirac’s “unfortunate electron” to be “deceived sometimes by chance coincidences which may easily happen when there are only [very few] particles concerned”. To my knowledge, this rationalisation of Dirac’s prediction fails to actually connect the settings in which positrons are produced in the theory to those in which entropy fluctuates the wrong way as per Eddington. Nonetheless, Eddington is categorical: “You must understand that they only do this in the mathematical problems, not in real life”; “Dirac’s theory predicts things which never happen, simply because it is applied to problems which never occur in Nature”. Alas for Eddington, the positron was experimentally discovered the following year.
Eddington accidentally discovers the Big Bang – then struggles to undiscover it
So Eddington holds that time is just like any other dimension, except that entropy (almost) always increases in one special direction (from the past to the future). Therefore, we (and our brains) are justified in treating this particular direction of time as the “right” one, even if at a fundamental level past and future are no more significant than left and right.
In the article, Eddington only brings this up because he is ostensibly speculating about whether the Universe will come to and end, and just wants to justify calling what he’s looking at “the future”. As a result of these considerations, he unexpectedly finds himself considering whether it had a beginning too – and ends up spending most of his time trying to convince himself that his argument for why it did is wrong.
Basically, Eddington says: see in which time direction entropy is growing and call that the “future”; then look in the opposite direction (the “past”) and you should see less and less entropy until if you look far enough you should see “the maximum possible organisation”. In Eddington’s own words: “To go back further is impossible. We have come to an abrupt end of space-time – only we generally call it the ‘beginning’.” Since in this worldview past and future only make sense in terms of how entropy is changing, finding a point where entropy can decrease no longer is equivalent to finding the furthest past there is, the beginning of time itself.
Shock! Horror! Does this mean Eddington and not Lemaître is the real father of the Big Bang?! Eddington makes it very clear that, if this is the case, he’s a most unwilling absentee father:
“Philosophically, the notion of a beginning of the present order of Nature is repugnant to me. I am simply stating the dilemma to which our present fundamental conception of physical law leads us. I see no way round it; but whether future developments of science will find an escape I cannot predict.”
Arthur Eddington, The End of the World: from the Standpoint of Mathematical Physics, NATURE 127:427-453, MARCH 1931
Eddington leaves no room for doubt: this conclusion offends the very core of his being. Although he has no idea of how to avoid it4, he refuses to accept it and hopes for a future refutation which he cannot offer at the time. Instead, he offers an attempt at an explanation for why such a “repugnant” feature has found its way into his model of the Universe.
This explanation is based on the observation that “our surroundings” look far too different from what you’d expect from a “fortuitous concourse of atoms”. The structures we see around us display such organization that the odds that they arose by chance are “multimillions to 1” (where by “multimillions” Eddington means numbers of order 1010000000000), a feature which less scrupulous theists “would like to call (…) purpose or design”. But not Eddington. Eddington “non-committally [calls it] anti-chance”. The problem, he says, is that there is naturally no place for anti-chance in the laws of physics (because it is never observed in actual experiments); therefore in order to produce a valid model of the highly unlikely world we observe these laws should require a highly improbable configuration of particles at some suitably distant point in the past.
“By sweeping [anti-chance] far enough away from the sphere of our current physical problems, we fancy we have got rid of it. It is only when some of us are so misguided as to try to get back billions of years into the past that we find the sweepings all piled up like a high wall and forming a boundary – a beginning of time – which we cannot climb over.”
Arthur Eddington, The End of the World: from the Standpoint of Mathematical Physics, NATURE 127:427-453, MARCH 1931
To be clear, I think Eddington’s take on anti-chance is dubious at best. As far as I can see, that which I call the laws of physics (though not necessarily all complex models built upon them) take into account entire probability distributions whenever probabilities play a role. That is to say extremely unlikely events certainly are factored into our predictions as much as more likely ones; just with an appropriately vanishing weight, proportional to their improbability. I suppose Eddington might argue that when probabilities are too small then you can’t trust these predictions too much because you’re talking about events which have never been observed in the entire history of humanity (or the Universe); and that is a fair observation, but doesn’t change the fact that highly unlikely events which are required of our models by logical consistency are still taken into account. And I’d certainly argue that events of vanishingly small probability which are logically required by models built on assumptions which have been properly tested in other settings (i.e., by their higher-probability predictions) should be treated with the same epistemic confidence as any other untested prediction of these models. Eddington’s stance here reminds me of more modern attempts to dismiss experts who rely on models to predict the chance of events with actual real-world consequences (see, e.g., the review of “Book Three” here).
Then again, Eddington doesn’t suggest he comprehends exactly what is going on, and by his own admission the whole argument is very handy-wavy and all he’s doing is letting the reader know exactly how his brain is flailing to try and reject a “repugnant” conclusion. And to be completely fair there do seem to be very many aspects of the way the Universe works and is organised that seem incompatible with sheer chance. Modern approaches to these features, however, tend to rely more on anthropic arguments and building theories in which the world we see is basically unavoidable (often at the cost of predictability). That being said, there is still no satisfying solution to this problem, so fans of Eddington may still hold out hope that future developments will vindicate him in some form.
Lemaître and the primeval atom
Basically, Lemaître saw Eddington’s rejected derivation of a beginning of time and said “no, wait, this can actually make sense physically!”. To start things off, Lemaître points out a quantum version of Eddington’s argument: not only must entropy decrease as you look back further and further into the past, but it should do so in discrete (rather than continuous) amounts. This matters because presumably you could try to evade Eddington’s classical conclusion by making entropy decrease less and less, à la Zeno. Since there is a minimum amount by which entropy à la Lemaître is allowed to change, this can’t happen. (I expect Eddington didn’t mention this sort of potential loophole because he implicitly assumed this same sort of view; and possibly thought it too contrived anyway.) This way, if you go back far enough you eventually have to find yourself at minimum entropy. Lemaître envisions this state of maximum organization as one in which all the matter (and energy) in the Universe is packed into a single colossal atom – “primeval atom” is the term Lemaître used to refer both to this atom and to his theory. (The term “Big Bang” came into use much later, taken from a pejorative use in a BBC radio interview to one of the theory’s most ardent opponents. As it turns out, “Big Bang” was a much better term: much catchier and less tied to the details of Lemaître’s original model.)
Lemaître further points out that in a situation in which the entire Universe is concentrated in a single atom there is no meaningful sense in which one can speak of space and time5. The fact that this initial state makes physical sense (we all know what atoms are) and yet makes notions of space and time obsolete provide him with all the philosophical soothing he needs:
“If this suggestion is correct, the beginning of the world happened a little before the beginning of space and time. I think that such a beginning of the world is far enough from the present order of Nature to be not at all repugnant.”
GEORGES LEMAÎTRE, THE BEGINNING OF THE WORLD FROM THE POINT OF VIEW OF QUANTUM THEORY, NATURE 127:706, MAY 1931
Finally, Lemaître makes it very clear that he does not have a detailed model of how any of this should work. Nevertheless, he happily discusses the consequences of seriously accepting this intuitive picture, in which the entire Cosmos starts as a single atom whose atomic weight is the total mass of the Universe: this primeval atom should be extremely unstable and quickly “divide in smaller and smaller atoms by a kind of super-radioactive process”. And boom! (or bang?), that’s how regular atoms (and the Universe as we know it) are born!
So who’s the father?
At the end of the day, I’m convinced Lemaître is rightly remembered as the father of the Big Bang. Eddington clearly never saw that result as physically meaningful (in fact, he opposed it his entire life) and Lemaître was clearly the first to take it seriously and to try and come up with a working model for what a “Big Bang” should look like and how it could develop into the Universe we observe today. Needless to say, our modern Big Bang theory6 shares only the spirit of Lemaître’s original model (these days nobody talks of primeval atoms), but he defended the basic idea of a physical beginning of spacetime (not just his particular model for it) through times when it was immensely unpopular with his peers (who presumably didn’t find his being a priest unrelated to his trying to convince them the Universe might have had a beginning). Notwithstanding, Eddington’s contribution to the birth of this theory should be better appreciated, and considered in the same light as Planck’s contribution to quantum mechanics and Lorentz’s contribution to relativity.
Footnotes
1. Mind you, it still is very much obsessed with immutable laws. After all, you could probably define physics as simply the study of universal immutable laws of Nature. What advances like the Big Bang were important in cementing was the understanding that immutable laws do not imply an immutable world.
2. Lemaître was a Catholic priest. Eddington had trouble with the British government during the second part of World War I owing to his refusal to fight, based on (Quaker) religious grounds.
3. In Eddington’s example, he puts a kettle of water on the fire and the water freezes.
4. He does point out how some people have proposed a “loophole” which almost avoids it but ultimately fails to appease Eddington’s epistemological conscience. The idea is to make use of a mathematical result which says that (under certain more or less general assumptions) random configurations of particles can be expected to, over the course of a ridiculously long period of time (by which I mean typically longer than the accepted age of the Universe), go through all possible configurations, including those corresponding to minimum entropy states. This way, you could envision a Universe that doesn’t really have a beginning in the time dimension, but more or less periodically returns to a state of minimum entropy which evolves into something like the Universe in which we live, presumably to reach a state of heat death before eventually (after an almost literal eternity) going through the whole cycle again. The problem with this is that it relies on having periods in which entropy is decreasing as time goes “forward”, which Eddington goes to great lengths to dismiss as unphysical (as in our long tangent above).
5. Lemaître isn’t just saying that if everything is in one place then space and time can’t be invoked as describing relations between objects. Rather, he seems to see space and time as “no more than statistical notions” in quantum mechanical settings, such that they must “fade out when applied to individual phenomena involving but a small number of quanta”. In this point, I believe he is conflating the notions of space and position in space (as well as time and position in time). Certainly in standard quantum mechanics (i.e., non-relativistic quantum mechanics, which is the only kind possibly known to Lemaître) particle positions in spacetime are given by a probability density and the values we classically assign to them can be seen as mere averages; but space and time still feature in our equations as absolute quantities.
6. The details of our modern account of the origin of the Universe are left for a future post (I think this one is long enough as is!).
Aetherial Porosity 