While The Theory of Everything , the Stephen Hawking biopic film, is topical, I interrupt this blog, which is the outcome of one Eureka Moment, to tell you the tale of another such experience. This Eureka Moment was concerned with a theory of the Universe with a capital U, which I may occasionally refer to as It with a capital I.
There is a scene in the movie, set in 1963, when Ph.D. student Hawking passes a page of calculations over the table to his Cambridge University supervisor, Dennis Sciama, who immediately knows he has a big talent to nurture. Well, just two years later, as a second-year undergraduate, I passed my calculations on The Theory of Everything across the desk to Dennis Sciama, but his reaction was rather different. Here’s a picture of Sciama as I remember him.
Great notions often arise when two quite different lines of thought are mixed together. The first mental ingredient to my Big Idea was provided by Dennis Sciama’s fascinating series of introductory lectures on cosmology which I was attending. I must still have the notes somewhere. The other ingredient was a pervading anxiety of the 1960’s. The world was in the middle of the Cold War nuclear missile race between Russia and the West. In Britain, the Campaign for Nuclear Disarmament was marching on the Aldermaston weapons research establishment every year and the CND ban-the-bomb badge was perhaps the most widely recognised of all logos. The Cuban Missile Crisis of 1962 had left a feeling that although we had escaped “mutually assured destruction” this time, there would surely be a next time when our luck might run out. Because man now had the power to destroy himself, however small the probability of this happening in a particular year, sooner or later it was bound to happen. Not, I hope, quite a sound piece of reasoning, but that’s how many people felt then and a nagging unease at this situation was hard to shake off. Nowadays worry about the nuclear arsenals that are still with us seems to have been elbowed aside by anxieties concerning terrorism and global warming. Over a bread-and-cheese lunch in my Lyndewode Road digs after one of Sciama’s morning lectures, cosmology and mutually-assured destruction met and fused. The Universe is still here I mused – after billions of years – perhaps it hasn’t the power to destroy itself! If It had, It would have done so by now. What limit does that notion place on the total energy embodied in the Universe? Over a mug of Nescafe (this was 1965) I settled down to some back-of-the-envelope calculations. We don’t know how far the Universe extends but we can do the sums for the big bit, the Hubble Volume, we can observe and claim that, by the Principle of Mediocrity (since elbowed aside by the Anthropic Principle), the result will apply to It all. How much energy, E, is there in the Universe? Einstein’s most famous of all formulas (perhaps the ONLY formula that most people can quote) gives an answer:
E = Mc2
where M is the mass of the observable Universe and c the velocity of light. How much energy, E, is needed to utterly destroy the Universe? What would “destroy” mean in such a general case? I decided that destruction would mean blowing everything utterly apart, dispersing the stuff of the stars and galaxies to an infinite separation, leaving space empty. The energy, E, needed to do that would have to overcome the mutual gravitational attraction of the Universe’s matter and would equal It’s gravitational potential energy, which can be expressed by another profound, but less well known equation:
E = GM2 /R
where R is the radius of the observable Universe, Its Hubble Radius at which galaxies are receding from us with the speed of light, and G is the Gravitational Constant, a fundamental constant defined as the attraction between unit masses at unit separation and often called “big G” although it’s strength is rather small. The limit or upper bound on the amount of energy in the Universe would be the quantity which is JUST ENOUGH to disperse and destroy the Universe, implying that
E = E , so that we can put
Mc2 = GM2 /R
Another way of looking at this equation is to write it
Mc2 – GM2 /R = 0
which is to say that the total net energy in the Universe is ZERO! The positive energy contained within its mass is balanced by the negative energy required to pull it apart to a state of empty space. Obviously, this is a very blunt calculation, leaving out kinetic and electromagnetic energy, let alone the more recently posited “dark” matter and energy. Cancelling out and re-arranging we obtain:
M = Rc2/G
By pouring over several weeks’ lecture notes, I managed to find Sciama’s estimates for the number of stars in a typical galaxy, the average mass of stars, the separation of galaxies and the radius of the observable Universe, so I had R and could work out an estimate for M; the value of c I knew off by heart. So I decided to reshape the equation in the form of an estimate of the value of big G, which I didn’t know. So I worked out:
G = c2R/M
The value I obtained was G = 4 x 10-11 m3kg-1s-2 I then went over to my textbook shelf, the internet not having been thought of, and looked up the accepted value, which I found to be:
G = 6.67 X 10-11 m3kg-1s-2
My estimate wasn’t just in the same approximate ballpark, not just of the same order of magnitude – I was out by a factor of less than two.
What a moment. Insofar as I can now recall, the successful result came as a sharp physical shock, followed by a feeling of dizziness and then euphoria fading to boosted self-confidence. Was this the answer to The Ultimate Question of Life, the Universe and Everything? … to quote Douglas Adams, whose footprints are often to be found on what you had hoped would be a virgin beach. I seemed to have shown that the Universe had zero net energy, but
Mc2 – GM2 /R = 0
Has still more amazing implications. We can rewrite it once again as:
M = Rc2/G
Which means that the mass of the Universe is proportional to Its radius. Applying the Principle of Mediocrity once again, we should not expect there to be anything particularly special about our own time, so as we reel history backwards towards the Big Bang and R decreases, although the density of the Universe (M/R3) increases without limit, the MASS of the Universe, M, tends to zero as R tends to zero. Not only does the Universe have zero net energy at all times, at the Big Bang it had zero mass! Truly a Universe created from nothing!! All existence is seen as a sort of separating out of pluses and minuses which added together still equal the zero of non-existence. It’s as if we were living off a nest egg of billions in our savings account despite all this wealth been cancelled out by exactly the same size of overdraft in another account. Surely this everything is nothing hypothesis has a must-be-so feeling and a beauty matching that of Fred Hoyle’s discredited Steady State Theory of the Universe. Intriguingly, my theory, like Hoyle’s, requires the continuous creation of matter to keep the Universe in balance as it expands. A week or two later, I took my calculations to Dennis Sciama rather as Stephen Hawking had done some two years previously, as depicted in The Theory of Everything, but I was no Hawking. Sciama gave me a warm, but rather sad smile.
“I’m afraid all the simple ideas have long been had.”
he replied, pulling an offprint of one of his own papers from a desk drawer. The value of the dimensionless expression c2R/GM had long been discussed in connection with the curvature of hyperbolic space – positive, zero or negative – which is deemed to determine whether the Universe is “open” or “closed” or whether or not It would expand for ever or collapse back into The Big Crunch. You have heard all that stuff, ad infinitum (sorry) on BBC Horizon programmes! I was too chastened to put all my Universe-from-nothing interpretations to the great man and left clutching his offprint. Two years later, having given up physics in favour of a passionate but doomed journey towards the insoluble problems of town and country planning, I had the privilege of sitting next to the brilliant and infamous mathematical physicist Fred Hoyle at a St John’s College Foundation Dinner, but was too demoralised to tell him of my Eureka calculation. Instead he led me into a hopeless challenge to his thesis that no-one with any intelligence should waste their time thinking about insoluble problems. My arguments were not enhanced by the port and I have never been invited “in rotation” to another such Foundation Feast. Fred Hoyle was even less photogenic and more ill at ease in front of a camera than I – no single image on the web quite catches his essence at about the time I met him. Here’s an approximation.
So there is my tale. My Eureka Moment. Surely I could scarcely have experienced one more satisfying. I now realise that Sciama handed me an opportunity to investigate further and that it was my lack of real interest and ability in physics that stopped me taking up the opportunity. Instead, I was so demoralised by the humbling of my Big Idea that I thought the narrative too trivial to tell to Fred Hoyle when granted the ideal opportunity. That demoralisation was my own doing, not Sciama’s. So there is one moral to this tale. Another might be a reminder to encourage the young when they joyfully reinvent the wheel! Just occasionally, it will be a brand new wheel or, these days, more likely a brand new APP. It would be great to hear of blog-visitors’ Eureka Moments and how they played out – there’s probably a Eureka Blog somewhere.
P.S. My estimate for G really was too good to be true – had Sciama cooked the books?