Level 3 parallel worlds are those that appear in the many-worlds interpretation of quantum mechanics, which we’ve already tackled.
Everything described so far pales into insignificance when we come to level 4. Here, the various universes involved can have radically different laws of physics from each other. All conceivable mathematical structures, Tegmark tells us, exist here:
How about a universe that obeys the laws of classical physics, with no quantum effects? How about time that comes in discrete steps, as for computers, instead of being continuous? How about a universe that is simply an empty dodecahedron? In the level IV multiverse, all these alternative realities actually exist.
But do they?
In science, you get evidence from observations or from experiments.
Direct observational tests of Tegmark’s hypothesis are completely out of the question, at least until some remarkable spacefaring technology comes into being. The observable universe extends no more than 27plex metres from the Earth. An object (even the size of our visible universe) that is 118plexplex metres away cannot be observed now, and no conceivable improvement on technology can get round that. It would be easier for a bacterium to observe the entire known universe than for a human to observe an object 118plexplex metre away.
We are sympathetic to the argument that the impossibility of direct experimental tests does not make the theory unscientific. There is no direct way to test the previous existence of dinosaurs, or the timing (or occurrence) of the Big Bang. We infer these things from indirect evidence. So what indirect evidence is there for infinite space and distant copies of our own world?
Space is infinite, Tegmark says, because the cosmic microwave background tells us so. If space were finite, then traces of that finitude would show up in the statistical properties of the cosmic background and the various frequencies of radiation that make it up.
This is a curious argument. Only a year or so ago, some mathematicians used certain statistical features of the cosmic microwave background to deduce that not only is the universe finite, but that it is shaped a bit like a football.8 There is a paucity of very long-wavelength radiation, and the best reason for not finding it is that the universe is too small to accommodate such wavelengths. Just as a guitar string a metre long cannot support a vibration with a wavelength of 100 metres – there isn’t room to fit the wave into the available space.
The main other item of evidence is of a very different nature – not an observation as such, but an observation about how we interpret observations. Cosmologists who analyse the microwave background to work out the shape and size of the universe habitually report their findings in the form ‘there is a probability of one in a thousand that such and such a shape and size could be consistent with the data’. Meaning that with 99.9 per cent probability we rule out that size and shape. Tegmark tells us that one way to interpret this is that at most one Hubble volume in a thousand, of that size and shape, would exhibit the observed data. ‘The lesson is that the multiverse theory can be tested and falsified even when we cannot see the other universes. The key is to predict what the ensemble of parallel universes is and to specify a probability distribution over that ensemble.’
This is a remarkable argument. Fatally, it confuses actual Hubble volumes with potential ones. For example, if the size and shape under consideration is ‘a football about 27plex metres across’ – a fair guess for our own Hubble volume – then the ‘one in a thousand’ probability is a calculation based on a potential array of one thousand footballs of that size. These are not part of a single infinite universe: they are distinct conceptual ‘points’ in a phase space of big footballs. If you lived in such a football and made such observations, then you’d expect to get the observed data on about one occasion in a thousand.
There is nothing in this statement that compels us to infer the actual existence of those thousand footballs – let alone to embed the lot in a single, bigger space, which is what we are being asked to do. in effect, Tegmark is asking us to accept a general principle: that whenever you have a phase space (statisticians would say a sample space) with a well-defined probability distribution, then everything in that phase space must be real.
This is plain wrong.
A simple example shows why. Suppose that you toss a coin a hundred times. You get a series of tosses something like HHTTTHH … TTHH. The phase space of all possible such tosses contains precisely 2100 such sequences. Assuming the coin is fair, there is a sensible way to assign a probability to each such sequence – namely the chance of getting it is one in 2100. And you can test that ‘distribution’ of probabilities in various indirect ways. For instance, you can carry out a million experiments, each yielding a series of 100 tosses, and count what proportion has 50 heads and 50 tails, or 49 heads and 51 tails, whatever. Such an experiment is entirely feasible.
If Tegmark’s principle is right, it now tells us that the entire phase space of coin-tossing sequences really does exist. Not as a mathematical concept, but as physical reality.
However, coins do not toss themselves. Someone has to toss them.
If you could toss 100 coins every second, it would take about 24plex years to generate 2100 experiments. That is roughly 100 trillion times the age of the universe. Coins have been in existence for only a few thousand years. The phase space of all sequences of 100 coin tosses is not real. It exists only as potential.
Since Tegmark’s principle doesn’t work for coins, it makes no sense to suppose that it works for universes.
The evidence advanced in favour of level 4 parallel worlds is even thinner. It amounts to a mystical appeal to Eugene Wigner’s famous remark about ‘the unusual effectiveness of mathematics’ as a description of physical reality. In effect, Tegmark tells us that if we can imagine something, then it has to exist.
We can imagine a purple hippopotamus riding a bicycle along the edge of the Milky Way while singing Monteverdi. It would be lovely if that meant it had to exist, but at some point a reality check is in order.
We don’t want to leave you with the impression that we enjoy pouring cold water over every imaginative attempt to convey a feeling for some of the remarkable concepts of modern cosmology and physics. So we’ll end with a very recent addition to the stable of parallel worlds, which has quite a few things going for it. Perhaps unsurprisingly, the main thing not currently going for it is a shred of experimental evidence.
The new theory on the block is string theory. It provides a philosophically sensible answer to the age-old question: why are we here? And it does so by invoking gigantic numbers of parallel universes.
It is just much more careful how it handles them.
Our source is an article, ‘The String Theory Landscape’ by Raphael Bousso and Joseph Polchinski, in the September 2004 issue of Scientific American – a special issue on the theme of Albert Einstein.
If there is a single problem that occupies the core of modern physics, it is that of unifying quantum mechanics with relativity. This search for a ‘theory of everything’ is needed because although both of those theories are extraordinarily successful in helping us to understand and predict various aspects of the natural world, they are not totally consistent with each other. Finding a consistent, unified theory is hard, and we don’t yet have one. But there’s one mathematically attractive attempt, string theory, which is conceptually appealing even though there’s no observational evidence for it.
String theory holds that what we usually consider to be individual points of spacetime, dimensionless dots with no interesting structure of their own, are actually very, very tiny multidimensional surfaces with complicated shapes. The standard analogy is a garden hose. Seen from some way off, a hose looks like a line, which is a one-dimensional space – the dimension being distance along the hose. Look more closely, though, and you see that the hose has two extra dimensions, at right angles to that line, and that its shape in those directions is a circular band.
Maybe our own universe is a bit
like that hosepipe. Unless we look very closely, all we see is three dimensions of space plus one of time – relativity. An awful lot of physics is observed in those dimensions alone, so phenomena of that type have a nice four-dimensional description – relativity again. But other things might happen along extra ‘hidden’ dimensions, like the thickness of the hose. For instance, suppose that at each point of the apparent four-dimensional spacetime, what seems to be a point is actually a tiny circle, sticking out at right angles to spacetime itself. That circle could vibrate. If so, then it would resemble the quantum description of a particle. Particles have various ‘quantum numbers’ such as spin. These numbers occur as whole number multiples of some basic amount. So do vibrations of a circle: either one wave fits into the circle, or two, or three … but not two and a quarter, say.
This is why it’s called ‘string theory’. Each point of spacetime is replaced by a tiny loop of string.
In order to reconstruct something that agrees with quantum theory, however, we can’t actually use a circular string. There are too many distinct quantum numbers, and plenty of other problems that have to be overcome. The suggestion is that instead of a circle, we have to use a more complicated, higher-dimensional shape, known as a ‘brane’.9 Think of this as a surface, only more so. There are many distinct topological types of surface: a sphere, a doughnut, two doughnuts joined together, three doughnuts … and in more dimensions than two, there are more exotic possibilities.
Particles correspond to tiny closed strings that loop around the brane. There are lots of different ways to loop a string round a doughnut – once through the hole, twice, three times … The physical laws depend on the shape of the brane and the paths followed by these loops.
The current favourite brane has six dimensions, making ten in all. The extra dimensions are thought to be curled up very tightly, smaller than the Planck length, which is the size at which the universe becomes grainy. It is virtually impossible to observe anything that small, because the graininess blurs everything and the fine detail cannot be seen. So there’s no hope of observing any extra dimensions directly. However, there are several ways to infer their presence indirectly. In fact, the recently discovered acceleration in the rate of expansion of the universe can be explained in that manner. Of course, this explanation may not be correct: we need more evidence.
The ideas here change almost by the day, so we don’t have to commit ourselves to the currently favoured six-dimensional set-up. We can contemplate any number of different branes and differently arranged loops. Each choice – call it a loopy brane – has a particular energy, related to the shape of the brane, how tightly it is curled up, and how tightly the loops wind round it. This energy is the ‘vacuum energy’ of the associated physical theory. In quantum mechanics, a vacuum is a seething mass of particles and anti-particles coming into existence for a brief instant before they collide and annihilate each other again. The vacuum energy measures how violently they seethe. We can use the vacuum energy to infer which loopy brane corresponds to our own universe, whose vacuum energy is extraordinarily small. Until recently it was thought to be zero, but it’s now thought to be about l/120plex units, where a unit is one Planck mass per cubic Planck length, which is a googol grammes per cubic metre.
We now encounter a cosmic ‘three bears’ story. Macho Daddy Bear prefers a vacuum energy larger than +l/118plex units, but such a spacetime would be subject to local expansions far more energetic than a supernova. Wimpy Mummy Bear prefers a vacuum energy smaller than −l/120plex units (note the minus sign), but then spacetime contracts in a cosmic crunch and disappears. Baby Bear and Goldilocks like their vacuum energy to be ‘just right’: somewhere in the incredibly tiny range between +1/118plex and − 1/120plex units. That is the Goldilocks zone in which life as we know it might possibly exist.
It is no coincidence that we inhabit a universe whose vacuum energy lies in the Goldilocks zone, because we are life as we know it. If we lived in any other kind of universe, we would be life as we don’t know it. Not impossible, but not us.
This is our old friend the anthropic principle, employed in an entirely sensible way to relate the way we function to the kind of universe that we need to function in. The deep question here is not ‘why do we live in a universe like that?’, but ‘why does there exist a universe like that, for us to live in?’ This is the vexed issue of cosmological fine-tuning, and the improbability of a random universe hitting just the right numbers is often used to prove that something – they always say ‘We don’t know, could be an alien,’ but what they’re all thinking is: ‘God’ – must have set our universe up to be just right for us.
The string theorists are made of sterner stuff, and they have a more sensible answer.
In 2000 Bousso and Polchinski combined string theory with an earlier idea of Steven Weinberg to explain why we shouldn’t be surprised that a universe with the right level of vacuum energy exists. Their basic idea is that the phase space of possible universes is absolutely gigantic. It is bigger than, say, 500plex. Those 500plex universes distribute their vacuum energies densely in the range -1 to +1 units. The resulting numbers are much more closely packed than the l/118plex units that determine the scale of the ‘acceptable’ range of vacuum energies for life as we know it. Although only a very tiny proportion of those 500plex universes fall inside that range, there are so many of them that that a tiny proportion is still absolutely gigantic – here, around 382plex. So a whacking great 382plex universes, from a phase space of 500plex loopy branes, are capable of supporting our kind of life.
However, that’s still a very small proportion. If you pick a loopy brane at random, the odds are overwhelmingly great that it won’t fall inside the Goldilocks range.
Not a problem. The string theorists have an answer to that. If you wait long enough, such a universe will necessarily come into being. In fact, all universes in the phase space of loopy branes will eventually become the ‘real’ universe. And when the real universe’s loopy brane gets into the Goldilocks range, the inhabitants of that universe will not know about all that waiting. Their sense of time will start from the instant when that particular loopy brane first occurred.
String theory not only tells us that we’re here because we’re here – it explains why a suitable ‘here’ must exist.
The reason why all of those 500plex or so universes can legitimately be considered ‘real’ in string theory stems from two features of that theory. The first is a systematic way to describe all the possible loopy branes that might occur. The second invokes a bit of quantum to explain why, in the long run, they will occur. Briefly: the phase space of loopy branes can be represented as an ‘energy landscape’, which we’ll name the branescape. Each position in the landscape corresponds to one possible choice of loopy brane; the height at that point corresponds to the associated vacuum energy.
Peaks of the branescape represent loopy branes with high vacuum energy, valleys represent loopy branes with low vacuum energy. Stable loopy branes lie in the valleys. Universes whose hidden dimensions look like those particular loopy branes are themselves stable … so these are the ones that can exist, physically, for more than a split second.
In hilly districts of the branescape, the landscape is rugged, meaning that it has a lot of peaks and valleys. They get closer together than elsewhere, but they are still generally isolated from each other. The branescape is very rugged indeed, and it has a huge number of valleys. But all of the valleys’ vacuum energies have to fit inside the range from -1 units to +1 units. With so many numbers to pack in, they get squashed very close together.
In order for a universe to support life as we know it, the vacuum energy has to lie in the Goldilocks zone where everything is just right. And there are so many loopy branes that a huge number of them must have vacuum energies that fall inside it:
Vastly more will fall outside that range, but never mind.
The theory has one major advantage: it explains why our universe has such a
small vacuum energy, without requiring it to be zero – which, we now know, it isn’t.
The upshot of all the maths, then, is that every stable universe sits in some valley of the branescape, and an awful lot of them (though a tiny proportion of the whole) lie in the Goldilocks range. But all of those universes are potential, not actual. There is only one real universe. So if we merely pick a loopy brane at random, the chance of hitting the Goldilocks zone is pretty much zero. You wouldn’t bet on a horse at those odds, let alone a universe.
Fortunately, good old quantum gallops to our rescue. Quantum systems can, and do, ‘tunnel’ from one energy valley to another. The uncertainty principle lets them borrow enough energy to do that, and then pay it back so quickly that the corresponding uncertainty about timing prevents anyone noticing. So, if you wait long enough – umptyplexplexplex years, perhaps, or umptyplexplexplexplex if that’s too short – then a single quantum universe will explore every valley in the entire branescape. Along the way, at some stage it finds itself in a Goldilocks valley. Life like ours then arises, and wonders why it’s there.
It’s not aware of the umptyplexplexplexplex years that have already passed in the multiverse: just of the few billion that have passed since the wandering universe tunnelled its way into the Goldilocks range. Now, and only now, do its human-like inhabitants start to ask why it’s possible for them to exist, given such ridiculous odds to the contrary. Eventually, if they’re bright enough, they work out that thanks to the branescape and quantum, the true odds are a dead certainty.
It’s a beautiful story, even if it turns out to be wrong.
1 To see why, double it: the result now is 2 + 1 + ½ + ¼ + ⅛ + + … which is 2 more than the original sum. What number increases by 2 when you double it? There’s only one such number, and it’s 2.
2 If you’ve never encountered the mathematical joke, here it is. Problem 1: a kettle is hanging on a peg. Describe the sequence of events needed to make a pot of tea. Answer: take the kettle off the peg, put it in the sink, turn on the tap, wait till the kettle fills with water, turn the tap off … and so on. Problem 2: a kettle is sitting in the sink. Describe the sequence of events needed to make a pot of tea. Answer: not ‘turn on the tap, wait till the kettle fills with water, turn the tap off … and so on’. Instead: take the kettle out of the sink and hang it on the peg; then proceed as before. This reduces the problem to one that has already been solved. (Of course the first step puts it back in the sink – that’s why it’s a joke.)
Science of Discworld III Page 20