It is my contention that space time describes the impressed geometry of existence rather well and is very important. Yet it does not describe existence itself.
More simply, existence is an action that initiates a 3D pendulum that thus produces time. While mathematical continuity exists nicely enough in 3D, it does not at all along a proposed time axis. continuity is reflected internally instead and all objects occupy a different space through each step in time.
Recall that photons are bundles of 2D space as well. All this tracks discontinuously through space.
The point is that my formulation absorbs our present understanding of physics while only using the geometry of 3D which we actually experience. The driving metric is the nth ordered Pythagorean in which n is the number of axis involved. For n > 2, the observed scales spiral down.
All this becomes apparent with the mathematics of my third and forth order Pythagorean and understanding my work on the Space Time Pendulum.
These higher ordered Pythagoreans naturally produce our particle universe and its space time geometry as well without any higher dimensions at all.
.
Radical dimensions
Relativity says we live in four dimensions. String theory says it’s 10. What are ‘dimensions’ and how do they affect reality?
Illustration by Richard Wilkinson
Writing away at my desk, I reach my hand up to turn on a lamp, and down to open a drawer to take out a pen. Extending my arm forward, I brush my fingers against a small, strange figurine given to me by my sister as a good-luck charm, while reaching behind I can pat the black cat snuggling into my back. Right leads to the research notes for my article, left
to my pile of ‘must-do’ items (bills and correspondence). Up, down,
forward, back, right, left: I pilot myself in a personal cosmos of
three-dimensional space, the axes of this world invisibly pressed upon
me by the rectilinear structure of my office, defined, like most Western
architecture, by three conjoining right angles.
Our architecture, our education and our dictionaries tell us that space is three-dimensional. The OED defines it as ‘a continuous area or expanse which is free, available or unoccupied … The dimensions of height, depth and width, within which all things exist and move.’ In the 18th century, Immanuel Kant argued that three-dimensional Euclidean space is an a priori necessity and, saturated as we are now in computer-generated imagery and video games, we are constantly subjected to representations of a seemingly axiomatic Cartesian grid. From the perspective of the 21st century, this seems almost self-evident.
Yet the notion that we inhabit a space with any mathematical structure is a radical innovation of Western culture, necessitating an overthrow of long-held beliefs about the nature of reality. Although the birth of modern science is often discussed as a transition to a mechanistic account of nature, arguably more important – and certainly more enduring – is the transformation it entrained in our conception of space as a geometrical construct.
Over the past century, the quest to describe the geometry of space has become a major project in theoretical physics, with experts from Albert Einstein onwards attempting to explain all the fundamental forces of nature as byproducts of the shape of space itself. While on the local level we are trained to think of space as having three dimensions, general relativity paints a picture of a four-dimensional universe, and string theory says it has 10 dimensions – or 11 if you take an extended version known as M-Theory. There are variations of the theory in 26 dimensions, and recently pure mathematicians have been electrified by a version describing spaces of 24 dimensions. But what are these ‘dimensions’? And what does it mean to talk about a 10-dimensional space of being?
In order to come to the modern mathematical mode of thinking about space, one first has to conceive of it as some kind of arena that matter might occupy. At the very least, ‘space’ has to be thought of as something extended. Obvious though this might seem to us, such an idea was anathema to Aristotle, whose concepts about the physical world dominated Western thinking in late antiquity and the Middle Ages.
Strictly speaking, Aristotelian physics didn’t include a theory of space, only a concept of place. Think of a cup sitting on a table. For Aristotle, the cup is surrounded by air, itself a substance. In his world picture, there is no such thing as empty space, there are only boundaries between one kind of substance, the cup, and another, the air. Or the table. For Aristotle, ‘space’ (if you want to call it that), was merely the infinitesimally thin boundary between the cup and what surrounds it. Without extension, space wasn’t something anything else could be in.
Centuries before Aristotle, Leucippus and Democritus had posited a theory of reality that invoked an inherently spatialised way of seeing – an ‘atomistic’ vision, whereby the material world is composed of minuscule particles (or atoms) moving through a void. But Aristotle rejected atomism, claiming that the very concept of a void was logically incoherent. By definition, he said, ‘nothing’ cannot be. Overcoming Aristotle’s objection to the void, and thus to the concept of extended space, would be a project of centuries. Not until Galileo and Descartes made extended space one of the cornerstones of modern physics in the early 17th century does this innovative vision come into its own. For both thinkers, as the American philosopher Edwin Burtt put it in 1924, ‘physical space was assumed to be identical with the realm of geometry’ – that is, the three-dimensional Euclidean geometry we are now taught in school.
Our architecture, our education and our dictionaries tell us that space is three-dimensional. The OED defines it as ‘a continuous area or expanse which is free, available or unoccupied … The dimensions of height, depth and width, within which all things exist and move.’ In the 18th century, Immanuel Kant argued that three-dimensional Euclidean space is an a priori necessity and, saturated as we are now in computer-generated imagery and video games, we are constantly subjected to representations of a seemingly axiomatic Cartesian grid. From the perspective of the 21st century, this seems almost self-evident.
Yet the notion that we inhabit a space with any mathematical structure is a radical innovation of Western culture, necessitating an overthrow of long-held beliefs about the nature of reality. Although the birth of modern science is often discussed as a transition to a mechanistic account of nature, arguably more important – and certainly more enduring – is the transformation it entrained in our conception of space as a geometrical construct.
Over the past century, the quest to describe the geometry of space has become a major project in theoretical physics, with experts from Albert Einstein onwards attempting to explain all the fundamental forces of nature as byproducts of the shape of space itself. While on the local level we are trained to think of space as having three dimensions, general relativity paints a picture of a four-dimensional universe, and string theory says it has 10 dimensions – or 11 if you take an extended version known as M-Theory. There are variations of the theory in 26 dimensions, and recently pure mathematicians have been electrified by a version describing spaces of 24 dimensions. But what are these ‘dimensions’? And what does it mean to talk about a 10-dimensional space of being?
In order to come to the modern mathematical mode of thinking about space, one first has to conceive of it as some kind of arena that matter might occupy. At the very least, ‘space’ has to be thought of as something extended. Obvious though this might seem to us, such an idea was anathema to Aristotle, whose concepts about the physical world dominated Western thinking in late antiquity and the Middle Ages.
Strictly speaking, Aristotelian physics didn’t include a theory of space, only a concept of place. Think of a cup sitting on a table. For Aristotle, the cup is surrounded by air, itself a substance. In his world picture, there is no such thing as empty space, there are only boundaries between one kind of substance, the cup, and another, the air. Or the table. For Aristotle, ‘space’ (if you want to call it that), was merely the infinitesimally thin boundary between the cup and what surrounds it. Without extension, space wasn’t something anything else could be in.
Centuries before Aristotle, Leucippus and Democritus had posited a theory of reality that invoked an inherently spatialised way of seeing – an ‘atomistic’ vision, whereby the material world is composed of minuscule particles (or atoms) moving through a void. But Aristotle rejected atomism, claiming that the very concept of a void was logically incoherent. By definition, he said, ‘nothing’ cannot be. Overcoming Aristotle’s objection to the void, and thus to the concept of extended space, would be a project of centuries. Not until Galileo and Descartes made extended space one of the cornerstones of modern physics in the early 17th century does this innovative vision come into its own. For both thinkers, as the American philosopher Edwin Burtt put it in 1924, ‘physical space was assumed to be identical with the realm of geometry’ – that is, the three-dimensional Euclidean geometry we are now taught in school.
Long before physicists embraced
the Euclidean vision, painters had been pioneering a geometrical
conception of space, and it is to them that we owe this remarkable leap
in our conceptual framework. During the late Middle Ages, under a newly
emerging influence deriving from Plato and Pythagoras, Aristotle’s prime
intellectual rivals, a view began to percolate in Europe that God had
created the world according to the laws of Euclidean geometry. Hence, if
artists wished to portray it truly, they should emulate the Creator in
their representational strategies. From the 14th to the 16th centuries,
artists such as Giotto, Paolo Uccello and Piero della Francesca
developed the techniques of what came to be known as perspective
– a style originally termed ‘geometric figuring’. By consciously
exploring geometric principles, these painters gradually learned how to
construct images of objects in three-dimensional space. In the process,
they reprogrammed European minds to see space in a Euclidean fashion.
The historian Samuel Edgerton recounts this remarkable segue into modern science in The Heritage of Giotto’s Geometry
(1991), noting how the overthrow of Aristotelian thinking about space
was achieved in part as a long, slow byproduct of people standing in
front of perspectival paintings and feeling, viscerally, as if they were
‘looking through’ to three-dimensional worlds on the other side of the
wall. What is so extraordinary here is that, while philosophers and
proto-scientists were cautiously challenging Aristotelian precepts about
space, artists cut a radical swathe through this intellectual territory
by appealing to the senses. In a very literal fashion, perspectival
representation was a form of virtual reality that, like today’s VR
games, aimed to give viewers the illusion that they had been transported
into geometrically coherent and psychologically convincing other worlds.
The structure of the ‘real’ went from a philosophical and theological question to a geometrical proposition
The
illusionary Euclidean space of perspectival representation that
gradually imprinted itself on European consciousness was embraced by
Descartes and Galileo as the space of the real world. Worth adding here
is that Galileo himself was trained in perspective. His ability to
represent depth was a critical feature in his groundbreaking drawings of
the Moon, which depicted mountains and valleys and implied that the
Moon was as solidly material as the Earth.
By adopting the space
of perspectival imagery, Galileo could show how objects such as
cannonballs moved according to mathematical laws. The space itself was
an abstraction – a featureless, inert, untouchable, un-sensable void,
whose only knowable property was its Euclidean form. By the end of the
17th century, Isaac Newton had expanded this Galilean vision to
encompass the universe at large, which now became a potentially infinite
three-dimensional vacuum – a vast, quality-less, emptiness extending
forever in all directions. The structure of the ‘real’ had thus been
transformed from a philosophical and theological question into a
geometrical proposition.
Where
painters had used mathematical tools to develop new ways of making
images, now, at the dawn of the ‘scientific revolution’, Descartes
discovered a way to make images of mathematical relations in and of
themselves. In the process, he formalised the concept of a dimension,
and injected into our consciousness not only a new way of seeing the
world but a new tool for doing science.
Almost everyone today
recognises the fruits of Descartes’s genius in the image of the
Cartesian plane – a rectangular grid marked with an x and y axis, and a coordinate system.
By definition, the Cartesian plane is a two-dimensional space because we need two
coordinates to identify any point within it. Descartes discovered that
with this framework he could link geometric shapes and equations. Thus, a
circle with a radius of 1 can be described by the equation x2 + y2 =1.A
vast array of figures that we can draw on this plane can be described
by equations, and such ‘analytic’ or ‘Cartesian’ geometry would soon
become the basis for the calculus developed by Newton and G W
Leibniz to further physicists’ analysis of motion. One way to understand
calculus is as the study of curves; so, for instance, it enables us to
formally define where a curve is steepest, or where it reaches a local
maximum or minimum. When applied to the study of motion, calculus gives
us a way to analyse and predict where, for instance, an object thrown
into the air will reach a maximum height, or when a ball rolling down a
curved slope will reach a specific speed. Since its invention, calculus
has become a vital tool for almost every branch of science.
Considering
the previous diagram, it’s easy to see how we can add a third axis.
Thus with an x, y and z axis, we can describe the surface of a sphere –
as in the skin of a beach ball. Here the equation (for a sphere with a
radius of 1 ) becomes: x2 + y2 + z2 = 1
With
three axes, we can describe forms in three-dimensional space. And
again, every point is uniquely identified by three coordinates: it’s the
necessary condition of three-ness that makes the space three-dimensional.
But
why stop there? What if I add a fourth dimension? Let’s call it ‘p’.
Now I can write an equation for something I claim is a sphere sitting in
four-dimensional space: x2 + y2 + z2 + p2
= 1. I can’t draw this object for you, yet mathematically the addition
of another dimension is a legitimate move. ‘Legitimate’ meaning there’s
nothing logically inconsistent about doing so – there’s no reason I can’t.
A ‘dimension’ becomes a purely symbolic concept not necessarily linked to the material world at all
And
I can keep on going, adding more dimensions. So I define a sphere in
five-dimensional space with five coordinate axes (x, y, z, p, q) giving
us the equation: x2 + y2 + z2+ p2 + q2 = 1. And one in six-dimensions: x2 + y2 + z2 + p2 + q2 + r2 = 1, and so on.
Although
I might not be able to visualise higher-dimensional spheres, I can
describe them symbolically, and one way of understanding the history of
mathematics is as an unfolding realisation about what seemingly sensible
things we can transcend. This is what Charles Dodgson, aka Lewis
Carroll, was getting at when, in Through the Looking Glass, and What Alice Found There (1871), he had the White Queen assert her ability to believe ‘six impossible things before breakfast’.
Mathematically,
I can describe a sphere in any number of dimensions I choose. All I
have to do is keep adding new coordinate axes, what mathematicians call
‘degrees of freedom’. Conventionally, they are named x1, x2, x3, x4, x5, x6 et cetera.
Just as any point on a Cartesian plane can be described by two (x, y)
coordinates, so any point in a 17-dimensional space can be described by
set of 17 coordinates (x1, x2, x3, x4, x5, x6 … x15, x16, x17). Surfaces like the spheres above, in such multidimensional spaces, are generically known as manifolds.
From
the perspective of mathematics, a ‘dimension’ is nothing more than
another coordinate axis (another degree of freedom), which ultimately
becomes a purely symbolic concept not necessarily linked at all to the
material world. In the 1860s, the pioneering logician Augustus De
Morgan, whose work influenced Lewis Carroll, summed up the increasingly
abstract view of this field by noting that mathematics is purely ‘the
science of symbols’, and as such doesn’t have to relate to anything
other than itself. Mathematics, in a sense, is logic let loose in the
field of the imagination.
Unlike
mathematicians, who are at liberty to play in the field of ideas,
physics is bound to nature, and at least in principle, is allied with
material things. Yet all this raises a liberating possibility, for if
mathematics allows for more than three dimensions, and we think
mathematics is useful for describing the world, how do we know that
physical space is limited to three? Although Galileo, Newton and Kant
had taken length, breadth and height to be axiomatic, might there not be
more dimensions to our world?
Again, the idea of a
universe with more than three dimensions was injected into public
consciousness through an artistic medium, in this case literary
speculation, most famously in the mathematician Edwin A Abbott’s Flatland
(1884). This enchanting social satire tells the story of a humble
Square living on a plane, who is one day visited by a three-dimensional
being, Lord Sphere, who propels him into the magnificent world of
Solids. In this volumetric paradise, Square beholds a three-dimensional
version of himself, the Cube, and begins to dream of pushing on to a
fourth, fifth and sixth dimension. Why not a hypercube? And a
hyper-hypercube, he wonders?
Sadly, back in Flatland, Square is
deemed a lunatic, and locked in an insane asylum. One of the virtues of
the story, unlike some of the more saccharine animations and adaptations
it has inspired, is its recognition of the dangers entailed in
flaunting social convention. While Square is arguing for other
dimensions of space, he is also making a case for other dimensions of
being – he’s a mathematical queer.
In the late 19th and early 20th
centuries, a raft of authors (H G Wells, the mathematician and sci-fi
writer Charles Hinton, who coined the word ‘tesseract’ for the 4D cube),
artists (Salvador Dalí) and mystical thinkers (P D Ouspensky), explored
ideas about the fourth dimension and what it might mean for humans to
encounter it.
Then in 1905, an unknown physicist named Albert
Einstein published a paper describing the real world as a
four-dimensional setting. In his ‘special theory of relativity’, time
was added to the three classical dimensions of space. In the
mathematical formalism of relativity, all four dimensions are bound
together, and the term spacetime entered our lexicon. This
assemblage was by no means arbitrary. Einstein found that, by going down
this path, a powerful mathematical apparatus came into being that
transcended Newton’s physics and enabled him to predict the behaviour of
electrically charged particles. Only in a 4D model of the world can
electromagnetism be fully and accurately described.
Relativity was
a great deal more than another literary game, especially once Einstein
extended it from the ‘special’ to the ‘general’ theory. Now
multidimensional space became imbued with deep physical meaning.
In
Newton’s world picture, matter moves through space in time under the
influence of natural forces, particularly gravity. Space, time, matter
and force are distinct categories of reality. With special relativity,
Einstein demonstrated that space and time were unified, thus reducing
the fundamental physical categories from four to three: spacetime,
matter and force. General relativity takes a further step by enfolding
the force of gravity into the structure of spacetime itself. Seen from a
4D perspective, gravity is just an artifact of the shape of space.
To
comprehend this remarkable situation, let’s imagine for the moment its
two-dimensional analogue. Think of a trampoline, and imagine we draw on
its surface a Cartesian grid. Now put a bowling ball onto the grid.
Around it, the surface will stretch and warp so some points become
further away from each other. We’ve disturbed the inherent measure of
distance within the space, making it uneven. General relativity says
that this warping is what a heavy object, such as the Sun, does to
spacetime, and the aberration from Cartesian perfection of the space itself gives rise to the phenomenon we experience as gravity.
Whereas
in Newton’s physics, gravity comes out of nowhere, in Einstein’s it
arises naturally from the inherent geometry of a four-dimensional
manifold; in places where the manifold stretches most, or deviates most
from Cartesian regularity, gravity feels stronger. This is sometimes
referred to as ‘rubber-sheet physics’. Here, the vast cosmic force
holding planets in orbit around stars, and stars in orbit around
galaxies, is nothing more than a side-effect of warped space. Gravity is
literally geometry in action.
If moving into four dimensions helps to explain gravity, then might thinking in five dimensions have any scientific advantage? Why not give it a go?
a young Polish mathematician named Theodor Kaluza asked in 1919,
thinking that if Einstein had absorbed gravity into spacetime, then
perhaps a further dimension might similarly account for the force of
electromagnetism as an artifact of spacetime’s geometry. So Kaluza added
another dimension to Einstein’s equations, and to his delight found
that in five dimensions both forces fell out nicely as artifacts of the
geometric model.
You’re an ant running on a long, thin hose, without ever being aware of the tiny circle-dimension underfoot
The
mathematics fit like magic, but the problem in this case was that the
additional dimension didn’t seem to correlate with any particular
physical quality. In general relativity, the fourth dimension was time; in Kaluza’s theory, it wasn’t anything you could point to, see, or feel: it was just there in the mathematics. Even Einstein balked at such an ethereal innovation. What is it? he asked. Where is it?
In
1926, the Swedish physicist Oskar Klein answered this question in a way
that reads like something straight out of Wonderland. Imagine, he said,
you are an ant living on a long, very thin length of hose. You could
run along the hose backward and forward without ever being aware of the
tiny circle-dimension under your feet. Only your ant-physicists with
their powerful ant-microscopes can see this tiny dimension. According to
Klein, every point in our four-dimensional spacetime has a
little extra circle of space like this that’s too tiny for us to see.
Since it is many orders of magnitude smaller than an atom, it’s no
wonder we’ve missed it so far. Only physicists with super-powerful
particle accelerators can hope to see down to such a minuscule scale.
Once
physicists got over their initial shock, they became enchanted by
Klein’s idea, and during the 1940s the theory was elaborated in great
mathematical detail and set into a quantum context. Unfortunately, the
infinitesimal scale of the new dimension made it impossible to imagine
how it could be experimentally verified. Klein calculated that the
diameter of the tiny circle was just 10-30 cm. By comparison, the diameter of a hydrogen atom is 10-8
cm, so we’re talking about something more than 20 orders of magnitude
smaller than the smallest atom. Even today, we’re nowhere close to being
able to see such a minute scale. And so the idea faded out of fashion.
Kaluza,
however, was not a man easily deterred. He believed in his fifth
dimension, and he believed in the power of mathematical theory, so he
decided to conduct an experiment of his own. He settled on the subject
of swimming. Kaluza could not swim, so he read all he could about the
theory of swimming, and when he felt he’d absorbed aquatic exercise in
principle, he escorted his family to the seaside and hurled himself into
the waves, where lo and behold he could swim. In Kaluza’s
mind, the swimming experiment upheld the validity of theory and, though
he did not live to see the triumph of his beloved fifth dimension, in
the 1960s string theorists resurrected the idea of higher-dimensional
space.
By the 1960s, physicists had discovered two additional forces of nature, both operating at the subatomic scale. Called the weak nuclear force and the strong nuclear
force, they are responsible for some types of radioactivity and for
holding quarks together to form the protons and neutrons that make up
atomic nuclei. In the late 1960s, as physicists began to explore the new
subject of string theory (which posits that particles are like
minuscule rubber bands vibrating in space), Kaluza’s and Klein’s ideas
bubbled back into awareness, and theorists gradually began to wonder if
the two subatomic forces could also be described in terms of spacetime geometry.
It turns out that in order to encompass both of these two forces, we have to add another five dimensions to our mathematical description. There’s no a priori
reason it should be five; and, again, none of these additional
dimensions relates directly to our sensory experience. They are just
there in the mathematics. So this gets us to the 10 dimensions of string
theory. Here there are the four large-scale dimensions of
spacetime (described by general relativity), plus an extra six ‘compact’
dimensions (one for electromagnetism and five for the nuclear forces),
all curled up in some fiendishly complex, scrunched-up, geometric
structure.
A great deal of effort is being expended by physicists
and mathematicians to understand all the possible shapes that this
miniature space might take, and which, if any, of the many alternatives
is realised in the real world. Technically, these forms are known as
Calabi-Yau manifolds, and they can exist in any even number of
higher dimensions. Exotic, elaborate creatures, these extraordinary
forms constitute an abstract taxonomy in multidimensional space; a 2D
slice through them (about the best we can do in visualising what they
look like) brings to mind the crystalline structures of viruses; they
almost look alive.
There
are many versions of string-theory equations describing 10-dimensional
space, but in the 1990s the mathematician Edward Witten, at the
Institute for Advanced Study in Princeton (Einstein’s old haunt), showed
that things could be somewhat simplified if we took an 11-dimensional
perspective. He called his new theory M-Theory, and enigmatically
declined to say what the ‘M’ stood for. Usually it is said to be
‘membrane’, but ‘matrix’, ‘master’, ‘mystery’ and ‘monster’ have also
been proposed.
Ours might be just one of many co-existing universes, each a separate 4D bubble in a wider arena of 5D space
So
far, we have no evidence for any of these additional dimensions – we
are still in the land of swimming physicists dreaming of a miniature
landscape we cannot yet access – but string theory has turned out to
have powerful implications for mathematics itself. Recently,
developments in a version of the theory that has 24 dimensions has shown
unexpected interconnections between several major branches of
mathematics, which means that, even if string theory doesn’t pan out in
physics, it will have proven a richly rewarding source of purely
theoretical insight.
In mathematics, 24-dimensional space is rather special – magical things
happen there, such as the ability to pack spheres together in a
particularly elegant way – though it’s unlikely that the real world has
24 dimensions. For the world we love and live in, most string theorists
believe that 10 or 11 dimensions will prove sufficient.
There is
one final development in string theory that warrants attention. In 1999,
Lisa Randall (the first woman to get tenure at Harvard as a theoretical
physicist) and Raman Sundrum (an Indian-American particle theorist) proposed
that there might be an additional dimension on the cosmological scale,
the scale described by general relativity. According to their ‘brane’
theory – ‘brane’ being short for ‘membrane’ – what we normally call our Universe
might be embedded in a vastly bigger five-dimensional space, a kind of
super-universe. Within this super-space, ours might be just one of a
whole array of co-existing universes, each a separate 4D bubble within a
wider arena of 5D space.
It is hard to know if we’ll ever be able
to confirm Randall and Sundrum’s theory. However analogies have been
drawn between this idea and the dawn of modern astronomy. Europeans 500
years ago found it impossible to imagine other physical ‘worlds’ beyond
our own, yet now we know that the Universe is populated by billions of
other planets orbiting around billions of other stars. Who
knows, one day our descendants could find evidence for billions of other
universes, each with their own unique spacetime equations.
The
project of understanding the geometrical structure of space is one of
the signature achievements of science, but it might be that physicists
have reached the end of this road. For it turns out that, in a sense,
Aristotle was right – there are indeed logical problems with the notion
of extended space. For all the extraordinary successes of relativity, we
know that its description of space cannot be the final one because at
the quantum level it breaks down. For the past half-century, physicists
have been trying without success to unite their understanding of space
at the cosmological scale with what they observe at the quantum scale,
and increasingly it seems that such a synthesis could require radical
new physics.
After Einstein developed general relativity, he spent
much of the rest of his life trying to ‘build all of the laws of nature
out of the dynamics of space and time, reducing physics to pure
geometry’, as Robbert Dijkgraaf, director of the Institute for Advanced
Study at Princeton, put it recently. ‘For [Einstein], space-time was the
natural “ground-level” in the infinite hierarchy of scientific
objects.’ Like Newton’s world picture, Einstein’s makes space the
primary grounding of being, the arena in which all things happen. Yet at
very tiny scales, where quantum properties dominate, the laws of
physics reveal that space, as we are used to thinking about it, might
not exist.
A view is emerging among some theoretical physicists
that space might in fact be an emergent phenomenon created by something
more fundamental, in much the same way that temperature emerges
as a macroscopic property resulting from the motion of molecules. As
Dijkgraaf put it: ‘The present point of view thinks of space-time not as
a starting point, but as an end point, as a natural structure that
emerges out of the complexity of quantum information.’
A leading proponent of new ways of thinking about space is the cosmologist Sean Carroll at Caltech, who recently said
that classical space isn’t ‘a fundamental part of reality’s
architecture’, and argued that we are wrong to assign such special
status to its four or 10 or 11 dimensions. Where Dijkgraaf makes an
analogy with temperature, Carroll invites us to consider ‘wetness’, an
emergent phenomenon of lots of water molecules coming together. No
individual water molecule is wet, only when you get a bunch of them
together does wetness come into being as a quality. So, he says, space emerges from more basic things at the quantum level.
Carroll writes that, from a quantum perspective, the Universe ‘evolves in a mathematical realm with more than 10(10^100) dimensions’ – that’s 10 followed by a googol
of zeroes, or 10,000 trillion trillion trillion trillion trillion
trillion trillion trillion zeroes. It’s hard to conceive of this almost
impossibly vast number, which dwarfs into insignificance the number of
particles in the known Universe. Yet every one of them is a separate
dimension in a mathematical space described by quantum equations; every
one a new ‘degree of freedom’ that the Universe has at its disposal.
Even
Descartes might have been stunned by where his vision has taken us, and
what dazzling complexity has come to be contained in the simple word
‘dimension’.
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