Extra dimensions – and how to hide them

Why our universe could possess dimensions beyond length, width and depth – and why those dimensions need not be noticeable in everyday life

An article by Stefan Theisen

How many dimensions does the world have? The answer, it seems, is obvious: Space has three dimensions. One indication is that, to define a location in space, we need exactly three numbers – for instance geographic latitude and longitude as well as height above the earth’s surface. Another is volume: the volume of a cube that encompasses a certain portion of space is equal to the length of the cube’s sides, to the third power. (In special relativity as well as in general relativity, time is added as a fourth coordinate, resulting in four-dimensional spacetime. This special and very natural extra dimension will not play a role in the following.)

Putting the question differently: How sure can we be that space has three dimensions? Surely there cannot be less than three, but might there be more? At first glance, this question might seem absurd. But in fact, theoretical physicists have been discussing such extra dimensions for a good number of years now.

Space from afar

Imagine a rope, used in some backyard as a washing line, and imagine that you observe that scene from afar and then zoom in until you notice an ant moving around on the rope’s surface:

Rope with ant moving around on it's surface

Viewed from a distance, the rope looks like a one-dimensional object, akin to a line, with just one possible set of directions of motion: forward or backward along the line (represented by the cyan double arrow):

Rope viewed as one-dimensional line

The ant’s view of space

Only if you look closer you will notice that the rope has a certain circumference – a two-dimensional surface. Insects such as the aforementioned ant, which move on its cylindrical surface, will experience their surroundings as two-dimensional: at every location, they can move in two sets of directions, not just one: backwards and forwards along the rope (cyan double arrow), but also orthogonal to that direction, along the rope’s circumference (black double arrow):

Ant on rope – a two-dimensional surface

An intelligent ant might also notice that this second dimension is rolled up, so to speak: If they wander straight along the circumference without changing their direction, they will end up where they started – in an ordinary plane, that could never happen.

The picture of the ant on the surface of the rope can be generalized to three-dimensional space. Experience tells us that there are no more than three extended dimensions. But it is still possible that space has hidden extra dimensions that are “rolled up” in a similar way as the second dimension of the rope: extra dimensions with but a tiny circumference, too tiny to be noticeable to our senses or to usual measuring instruments.

There are, in fact, physical theories that predict the existence of such extra dimensions. In string theory, for instance, the natural number of space dimensions for our universe is nine or even ten. Rolling up the extra dimensions – either in the same simple way as with the garden hose or in much more complicated shapes – is an important tool for fashioning, from these theories, models for our world with its mere three extended space dimensions.

Further Information

The relativistic background of this spotlight topic is explained in Elementary Einstein, especially on the page Superstrings and universal harmony in the chapter Relativity and the quantum.

For more information about extra dimensions, check out the spotlight topics Hunting for extra dimensions, The embedded universe and Simplicity in higher dimensions. Related spotlight topics can be found in the section Relativity and the quantum.

Colophon

Stefan Theisen

is a research group leader at Albert Einstein Institute with a research focus on string theory.

Citation

Cite this article as:
Stefan Theisen, “Extra dimensions – and how to hide them” in: Einstein Online Band 04 (2010), 03-1013

Definitioner

time

It is a fact of life that not all events in our universe happen concurrently - instead, there is a certain order. Defining a time coordinate or defining time, the way physicists do it, is to define a prescription to associate with each event a number so as to reflect that order - if event B happens after event A, then the number associated with B should be larger than that associated with A. The first step of this definition is to construct a clock: Choose a simple process that repeats regularly. (What is "regular"? Luckily, in our universe, all elementary processes such as a swinging pendulum, the oscillations of atoms or of electronic circuits lead to the same concept of regularity.) As a second step, install a counter: A mechanism that, with every repetition of the chosen process, raises the count by one. With this definition, one can at least assign a time (the numerical value of the counter) to events happening at location of the clock. For events at different locations, an additional definition is necessary: One needs to define simultaneity. After all, the statement that some far-away event A happens at 12 o'clock is the same as saying that event A and "our clock counter shows 12:00:00" are simultaneous. The how and why of defining simultaneity - a centre-piece of Einstein's special theory of relativity - are described in the spotlight topic Defining "now". With all these preparations, physicists can, in principle, assign a time coordinate value ("a time") to any possible events, and describes how fast or how slow processes happen, compared to that time coordinate.

surface

Geometric space with two dimensions. Examples include the plane or the surface of a sphere.

string theory

Candidate for a theory of quantum gravity; a quantum theory, where the fundamental building blocks are tiny, one-dimensional, oscillating entities, called strings. A brief description can be found on the page String theory in the chapter Relativity and the quantum of Elementary Einstein.

string (string theory)

special relativity

Albert Einstein's theory of the fundamentals of space, time, and movement (but not gravity). For a brief introduction, check out the chapter Special relativity of Elementary Einstein. Selected aspects of special relativity are described in the category Special relativity of our Spotlights on relativity.

space

In a strict sense: Space as we know it from everyday life: the totality of all locations in which objects can sit, with three dimensions. In a more general sense used by mathematicians, all kinds of sets of points are spaces - a line for instance, which has but a single dimension, or a two-dimensional surface, but also higher-dimensional spaces. Also, in such more general spaces, geometry can be different from the standard Euclidean geometry taught in high schools - such spaces can be curved.

spacetime

Already in special relativity, observers in motion relative to each other will not, in general, agree as to whether two events happen simultaneously, or as to how great is the distance between two objects. They do, however, agree as to what events there are, although not to when and where they happen. This observer-independent totality of all events is called spacetime. How spacetime is split into space and time can differ from observer to observer. Every-day space has three dimensions. Adding time adds another dimension - spacetime has four dimensions, all in all. We are used to the idea of a point in space - an object that has only one location and is completely defined once its space coordinates are given. In spacetime, a spacetime point is an object defined completely once its space coordinates and its time coordinate are given - which makes a spacetime point nothing but an elementary event. The idea of spacetime is, in addition to its role in special relativity, a building block of general relativity. Analogous to how a plane is flat, but the surface of a sphere is curved, in general relativity, curved or distorted versions of the simple, flat spacetime of special relativity play a role. Spacetime curvature, in general relativity, is intimately connected with gravity. For an introduction to the basics of both theories of relativity, check out the chapters Special relativity and General relativity in Elementary Einstein. Sometimes, it can be helpful to view spacetime in analogy to ordinary space - such analogies are explored in the spotlight topics Time dilation on the road (for time dilation) and Twins on the road (for the twin effect).

second

In the International System of units: the basic unit of time. Defined as a certain multiple of the oscillation period of electromagnetic radiation set free in a certain transition within the electron shell of atoms of the type Cesium-133.

relativistic

Models, effects or phenomena in which special relativity or general relativity play a crucial role are called relativistic. Examples are relativistic quantum field theories as theories based on special relativity, or the relativistic perihelion shift as a consequence of general relativity. In addition, conditions under which the difference between relativistic physics and ordinary, classical physics are especially pronounced, are also called relativistic. For instance, when material objects reach speeds close to speed of light, one talks of relativistic speeds, while speeds that are so small compared to light as to make relativistic effects undetectably small are non-relativistic.

plane

A surface within which the axioms of Euclidean geometry (synonym: plane geometry) hold - the rules of geometry as they are taught in high school, with well-known formulae such as Pythagoras's theorem and "the perimeter of a circle is 2 times pi times its radius" hold.

relativistic (perihelion advance, relativistic)

For planetary orbits, there is a small difference between the predictions of Newtonian gravity and general relativity. For instance, in Newton's theory, the orbital curve of a lonely planet orbiting a star is an ellipse. In general relativity, it is a kind of rose or rhodonea curve. Such a curve is similar to an ellipse curve, which shifts a bit with each additional orbit. The shift can be defined by looking at the point which is closest to the sun (perihelion) on each orbit. The additional relativistic shift is, hence, called relativistic perihelion shift or relativistic perihelion advance. A picture can be seen on the page A planet goes astray in the chapter General relativity of Elementary Einstein.

line

Geometric object with a single dimension. A line can either be an independent one-dimensional space (in the abstract mathematical space where a space does not need to have three dimensions), or it can be embedded into a more general space, like a line drawn onto a piece of paper (i.e. a surface).

general relativity (general theory of relativity)

Albert Einstein's theory of gravity; a generalization of his special theory of relativity. For information about the concepts and applications of this theory, we recommend the chapter general relativity of our introductory section Elementary Einstein. Further information about many different aspects of general relativity and its applications can be found in our section Spotlights on relativity.

focus

In optics: A focus, also referred to as an image point, is the point where incoming, parallel light rays meet after traveling through a lens. In geometry: For an ellipse all points on the curve are located around two focal points, such that the sum of the distances to each focal point is equal for all points.

extra dimension

According to some of the candidate models for a theory of quantum gravity, notably in string theory, our world should have extra dimensions - space dimensions in addition to the three we know from everday life. More information about extra dimensions, possibilities of observing their effects and their use for building physical models can be found in our section Spotlights on relativity, namely Extra dimensions and how to hide them, The hunt for extra dimensions and Simplicity in higher dimensions.

extra dimensions (extra dimension)

dimension

The number of independent directions within a set of points, alternatively: the number of coordinates needed to give each point a unique name. This is rather abstract - time for some examples: A line is one-dimensional. There's only one direction to go on the line (the opposite direction isn't counted extra): Back-forth. A single number is sufficient to define a point of the line. For instance, on a motorway, given the statement "the accident happened 4 kilometres from the beginning of the I95 (or M1, or whatever)" is sufficient information for the rescue workers to know exactly where to go. Surfaces are two-dimensional, as there are two independent directions: back-forth and left-right, say. On the earth's surface, the two coordinate numbers geographical longitude and latitude uniquely define each location. The space that surrounds us is three-dimensional. There are three independent directions, say back-forth, left-right and up-down. In order to define a location in space, one needs to specify three numbers - for instance, two to specify where a house is located on the earth's surface (latitude/longitude, see above) and one floor number (or, more precisely, the height above the earth's surface). Adding time to the three space coordinates (a must for defining an appointment - where and when?), the result is four-dimensional spacetime. In order to define an event in spacetime, one needs to give four numbers: three of them determine where in space the event happens, the fourth gives the time where it happens. According to some of the models that have been studied as candidates for a theory of quantum gravity, our world should have even more space dimensions than the usual three. Some information about these extra dimensions can be found in the spotlight topics "Extra dimensions, and how to hide them", "Hunting for extra dimensions" and Extra dimensions and simplicity.

Albert Einstein Institute

One of the research institutes of the Max Planck Society; an international centre for research on Einstein's theory of gravity - from the mathematical fundamental, astrophysics and gravitational waves to quantum gravity. Founded in 1995, the institute is situated in Golm near Potsdam in Germany. In 2002, the experimental branch of the institute was opened in Hannover. It is dedicated to research with the gravitational wave detector GEO 600 . AEI webpages Website of AEI-Hannover