What is a Dimension?
Most recent answer: 10/22/2007
Q:
Under what factors can something be considered a dimension.
Can u explain wad is a dimension, and why time is the fourth dimension too?
- Qihan and Wufan
- Qihan and Wufan
A:
Hi Qihan and Wufan,
That’s a very deep question! People have though quite a lot about this over the years and we’re still working on it. Let’s get some flavor for some of the ideas concerining dimensions.
Let’s start with math. We can devise "spaces" which consist of "points" which need N separate numbers to specify a location in space. We put these N numbers for each point inside parentheses like so:
(3.4, 5.3, 1.8, 2.2)
This may mean that a point is located 3.4 meters along one axis, then make a right angle turn and go 5.3 meters, make another right angle turn (perpendicular to both before) and go 1.8 meters, and then make yet another turn and go 2.2 meters, and there you are. This would be a point in 4-dimensional real space, and the set of all such points is the space. Another set of points has the same dimensionality as this if you can construct a continuous mapping between N-dimensional real space and the other set you are interested in (and such that each point in one corresponds to exactly one point in the other). For example, the surface of the earth is two-dimensional, even though the coordinates (latitude and longitude) have some problems at the poles. The dimensionality of a space is how many numbers you need in order to specify a point in it, and where this map is continuous.
Here’s an example of a map that’s not continuous -- suppose we consider the x-y plane. It’s two-dimensional. Pick a point (x,y) and write x and y out in decimal form. Then pick one digit at a time from x and one at a time from y, and make a new decimal number by alternating the digits of x and y. (for example x=1.23, y=4.56, and my new number is 1.42536). I can specify any place in the x-y plane with just one number! But the mapping is not continuous so this doesn’t count as the dimension. There are more rigorous definitions, but they can be obscure and hard to explain here.
Back to physics -- in the space we live in, you can only find two directions perpendicular to a third and perpendicular to each other. Trying to put a fourth axis down results in some of the angles not being 90 degrees. There are lots of other indications that there are only three dimensions. One is the heat capacity of gases. Molecules can only jiggle in three space dimensions, and so the amount of energy versus the temperature tells us how many different directions things can go. We’ve investigated some systems (like thin films of materials with electrons which can move in them) and found the heat capacities behave as if they can only live in two dimensions. Another is the inverse-square law for the strengths of electrostatic forces (and gravity). If the field lines spread out in space, the dimensionality of space affects the strength of the force as you get away from the source of the force. If we lived in more or fewer dimensions, electricity, magnetism, and gravity would act differently.
Time isn’t quite the same kind of dimension as space. Einstein’s special and general theories treat time and space together (you need three numbers to specify a location and one number to specify time to fully specify an event), but one of the coordinates is not like the others. The distance between two points in real space is given by the Pythagorean theorem: D=square root((x1-x2)^2+(y1-y2)^2 +(z1-z2)^2). With time included, a much more useful "distance" function which helps describe actual physical behavior is D=square root((x1-x2)^2+(y1-y2)^2+(z1-z2)^2-(c*t1-c*t2)^2). This minus sign makes all the difference! And the speed of light c comes in to set the scale of distance and time. This second "distance" is useful because it doesn’t depend on how fast an observer is moving, while the first "distance" does. So yes, it is important to treat time along with the three spatial coordinates, but not all four coordinates can be treated the same.
Adding in gravitational fields only makes things more complicated -- changing the "distance" formula. But always one dimension comes in with an opposite sign when computing the distance. We often say we live in "3+1 dimensions" rather than in four dimensions for this reason.
Some people propose that space actually has many more than three dimensions. These are needed in string theories of elementary particle physics or in other models with extra dimensions. Somehow we have to get rid of them to explain why we do not see them. They could be very very tiny and wrapped up -- you can only go a tiny distance in one before coming back to where you started. Then you might not notice that it’s there. Unless it affects the strengths of some forces, like gravity and electricity and magnetism. Electricity and magnetism are much stronger forces than gravity is, and the idea is that they could have the same strength if gravity is somehow spread out over more dimensions than electricity and magnetism, but that these extra ones are rolled up really tight. There is no experimental evidence for such extra dimensions, although people continue to search for them (the idea’s kooky, but then again, many of the ideas we have now we’d think of as kooky if it weren’t for all the evidence for them). We should treat such stuff with open-minded skepticism.
Tom
That’s a very deep question! People have though quite a lot about this over the years and we’re still working on it. Let’s get some flavor for some of the ideas concerining dimensions.
Let’s start with math. We can devise "spaces" which consist of "points" which need N separate numbers to specify a location in space. We put these N numbers for each point inside parentheses like so:
(3.4, 5.3, 1.8, 2.2)
This may mean that a point is located 3.4 meters along one axis, then make a right angle turn and go 5.3 meters, make another right angle turn (perpendicular to both before) and go 1.8 meters, and then make yet another turn and go 2.2 meters, and there you are. This would be a point in 4-dimensional real space, and the set of all such points is the space. Another set of points has the same dimensionality as this if you can construct a continuous mapping between N-dimensional real space and the other set you are interested in (and such that each point in one corresponds to exactly one point in the other). For example, the surface of the earth is two-dimensional, even though the coordinates (latitude and longitude) have some problems at the poles. The dimensionality of a space is how many numbers you need in order to specify a point in it, and where this map is continuous.
Here’s an example of a map that’s not continuous -- suppose we consider the x-y plane. It’s two-dimensional. Pick a point (x,y) and write x and y out in decimal form. Then pick one digit at a time from x and one at a time from y, and make a new decimal number by alternating the digits of x and y. (for example x=1.23, y=4.56, and my new number is 1.42536). I can specify any place in the x-y plane with just one number! But the mapping is not continuous so this doesn’t count as the dimension. There are more rigorous definitions, but they can be obscure and hard to explain here.
Back to physics -- in the space we live in, you can only find two directions perpendicular to a third and perpendicular to each other. Trying to put a fourth axis down results in some of the angles not being 90 degrees. There are lots of other indications that there are only three dimensions. One is the heat capacity of gases. Molecules can only jiggle in three space dimensions, and so the amount of energy versus the temperature tells us how many different directions things can go. We’ve investigated some systems (like thin films of materials with electrons which can move in them) and found the heat capacities behave as if they can only live in two dimensions. Another is the inverse-square law for the strengths of electrostatic forces (and gravity). If the field lines spread out in space, the dimensionality of space affects the strength of the force as you get away from the source of the force. If we lived in more or fewer dimensions, electricity, magnetism, and gravity would act differently.
Time isn’t quite the same kind of dimension as space. Einstein’s special and general theories treat time and space together (you need three numbers to specify a location and one number to specify time to fully specify an event), but one of the coordinates is not like the others. The distance between two points in real space is given by the Pythagorean theorem: D=square root((x1-x2)^2+(y1-y2)^2 +(z1-z2)^2). With time included, a much more useful "distance" function which helps describe actual physical behavior is D=square root((x1-x2)^2+(y1-y2)^2+(z1-z2)^2-(c*t1-c*t2)^2). This minus sign makes all the difference! And the speed of light c comes in to set the scale of distance and time. This second "distance" is useful because it doesn’t depend on how fast an observer is moving, while the first "distance" does. So yes, it is important to treat time along with the three spatial coordinates, but not all four coordinates can be treated the same.
Adding in gravitational fields only makes things more complicated -- changing the "distance" formula. But always one dimension comes in with an opposite sign when computing the distance. We often say we live in "3+1 dimensions" rather than in four dimensions for this reason.
Some people propose that space actually has many more than three dimensions. These are needed in string theories of elementary particle physics or in other models with extra dimensions. Somehow we have to get rid of them to explain why we do not see them. They could be very very tiny and wrapped up -- you can only go a tiny distance in one before coming back to where you started. Then you might not notice that it’s there. Unless it affects the strengths of some forces, like gravity and electricity and magnetism. Electricity and magnetism are much stronger forces than gravity is, and the idea is that they could have the same strength if gravity is somehow spread out over more dimensions than electricity and magnetism, but that these extra ones are rolled up really tight. There is no experimental evidence for such extra dimensions, although people continue to search for them (the idea’s kooky, but then again, many of the ideas we have now we’d think of as kooky if it weren’t for all the evidence for them). We should treat such stuff with open-minded skepticism.
Tom
(published on 10/22/2007)