# What do I need to read to understand dimensions and spacetime?

The concept of dimension seems to be:

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it.

According to wikipedia. But the fourth dimension seems to have also some kind of connection with spacetime which seems to be related to Minkowski spaces. I want to understand dimensionality and also learn about these issues about spacetime.

I'm searching for references on what I should read in order to understand this, I'm searching for a serious way (no pop-science) to understand it, I'm searching for a list of topics and also some recomendations on textbooks for it.

I hope I'm not asking too much, but I'm very curious to grasp this subject. I'm also realistic on this, I do not think it's something easy of fast to learn.

EDIT: I've found some books on Minkowski spaces, I guess I'll be able to understand from those in the near future.

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It is easy to be confused because the different meanings often given to the "fourth dimension". In the simplest an more natural case, the fourth dimension is just another spatial dimension, and you can have as many dimensions as you want. An introductory book on linear algebra should make it easy to understand (look for metric spaces in particular). This fourth dimension has nothing to do with time. But... In physics, relativity theory defines a relationship between space an time, in the sense that space and time can be interchanged if two observers move relative to each other. The equations for this relationship resemble those of metric spaces. The equations look as if time behaves like an "imaginary" (because in involves imaginary numbers) extra spatial dimension. In the end the mathematical structure can be described as something called Minkowsky space, although it doesnt behave like an actual spatial four dimensional spatial space. Because of this relationship between time and space, usually people consider time as the fourth dimension, but it is not a good analogy. A fourth spatial dimension would behave very differently, and moving into a 4D space from somebody inhabiting 3D space would be equivalent to moving into 3d space for somebody living in a 2d planar surface. You can read about hypercubes to get a better intuition. But remember, dont mix extra spatial dimensions with time as a fourth dimension, they are completely different stuff.

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There are two different concepts of the fourth dimension in what you talk about. Four dimensional Euclidean space, $\mathbb R^4$ has four equivalent coordinates. To find the distance between two points, you sum the squares of the differences of coordinates and take the square root. Many things are similar to your experience in $\mathbb R^3$

Spacetime is a different creature. The time axis is fundamentally different from the three space axes. Your normal experience should tell you this and it doesn't lie. Mathematically it is because the "metric" has a minus sign on the difference in time-$d^2=\Delta x^2+\Delta y^2+\Delta z^2 -c^2\Delta t^2$ This means "distances" are not positive definite. Anywhere a light ray goes is zero distance. If the distance between two points is greater than zero, they can't influence each other and there is a reference frame where they happen at the same time. If the distance is less than zero, one precedes the other and does in all reference frames.

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Distance between two points less than zero? Is it possible? How? I intuitively thought that the distance between two points could always be a positive number. –  Vÿska Feb 19 '13 at 5:44
Oh sorry, I guess you're saying something like: $D>0=$ something in my front and $D<0=$ something in my back. Is this it? –  Vÿska Feb 19 '13 at 5:50
Actually, what can be negative is the square of the distance, this is because in Minkowski space the time axis is imaginary, so the distance itself can be an imaginary number. But Gustavo, from my point of view, it is just a math trick that extends the intuitive definition of distance in real space to other physical system. I dont think there are many physicists that think that an imaginary axis is "real" (but I might be wrong) –  julian fernandez Feb 19 '13 at 5:58
@julianfernandez: In fact, there is a discussion in Wheeler's gravitation about how you can call the time axis imaginary (in that the coordinates are $ict$) in special relativity and make things work, but not in general relativity. You have to view the minus sign as coming from the metric. I don't remember the details. –  Ross Millikan Feb 19 '13 at 13:59
@Ross thanks, I wasn't aware of that!! –  julian fernandez Feb 19 '13 at 14:26

A big difference between 4d space and spacetime is its size. Consider the size (Cardinality) of all the sets which have a one to one mapping between the digits on my hand and their members to be 5. Define the Cardinality of the set of all integers as $\aleph_0$, said as aleph-null. Note that it is possible to produce a 1 to 1 mapping between the set of integers and the set of even integers. This is a property of infinite sets, they have 1 to 1 relationships with proper subsets of themselves, and these subsets have the same cardinality as the original set.

Note that $\aleph_0+\aleph_0=\aleph_0$. The set of Real numbers is composed of Integers + Fractions + Irrational + Transcendental numbers. The cardinality of Integers Fractions and Irrational numbers is $\aleph_0$. The cardinality of transendental numbers is $\aleph$ which is larger than $\aleph_0$ (probably $\aleph_1$ but not proven to be). $\aleph_0+\aleph_0+\aleph_0+C=C$ and this is the cardinality of the points on a line. A line is a proper subset of 2d space so the cardinality of 2d (and by extension any_d) space is C.

Spacetime may be imagined as your monitor consisting of C pixels each of which may represent any colour. Replacing the monitor with 3d space and the colour with time infers that spacetime has C points. This is F (probably $\aleph_2$ again not proven) because it is also the cardinality of the set of all single valued functions.

We have N < C < F. This makes it possible to resolve Zeno's paradoxes by having the arrow stationary in spacetime but moving in 3d.

This is called Relativity and would be the mathematical framework for a 3d universe moving through some other thing. Special Relativity and General Relativity continue the Relativity tag because it was thought that the universe would be explained based on this model. It turned out that realizing that the arrow and the observer have a different understanding of what is NOW is more fruitful.

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Well, but we still have $|\mathbb{R}^n| = \mathfrak{c}$ for every $n \in \mathbb{N}$. Even though cardinality is meant to talk about the size of a set, it's something pretty different than the notion of dimension that comes from linear algebra and differential geometry. –  user1620696 Aug 11 '13 at 13:47