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

Question

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.

Answer 1

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.

Answer 2

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.