What is the difference between a space dimension and a time dimension?

Question

The simplest case is obviously 4D spacetime composed of 3 space dimensions and 1 time dimension. In some talks i stumbled across physicists and mathematicians who talked about spaces in which you only work with space dimensions or only with time dimensions. So I woundered what excatly is the mathematical difference between space and time dimensions.

Answer 1

I feel like this should be more of a comment, but it's too long, so I'm posting this as an answer.

There's no strictly mathematical difference whatsoever, it all comes down to specific physical models.

In classical non-relativistic mechanics, time plays a special role since the equations of motion (Newton's/Lagrange's/Hamilton's/...) include derivatives with respect to time but not space:

$$m\frac{d^2x}{dt^2} = F$$

In classical continuum mechanics (heat/wave/diffusion equations, fluid mechanics) the equations typically contain partial derivatives with respect to all variables, so there's little distinction between time and space (time still bearing it's usual physical meaning, though). However, the specific form this derivatives take here differ for time and space: e.g. you'd typically have time derivative multiplied by one coefficient and spatial derivatives having another one, e.g. in wave equation:

$$\frac{d^2u}{dt^2} = c \frac{d^2u}{dx^2}$$

, or have first-order time derivative and second-order spatial derivatives, like in heat equation:

$$\frac{du}{dt} = c \frac{d^2u}{dx^2}$$

In special relativity, time and space are combined into a single entity, yet they differ via the metric: intervals in time and intervals in space have different signs (there are two conventions: time is negative and space is positive, or the other way round; it doesn't really matter). Note that in this case, due to coordinate independence, there's no way to say that a specific axis is the time axis, we can only say it's time-like or space-like.

In quantum mechanics, time plays a special role again: the equations (Schrödinger equation) have time derivative in a special position:

$$i\hbar \frac{\partial \psi}{\partial t} = \hat H \psi$$

They may also contain spatial derivatives for a different reason, as quantum counterparts of certain classical values, e.g. the kinetic energy operator $\hat T = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$ (for a 3-dimensional particle).

In quantum field theory, which tries to combine (among other things) special relativity and quantum mechanics, time and space are reunited, in the same way as in special relativity.

Answer 2

Loosely speaking, the number of dimensions is the number of coordinates it takes to specify a vector. So four dimensional space is the set of vectors $(x,y,z,t)$ where the entries are real numbers.

There is no way to call these "space" or "time" dimensions, and there's no need to limit their number to four.

In the ordinary two dimensional plane, the distance between to points is the square root of $$ (x_2-x_1)^2 + (y_2-y_1)^2 $$ (that's the Pythagorean theorem).

In four dimensional space the geometric ("space") distance would be the square root of $$ (x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2 + (t_2-t_1)^2 . $$

When modeling physical reality it takes four numbers to specify an event : three for "where" and one for "when". So the space of events is four dimensional. It turns out that the useful way to measure how far apart events are is to replace one $+$ sign by a $-$ sign to get this expression: $$ (x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2 - (t_2-t_1)^2 . $$

It's that minus sign that makes the $t$ coordinate time-like.

Answer 3

Time-like dimensions have an intrinsic direction, i.e:

$$ t_0 < t $$

Every moment leads naturally to the next moment: forwards in time. You can look backwards in time$^1$ by looking at the light from distant objects, but you can't go backwards in time. You can go to the future at a pace related to your speed (special relativity), but you cannot see into the future. All of this would be absurd with spatial dimensions - I can look left or right, I can move up or down, and I can almost always go back to the place I started.

$^1$_{Technically, everything we see is from the past}