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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.

Luxamba
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4 Answers4

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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.

lisyarus
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  • Newtonian/Lagrangian/Hamiltonian formalisms all have derivatives wrt velocity/momentum as well. – user121330 Apr 07 '22 at 18:14
  • A change of coordinates can set $c \to \pm 1$, though — just use $\xi = x/\sqrt{|c|}$. What really makes the difference in the wave equation is the fact that $c > 0$. In contrast, Laplace's equation has $c < 0$ when written in this form, and has fundamentally different properties as a PDE. – Michael Seifert Apr 07 '22 at 19:00
  • @user121330 Indeed, though they appear in different forms as well. I'll update the answer. – lisyarus Apr 08 '22 at 05:15
  • Actually, Potential energy back to force is explicitly a derivative wrt to position. Further, we expect things to move forwards and backwards in space, but not so in time. While we can do the math to go backwards in time, nothing (ignoring one interpretation of anti-matter) does. The temporal bi-directionality of many equations is a [helpful] mathematical artifact and does not represent reality. – user121330 Apr 08 '22 at 06:49
  • @user121330 Well, since the question is posted on math.se and not physics.se, I'm interpreting it as asking for purely mathematical differences; "(not) representing reality" doesn't seem to be such a difference. – lisyarus Apr 08 '22 at 14:33
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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.

Ethan Bolker
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  • "It's that minus sign that makes the $t$ coordinate time-like." It is different, but why does it make the coordinate time-like? – Sextus Empiricus Apr 07 '22 at 18:23
  • It's time like because that minus sign for the time coordinate makes the physics come out right. See https://en.wikipedia.org/wiki/Minkowski_space and https://en.wikipedia.org/wiki/Causal_structure . – Ethan Bolker Apr 07 '22 at 19:10
  • "because that makes the physics come out right" That is a very indirect explanation. The link to Minkowski space is also a bit handwaving, it doesn't solve the lack of explanation. – Sextus Empiricus Apr 07 '22 at 20:23
  • But maybe this question should be moved to physics instead of on mathematics, because indeed the first part is correct. Mathmatically, there is no way to call these "space" or "time" dimensions, and there's no need to limit their number to four. It is the physics that creates the differences and mathematics is only used to describe it. – Sextus Empiricus Apr 07 '22 at 20:37
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    The mathematical difference between dimensions called "spacelike" and dimensions called "timelike" is the minus sign. The reason those adjectives were chosen for the signs comes from the physical application. In that application there is just one timelike dimension. Mathematically, you can study a space with arbitrarily many of either kind. In those other contexts I don't think you call the ones with a minus sign timelike. – Ethan Bolker Apr 07 '22 at 20:38
  • The signs are more like a characteristic. It remains unclear how or why the signs make that difference between space and time. – Sextus Empiricus Apr 07 '22 at 20:40
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    My last comment in this back and forth. The mathematical model with one minus sign for the time coordinate is good physics: it makes experimentally verifiable predictions. The explanations on the wikipedia pages for Minkowski space and causation try to explain the "why" that makes the physics plausible. – Ethan Bolker Apr 07 '22 at 20:45
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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

user121330
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Look at the interval metric. The difference between time and space is that to get the absolute (Lorentz-invariant) distance, (the square of) elapsed time between 2 events is SUBTRACTED from (the squares of) the space distances x,y,z.

This compels you, whether you like it or not, to concede that elapsed time is negative spatial distance. This is why, when space distance and elapsed time are the same, the object exists on a null surface (light cone or event horizon) with a zero interval.

Miss Understands
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