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I assume the relation of the three spacial dimensions and the time dimension is handled purely in the mathematical domain, usually.

But is there any intuitive description of this relationship, how they fit together?

To get a better understanding of 4-dimensional spacetime, I'm trying to see similarities or relations between the space- and the time dimensions.

Getting a grasp on how they differ in areas where comparison makes sense should also help.

For example, an object can not accelerate in time - but something equivalent?

So the basic question is whether there are ways to get intuitive understanding of the relation of space and time as part of spacetime.


For possible approaches:
In another questions answer, I read a sensence like

I don't think anyone really has a good intuition about what it means to say that "time is a fourth dimension", despite being able to use the concept in practice.

If you agree, that could be actually the right answer, of course.

I see some naive models to visualize the 4 dimensions - would you think they are of help, or distracting?

Imagining a video as a stack of frames.
This uses the stack growing dimension to represent time, and accepts loosing one space dimension, working on 2D frames.

The other is keeping the three dimensions, used in some 3D visualisation software:

In the 3D space or a 2D projection of it on a screen, actual physical time is used for the time dimension by moving a time slider interactively.

Note how both are missing that time has a preferred direction - which seems to be a rather important property.

Are these approaches useful, or distracting, even misguiding?

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    $\begingroup$ I have tried, in currently learning general relativity, to picture spacetime in 4 dimensions and every time I tried to use this picture in working out a problem, I went wrong. With the math, it can be dry and you have to slog through it, but it's always correct and worth it for the understanding you get. You are trying to reverse all your life thinking in 3 D and its pretty impossible to stop thinking that way, to me at least. Hope this is of some use. $\endgroup$ – user81619 May 26 '15 at 21:20
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    $\begingroup$ Time is measured with a clock, space with a yardstick. That's really all the intuition you need for physics. Everything else is math and folklore. Both are as fun as they are irrelevant in this craft. I think it is pretty obvious that my opinion about the matter leans towards the "irrelevant" side. $\endgroup$ – CuriousOne May 26 '15 at 21:20
  • $\begingroup$ @CuriousOne It can only be irrelevant until you explicitly think about the 4D spacetime, which is the point here. $\endgroup$ – Volker Siegel May 26 '15 at 21:23
  • $\begingroup$ I have no idea how to attach a clock to a vector dreibein made from three yardsticks. Next time we see each other you will have to show me your and-clock-makes-four vierbein, then I buy you a beer. :-) $\endgroup$ – CuriousOne May 26 '15 at 21:25
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    $\begingroup$ It's not particularly clear what the question is, I'm afraid. $\endgroup$ – Emilio Pisanty May 26 '15 at 21:36
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In classical mechanics, we often deal with three vectors and their inner products, that is:

$a \cdot b = a^1b^1 + a^2b^2 + a^3b^3$

(note the superscripts above are not exponents, but indices)

Really, this is a specific kind of inner product, and one which only holds true in a flat, 3-dimensional space where all inner products are positive-definite. There is a more general way to write the inner product, which allows us to apply the concept to spacetime:

$a \cdot b = a^ib^jg_{ij}$

Where $g_{ij}$ is a special matrix-like object called the metric tensor. For Euclidean space (where I assume you've done most of your geometry so far), it is of the form $g_{ij} = \delta_{ij} = \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}$, that is, 1's along the diagonal, and zeros elsewhere. In (flat) spacetime, we use the Minkowski metric, defined as

$g_{ij} = \eta_{ij} = \begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix} $

and we deal with 4-vectors instead of 3-vectors, where the 0th dimension corresponds to the timelike quantity, and the others are spacelike. This allows us to write a position vector which describes exactly on point in space-time, or an event. With this definition, it is easy to see that it's the inner product of vectors which give them physical importance, and in that way, time-like components of spacetime can be thought of as being like imaginary, as their piece of the inner product is negative, but that is not always helpful.

I had to use some math here, but it is hard to explain these concepts without being at least a little bit rigorous.

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    $\begingroup$ We all know the math, but the math is made up. It carries you far, far away from the fact that a yardsticks is a dumb piece of metal that you can turn either way, while an atomic clock is complicated machine that can never turn its operation around. Nature is screaming at you not to put time and space in the same basket... and what do you do? $\endgroup$ – CuriousOne May 26 '15 at 22:07
  • $\begingroup$ @CuriousOne The point is, Einstein found they actually are in the same basket, even if nature managed to hide that for a while. But then, maybe nature knew we'd look for intuition, and was just trying to be nice to us ;) $\endgroup$ – Volker Siegel Jun 13 '15 at 19:25
  • $\begingroup$ @VolkerSiegel: Einstein didn't naively say that position and time coordinates are of the same kind, indeed, in his theory they behave very differently and are even marked as different by the metric. You need to pay attention to little things like a sign. $\endgroup$ – CuriousOne Jun 13 '15 at 19:36
  • $\begingroup$ @CuriousOne Heh ;) No doubt they are very different - I read "in the same basket" as "fundamentally related" - that does not mean "similar". Handling them as intiutively unrelated feels just to wrong to get used to it. $\endgroup$ – Volker Siegel Jun 13 '15 at 19:41

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