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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?

Volker Siegel
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1 Answers1

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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.

KidElephant
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