# How to explain (pedagogically) why there is 4 spacetime dimensions while we see only the 3 spatial dimesions?

I have been asked this question by a student, but I was able and in the same time incapable to give a good answer for this without equations, so do you have ideas how one can explain this in a simple way?

(Answers like we can take time as an imaginary, or our space is actually pseudo-Euclidean will be hard to grasp for new students.)

Note that the problem is not in visualizing the 4th dimension, that an easy thing to explain. The problem is more related to why we are in 3D that moving along 1D time dimension? In Differential geometry this interpreted by fiber bundles, but how to explain it to usual student.

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Yes I agree with anna, this question up till now had been edited for more than 4 times and by different people! –  TMS Feb 10 '13 at 8:29
You can roll back the edits . If you click on "edited 3 hours ago" the other versions are there –  anna v Feb 10 '13 at 8:32
TMS, you are getting heavily edited because you asked what seems to be an interesting question, but did it using sentences with unexpected words and missing verbs. For example, did you really mean "interpenetrated by fiber bundles", or did you perhaps mean "interpreted"? Please at least review your own text to make sure you have said what you really intended to say. –  Terry Bollinger Feb 11 '13 at 4:49
@Terry, corrected that, it was just auto correction. –  TMS Feb 11 '13 at 6:31

We do see the fourth dimension.

The difference between three dimensions and four dimensions is the difference between a (2d) snapshot image and a ("2d+t") video.

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A good student will immediately respond to that that means nothing more that we parametrized our series of pictures by parameter "time", even so the analogy not bad, but not enough for sure. –  TMS Nov 3 '12 at 14:45
And that would be correct from a good student. Actually time is not a dimension - it is a parameter. Which by some magic happens to look like 4th coordinate, because the interval in special relativity is conserved. That is all. Say like that, good student will understand :) –  Asphir Dom Nov 3 '12 at 21:15
Beleive it or not, but this is almost a question of religion. Catholics (especially French) were first who admitted time as a 4-th dimension. Protestants (Englishs) were considering the time as a parameter for very long time. Another catholic - Hamilton (Irish) - invented quaternions at the time when many protestants did not admit even complex numbers (imaginary unit $i$!). So this dispute has old roots... –  Murod Abdukhakimov Feb 8 '13 at 10:19

To stay well bound to the physics, I suggest to explain the concept of event, to point out that if we want to identify something that happens in our universe we need to locate it in space and time.

Now, the definition of "number of dimensions" is, roughly, how many numbers you need to identify an element (of the vector space). In this case, it's clear you need 3 spatial coordinate, plus "when".

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Some good ideas have been expressed here. My take on this is the following:

Imagine you have a laser gun and you send a laser pulse outwards in the outer space sending an image out there. The laser pulse travels at the speed of light. Now let us time the beam for a length of time $\delta t$. There is some distance that corresponds to this time and it is $c\delta t$ where c is the speed of light. This $c\delta t$ is the fourth dimension in the 4-D Minkowski space, corresponding to that short time $\delta t$. It tells us how far the image has travelled within this short time. So it is the speed of light that generates the fourth dimension, and also gives the length we call the fourth dimension. Therefore, the fourth dimension starts on our watch and $c\delta t$ is the length of it within the time $\delta t$. It makes sense only in the context of the speed of light. This is the whole point of space-time in special relativity. This is what it means when we say that an object is ‘so many light years away’. In a way, this is the distance that the image of an object is away (a galaxy for example) in the fourth dimension. The mathematical representation of this has been written in Leos Ondra reply. I hope this helps somewhat.

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This is a good way to define time as a dimension within special relativity. –  anna v Feb 10 '13 at 6:57

Note that the problem is not in visualizing the 4th dimension, that an easy thing to explain. The problem more related to why we are in 3D that moving along 1D time dimension?

You cannot avoid defining time in terms of change. In the same way that if there were no changes (dx/dy etc) in a terrain the map would be totally uniform and uninteresting, if the terrain did not change in time, time would be uniform and undefinable. Repetitive changes allow us to define time ( no need to go to entropy, the solar system, day/night etc are enough ) and clock it/ measure it.

Time projects into the 3 dimensional world. Geological strata ( and many other proxies) assign to each (x,y,z) a time value on the axis of t. Thus time can be projected into (x,y,z).In a similar way space dimensions project into time. The time taken to walk to the station has one to one correspondence with the distance in kilometers.

So time is a necessary dimension to describe the changes seen in three dimensions, in a similar way that a third space dimension is needed to describe the projections of a sphere to two dimensions.

Entropy must come in for a classical definition of the arrow of time and non reversibility. A rough discussion on disorder, broken glass not repairable etc should give them the concept.

Thus even without special relativity time can be thought as another dimension since it projects into the spatial ones. One can then go on to special relativity as surprising us with the different type of dimension ( pseudo euclidean) it turns out to be ,from experiments.

p.s. with this view of how time is defined in our experience we can also with assurance say that there is only one time dimension. If there were two time dimensions the functional dependence of changes in the space three dimensions would be complicated. It would be a many to one projection, similar to projecting a three space dimensional object to one space dimension.

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I want to add that my answer is an extension of the answer of @Bzazz –  anna v Feb 10 '13 at 6:49

You might introduce the thought of time being a fourth dimension by asking your students to contemplate the meaning of 'perpendicular'.

They will likely respond that length, width and height are perpendicular directions. If you push further and demand a defining characteristics of 'perpendicular' they will probably arrive at the non-mathematical and hand-waving characteristics that perpendicular directions are those that allow you to move in either of these dimensions, without making any movement in any of the other. At that stage you can ask them if you can move in time without moving in any of the three spatial dimensions.

Just leave them with that thought. They will come back with further questions...

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"They will come back with further questions..." Like the question: Is it possible to move only along x axis without moving a bit in y, z and time? –  Leos Ondra Nov 3 '12 at 12:11
Exactly. And that is the point at which you can reveal that time behaves different than the other directions, which causes us to observe time as distinct from the other three dimensions. –  Johannes Nov 3 '12 at 12:16
@Johannes: I added a post note in the question please check it –  TMS Nov 3 '12 at 14:49

Here's an attempt at a non-mathematical (and unorthodox) answer:

1) We can see in the three spatial dimensions because light travels through them, reflecting back from objects around us and eventually reaching your eye.

2) Everything (incl. light) travels "forward" through the time dimension (meaning, it is able to move in only one direction, and cannot move back and forth).

3) Light is therefore not able to reflect "back" from an object to your eye through the time dimension.

Thought experiment: take one spatial dimension, but everything is moving in one direction at the speed of light. Would you be able to see what's right next to you?

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I think one way is to illustrate it in 2 + 1 dimensions, and have them imagine it in 3 + 1. You could draw a cube, and explain that a slice along the cross section of the cube is a 'snapshot' of a 2D space, and the third dimension is time. So they are moving around in 2D, while they are "experiencing" the third.

It's hard to explain this exactly without a figure... if I can lay my hands on one or I can make one myself, I'll add it here when I do.

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Note that the problem is not in visualizing the 4th dimension, but why we humans can't "feel"/see it as the other 3. –  TMS Nov 3 '12 at 14:47
@TMS So what are you asking about? What is the difference between time and space? –  Leos Ondra Nov 3 '12 at 18:26
@Leos: I added post note in the question to describe the issue clearly. –  TMS Nov 3 '12 at 20:46

We see 3 dimensions because we ourselves are 3-dimensional. Imagine a 2d creature originally living in 2d space - an Euclidean plane. It naturally perceives only events which occur in its body, like a photon (assuming for a while something that something like light can exist in 2d) which interacts with its 2d cell in retina. If somehow in the course of evolution the third dimension is added to its flat world, it will still perceive only two dimensions. Now, hoverer, it can be rotated in 3d so its plane of living changes, and new strange things occur - a rod, which has always kept its length in the original flat 2d world, because the flat Pythagoras (and many others before him) proved that

$ds^2 = dx^2 + dy^2$

Now, the rod strangely contracts and lengthens, but one creature (called Einstein the Flat by some, and Lorentz or Fitzgerald by others) founds that there is still something like length which remains constant, namely

$ds^2 = dx^2 + dy^2 - dt^2$

which sounds strangely to others because

1. t has never been considered a dimension
2. There is minus sign before.
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Because it seems this question interesting to many people, I will tell you how currently I'm explaining this, the idea was came to me when I read about how Philosophers understands time, correct me if you think there is something wrong:

Even so we treat time as a dimensions, it's not that similar to the spatial one, and the reason is as follows:

We start with the most fundamental concept in physics: cause and effect, this concept enable us to sort events in a series : first event is cause then effect ... , that creates an "illusion" of the ability to number those events, what in turn provides as with ability to treat time as dimension and measure "distance" between events, anyway I said the illusion of numbering, because saying "numbering" tells us that we can do that in absolute way, which is wrong according to theory of relativity, because numbering events for one observer is not compatible generally speaking with others (here i explain how speed of light affects cause and effect), for that time is not the same as spatial dimension, and that why Menikowski space is pseudo-Euclidean, not Euclidean, then I add Anno2001's answer how to look at time.

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