How many seconds are a meter in the 4th dimension?

Question

The title really does say it all. I want to know how many seconds elapse between a meter in the 4th dimension?

I don't know if this question is naive, or my understand of the dimension is off.

I'm thinking of a 4d cube with 3d slices; the breadth of the 4d cube which lies along the 4th dimension is time, right?

Does this breadth have no length?

Perhaps it would be helpful to look at some examples of space-time diagrams to visualize things better. The axes are $x$ and $c t$. So we multiply the time with the speed of light for the vertical axis. This makes it so that one light year of time equals one light year of distance. — bleuofblue, Nov 12 '16 at 19:26
Related and possibly a duplicate: How far does a photon move in the 4th dimension when it travels one light second? — John Rennie, Nov 13 '16 at 11:30
@bleuofblue A purely space-like interval cannot be transformed into a purely time-like interval, so I prefer to say that an interval of 1 second has the same magnitude as a distance of 1 light-second. See here for further ramblings on this matter. :) — PM 2Ring, Nov 13 '16 at 18:38
It's completely unclear what this question is asking. Seconds do not "elapse" between a meter. — ACuriousMind, Nov 13 '16 at 18:51
@ACuriousMind As you can see from the answers, he tried to understand the relation of the timelike and the spacelike coordinates of the 4D Minkowski spacetime. It is a quite common question from curious laymans. — peterh, Nov 14 '16 at 01:04

score 13 · Answer 1 · edited Nov 12 '16 at 20:45

13

Light travels about $3\times 10^8$ meters in a second, so a meter is about $1/(3\times 10^8)$ seconds.

edited Nov 12 '16 at 20:45

auden

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answered Nov 12 '16 at 19:07

Didn't you mean $1/(3\times 10^8)$ seconds? Also, why should we globally assume the speed of light o calculate the time between a meter? Wouldn't it be different depending on the particle and its speed? – cinico Nov 12 '16 at 19:17
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@cinico: We use the speed of light because it's the fundamental speed of spacetime. I assume the OP is asking about that, given that the question mentions the fourth dimension (i.e. time). – Javier Nov 12 '16 at 19:55
@cinico fixed braino: thanks for spotting it. We assume $c$ is constant because experimentally it is, and it's an assumption of relativity that it should be. – Nov 12 '16 at 20:09
So does the 4th dimension grow at 3e8m/s? – Tobi Nov 13 '16 at 03:24
@Tobi that question doesn't mean anything. Dimensions are just a property of the spacetime manifold. – Nov 13 '16 at 09:36
If I drop a ball, I imagine this scene as a 4d cube of 3d slices in time, as the ball drops, does this cube grow in the 4th dimension at 3e8m/s – Tobi Nov 14 '16 at 15:33
@Tobi you seem to be assuming that there are four dimensions and time: that is not the case. It doesn't make sense to talk about how spacetime changes in time. – Nov 14 '16 at 19:52
I'm assuming there are 4 dimensions, the 4th being measurable in time and length. – Tobi Nov 14 '16 at 20:28

score 10 · Answer 2 · answered Nov 12 '16 at 20:32

10

A meter in the fourth dimension is $\frac{i}{c}$ seconds. A four dimensional Euclidean space with metric:

$$ds^2 = dx_1^2 + dx_2^2 + dx_3^2 + dx_4^2$$

can be obtained from the Lorentzian metric:

$$ds^2 = -c^2 dt^2 + dx_1^2 + dx_2^2 + dx_3^2$$

by putting $dt = \frac{i}{c} dx_4$

answered Nov 12 '16 at 20:32

Count Iblis

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To be honest, I don't get this - yet. However, I appreciate the answer. – Tobi Nov 13 '16 at 03:26
@Tobi all Count Iblis is doing there is substituting the given scale of the 4th dimension so we can see it as a function of time rather than distance. Strictly speaking it doesn't answer the question, but it is still a good answer as it aims to further your understanding of the relationship between the dimensions – Darren H Nov 13 '16 at 08:33
Amazing, I never thought about that 'i'. – Kartik Nov 13 '16 at 16:43
@Kartik In the early days of SR it was common to use $x_4 = ict$. It's a cute trick, but it leads to problems, especially when you try to use it in GR, as discussed in Special relativity and imaginary coefficient of the time coordinate. – PM 2Ring Nov 13 '16 at 18:33

AnoE · Answer 3 · 2016-11-13T17:28:16.043

Everybody so far interpreted the question as "how long does light take to move 1m". Let me interpret it in a different way.

How many seconds are a meter in the 4th dimension?

They are not.

The 3 dimensions of space are different than the 1 dimension of time.

Interpretation of the question

Consider this experiment (no rockets, spaceships or black holes involved, simply you sitting in front of your computer on earth):

Name our dimensions X, Y, Z and T. Lay a piece of standard paper on your desk. Draw a horizontal line on it. It will have a certain extent in X (let's call it 10cm), in Y (0.1cm - the width of the stroke of your pen) and Z (0.000001cm or however thick the layer of paint is). Label your desk with a "X" and "Y" axis, and if you wish a little "Z" along one of its legs. Do not label the paper with a coordinate system, we don't need it.

Now turn the paper 90° around the Z axis; it will now have 0.1cm length in X and 10cm in Y (remember, the axes are on your desk, not your paper - the desk is your coordinate system / reference system for this experiment). You did not change the line or the piece of paper at all. You merely rotated it around an axis in another spatial dimension. You can do the same with any kind of rotation, and nobody will be able to tell a difference, as long as you don't fold the paper. If you take very stiff paper and somehow glue it to your desk in a standing position, you can also add a Z length by rotating it "upwards".

Whatever kind of rotation you do, you will change the extent of the line in any of the dimensions (related to your desk, not the paper itself).

All of this was just a verbose setup to ask your question another way:

How many cm are a X-cm in the Y dimension?

The answer is: "the same". After rotation as above, if the line was 10cm in X before, it will be 10cm in Y. That is, the space dimensions (at least the 3 we know of for sure) have the same "spacing" if you so wish, at least in our everyday settings.

(Bear with me, all of this is totally trivial and I wrote it just so you can see how I understood your question.)

About time...

What length does the line have in T? Well, you set a timer when you draw it, and after playing around with it for 10s, you take an eraser and erase the line. This gives you a "length" of 10s in the T dimension.

Now: You can not do the "rotation trick" with T. You cannot turn the line so it is 0cm in X, Y, Z and some different duration in T - you cannot transform space into time. Not theoretically, not practically. No matter what you do (while staying at your desk and not in a rocket ship) will change the line with respect to T in any way whatsoever, as long as you don't go in there with a pen or an eraser, or you tear up the paper. That's also why we would usually call its extent in T something like "duration", not "length".

Space is space, and time is time, even in spacetime.

Addendum

To give you an idea past this "naive" stuff, check out https://en.wikipedia.org/wiki/Spacetime#Spacetime_intervals_in_flat_space for what happens if you put time and space together in a formula... the inherent difference between space and time dimensions gives rise to different types of spacetime-distances with fundamentally different properties (heck, one is even negative...).

score 2 · Answer 4 · answered Nov 13 '16 at 12:32

2

The speed of light is 299792458 meters per second, so it would take $1/299792458$ seconds for the photon to travel one meter.

Best of luck.

answered Nov 13 '16 at 12:32

Bob Jarvis - Слава Україні

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How exactly does this link into the question? – Tobi Nov 14 '16 at 20:33

How many seconds are a meter in the 4th dimension?

4 Answers4

Interpretation of the question

About time...

Addendum