What is the correct understanding of the ct dimension?

[XVX]

I'm confused about how to understand the ct dimension.


I cannot get passed thinking that physically real dimensions mean more degrees of motion.


If a flatlander went to the store to buy a can of Coke, the closed container would be a circle. The container is closed simply because there is no 3rd dimension.

As soon as the flatlander brings his circle of Coke into three dimensions, the Coke will leak out.

And if we live in four dimensions, then how did we create closed containers?


What is the correct understanding of the ct dimension?

 

[dst]

Actually, the coke will not leak out. Time is separate for a reason - things moving through time draw a "worldline". This is analogous to taking the 2d circle and letting it fall under (idealised flat earth) gravity to the ground, it will "sweep" out a cylinder and no fluid will leak out - no leakage occurs if you consider that fluid can only move (relative to an observer on the surface) in x and y dimensions.

Now imagine the WHOLE UNIVERSE is moving through this "dimension". How can you tell? You actually cannot because there is no reference point, everything is moving in one direction, at the same rate, regardless. In that same way, the 2D cup's whole two spatial dimensions are moving all at once in one direction, every single point on it has the same velocity in the z direction. This, I believe, is where we get the idea of energy conservation from (time translation symmetry).

Hopefully that made sense.

 

[XVX]

Thanks, it did.

But can't we argue that same picture with Newtonian mechanics?

We draw worldlines the same way with Newtonian mechanics, but the total displacement is different. With light, how can the total displacement be zero in such a flat plane that moves through a higher dimension sweeping out a cylinder?

And if the axis is ct, then moving through this dimension implies we are doing it at speed c.

So you would argue that it's just another normal spatial dimension?

 

[A.T.]

And if the axis is ct, then moving through this dimension implies we are doing it at speed c.

ct means c*t. You multiply coordinate time with c to get it into the same units as space. Explained in this thread:
https://www.physicsforums.com/showthread.php?t=217666

So you would argue that it's just another normal spatial dimension?

The proper time of moving objects has more similarity with the space dimensions. You can draw a diagram with a proper time axis. The coordinate time measured by the observer is then the length of every world line in the diagram:
http://www.adamtoons.de/physics/relativity.swf
In this type of diagram in fact everything moves at c trough space-time, only the direction changes.

 

[tiny-tim]

As soon as the flatlander brings his circle of Coke into three dimensions, the Coke will leak out.

And if we live in four dimensions, then how did we create closed containers?

Hi XVX!

Yes, our coke will leak out at the ends …

But the ends are in time … they are the beginning and the end, in time, of the can being sealed.

In fact, that's how CocaCola get the coke into the can … they do it far enough back along the ct axis, when the can isn't sealed!

(And you do much the same to drink it … except you go forward along the ct axis!)​

 

[belliott4488]

XVX - Are you familiar with formulations of Minkowski space-time where the fourth axis is ict, rather than ct? The reason people do this is that you can then apply "normal" coordinate transformations, but you'll pick up negative signs in funny places. Another way this is done is to use hyperbolic trig functions rather than circular trig functions when doing the transformations (which is equivalent).

The point is that when we use a fourth dimension to represent time, it's critically important to take into account that time is very different from spatial dimensions, so when you do "ordinary" geometric maneuvers, like coordinate rotations, evaluation of lengths, etc., you have to do them in a weird way. Basically, the time coordinate is always treated differently, often picking up a negative sign in places where the fourth coordinate of a Euclidean 4-D space would not.

Anyway, the point of all that is that simply taking the transformation from 2-d to 3-d, and then applying that thinking to go from 3-d to 4-d doesn't work. It would if your 4-d space were Euclidean, i.e. if it were spanned by four spatial dimensions, but it isn't.

 

[XVX]

Thanks belliott4488,

I did have a GR undergrad class with Hartle's book, Gravity, but we never did anything with imaginary time. I've thought about that before as it's easy to see how that can accommodate the minus sign in the line element. But then this seems to state that the 4th dimension is nothing but the imaginary axis.

Doing the 2D analogies definitely doesn't work, but what about the other way around? How should I imagine a 3D universe with relativity?


As stated by dst, I don't see how having a sheet of "universe" and moving it through a higher dimension results in the correct displacement of objects as given by GR.


A better 3D analogy of GR is to have a 2D spatial coordinate system with time as the imaginary axis? At least this way, I can kind of wrap my head around the total displacement of light being zero.

 

[belliott4488]

Well, yes, you can treat the time axis as being proportional to ict, but usually by the time you get to GR most people prefer just to let it measure ct, but to use the metric tensor to define things like vector norms, coordinate axis transformations (i.e. Lorentz transformations), etc.

It's not an easy trick to get used to thinking of space-time as one or two spatial dimensions plus a time dimension (and a suitable metric), and it's definitely harder still to go to three spatial dimension plus time. In any case, however, I think that the important step is getting comfortable with how the coordinate transformations work, even in 1 space + 1 time, more so than getting your head wrapped around the idea of four dimensions.

 

[JesseM]

Thanks, it did.

But can't we argue that same picture with Newtonian mechanics?

We draw worldlines the same way with Newtonian mechanics, but the total displacement is different. With light, how can the total displacement be zero in such a flat plane that moves through a higher dimension sweeping out a cylinder?

And if the axis is ct, then moving through this dimension implies we are doing it at speed c.

So you would argue that it's just another normal spatial dimension?

You can picture the same with Newtonian mechanics, but there are two important differences:

1. In Newtonian mechanics simultaneity is absolute, so if you picture the 3D spacetime for a 2D flatland universe, you have only one choice of the "angle" to slice this 3D spacetime into a stack of 2D "moments". In relativity simultaneity is relative, so you can slice up the 3D spacetime at a variety of angles, which will give different answers to whether two events happening at different points in space happened in the same "moment" or different "moments".

2. In relativity we have a quasi-geometric notion of the "distance" between two points in spacetime, given by $$\sqrt{c^2*dt^2 - dx^2 - dy^2}$$, analogous to the pythagorean formula for spatial distance between points in 2D space, $$\sqrt{dx^2 + dy^2}$$. In SR the spacetime distance will be the same regardless of which inertial coordinate system you use to calculate dt, dx, and dy, while the spatial distance will be different in different inertial coordinate systems. In Newtonian physics, there is no such notion of a "distance" in spacetime that is the same regardless of which inertial coordinate system you use, whereas if you pick two events which are simultaneous (and again, in Newtonian physics all inertial frames agree about whether two events are simultaneous), then all frames agree on the spatial distance between them as given by the Pythagorean formula.

 

[XVX]

Hey JesseM, I see your point with 2. but I think 1. is closest to my answer. 3D spacetime of a 2D flatland universe is exactly what got me stuck.

But I don't quite understand what your saying with 1. Can you please elaborate?


And if I ask you how many dimensions we live in, how will you answer?

3 spatial and 1 time?
4 spatial?
Call it all 4 spacetime?


How many dimensions do we live in?


hehehe