What is the 4th dimension?

Is it the additional dimension, Time, added to the Euclidean (x,y,z) 3-D space? Or does it even make sense to ask this question? And instead consider 4D space-time (Minkowski Space) as a manifold in which 3D is embedded?
I'm confused. :(
 
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My understanding of the 4th dimension is that it is just another degree of motion. We just perceive it as time. As you know the first 3 dimensions are just different ways of moving: up down, left right, forward and backwards. I usually just think of the 4th as future past. We call it a temporal dimension because we perceive it as time. And this time is very important to us as humans so we separate it from the other spacial dimensions.

Again this is just my understanding of the 4TH DIMENSION.
 
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What do you mean by 'dimension'?
(I guess it boils down to asking if a time directed vector is linearly independent with the set of whatever space dimensions we have)
 
What do you mean by 'dimension'?
(I guess it boils down to asking if a time directed vector is linearly independent with the set of whatever space dimensions we have)
I think I might understand it better now after reading an article describing a manifold: http://en.wikipedia.org/wiki/Manifold

So I guess by 'dimension' I mean, the dimension on the specific manifold, in which case the 4th dimension would be time in the $D Minkowski Space-time manifold?

What do you mean by linearly independent?

Thanks
 
Each dimension is orthogonal or perpendicular, they can't be described in terms of a linear combination of any other dimension. Time is one of 4 dimensions required to describe the position of something in space-time.

For example, if you want to tell someone where you party is, you say:

Go to the apartment complex on Dana Dr, and Hilltop (Dimensions X, and Y)
Then go to the 8th Floor (Dimension Z)
The party is at 9pm (Dimension T)

The 4 dimensions make a unique vector that describes where you party is located in space-time.

If you where to increase X, Y, Z, or T by too much the party will no longer be at that location. Say you increase Y by 100 meters, well you would end up on the wrong street. If you increase T by 10 hours, you would be far too late.
 
Each dimension is orthogonal or perpendicular, they can't be described in terms of a linear combination of any other dimension. Time is one of 4 dimensions required to describe the position of something in space-time.

For example, if you want to tell someone where you party is, you say:

Go to the apartment complex on Dana Dr, and Hilltop (Dimensions X, and Y)
Then go to the 8th Floor (Dimension Z)
The party is at 9pm (Dimension T)

The 4 dimensions make a unique vector that describes where you party is located in space-time.

If you where to increase X, Y, Z, or T by too much the party will no longer be at that location. Say you increase Y by 100 meters, well you would end up on the wrong street. If you increase T by 10 hours, you would be far too late.
OK. That's what I was trying to clarify (the highlighted statement above)
Thanks
 
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Is it the additional dimension, Time, added to the Euclidean (x,y,z) 3-D space? Or does it even make sense to ask this question? And instead consider 4D space-time (Minkowski Space) as a manifold in which 3D is embedded?
I'm confused. :(
It is meaningful to speak of Time as another dimension along with Space, not merely as a way of labeling independent degrees of freedom, but as part of an unbroken whole.

In particular, space and time can get mixed up with each other. Your time axis is "diagonal" to mine, containing part time and part space from my point of view.

Translating between reference frames is performing a rotation in 4 dimensions. The negative sign complicates things so you get hyperbolas instead of circles, but the math works.

Likewise, Maxwell's equations reduce to "one thing" in 4 dimensions, making it very simple, rather than having separate rules for E and B.

So yes, space-time is meaningfully a 4 dimensional entity.

--John
 
It is meaningful to speak of Time as another dimension along with Space, not merely as a way of labeling independent degrees of freedom, but as part of an unbroken whole.

In particular, space and time can get mixed up with each other. Your time axis is "diagonal" to mine, containing part time and part space from my point of view.

Translating between reference frames is performing a rotation in 4 dimensions. The negative sign complicates things so you get hyperbolas instead of circles, but the math works.

Likewise, Maxwell's equations reduce to "one thing" in 4 dimensions, making it very simple, rather than having separate rules for E and B.

So yes, space-time is meaningfully a 4 dimensional entity.

--John
This makes sense. Thanks!
 
Likewise, Maxwell's equations reduce to "one thing" in 4 dimensions, making it very simple, rather than having separate rules for E and B.

So yes, space-time is meaningfully a 4 dimensional entity.

--John
Curious... Can you post a link to these new equations? I might have read about this before, I can't remember. Does it do away with the B field?
 
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It still contains the same amount of information, just presented in a much nicer form. E and B are still encoded in the tensors.
 
The fourth dimension is obviously another direction, or 2 more directions in space. Time is a dimension. But, not the fourth dimension. When we have the first, second, and third established as space then all others will also be space, time may be a dimension, I for one would not call it the fourth dimension though.
The fourth dimension refers to a new direction of movement, unthinkable in our 3D programed brains.
 

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