What Exactly is the Fourth Dimension?

[Arthur d. S. Jr.]

What I've got it's an axis (let's call it "w") that together with the three-dimensional axis (x,y,z), forms a fourth dimension, on which all the four axis (x,y,z,w) are 90° from each other. We cannot see it this way because our sense of space and dimension is tridimensional, where only 3 axis can be 90 degrees from each other; like a two dimensional character from a game, it cannot see depth, and for him/her there is only x and y, in a 90° angle and it isn't possible to imagine a third axis in 90 degrees to it. In the same way, we, three-dimensional oriented beings, can not imagine another vector with ninety degrees other then x, y and z. But the problem is: some say the fourth dimension is time, some say it's not, Is it or not? And if it is, why? what does time have to do with x,y,z, coordinates or dimensions?

 

 

Thanks;

Arthur.

[Mordred]

Your looking at dimensions wrong.

 

the number of dimensions is the minimal number of coordinates needed to describe any location.

 

So in 2d x,y two coordinates. 3d x,y,z.

 

For 4d t,x,y,z 3 spatial components plus one of time. This is needed in GR as GR models time as a vector which requires both magnitude and direction.

[Phi for All]

What I've got it's an axis (let's call it "w") that together with the three-dimensional axis (x,y,z), forms a fourth dimension, on which all the four axis (x,y,z,w) are 90° from each other. We cannot see it this way because our sense of space and dimension is tridimensional, where only 3 axis can be 90 degrees from each other; like a two dimensional character from a game, it cannot see depth, and for him/her there is only x and y, in a 90° angle and it isn't possible to imagine a third axis in 90 degrees to it. In the same way, we, three-dimensional oriented beings, can not imagine another vector with ninety degrees other then x, y and z. But the problem is: some say the fourth dimension is time, some say it's not, Is it or not? And if it is, why? what does time have to do with x,y,z, coordinates or dimensions?

 

 

Thanks;

Arthur.

 

Time is a temporal dimension, the other three are spatial. Using the four together, we have a system for plotting events in our universe. Given x, y, z, and t coordinates, we can predict a meeting between you and I on a certain floor of a certain building in a certain city on a certain planet, etc.

 

If we follow the same rules for a fourth spatial dimension, starting with a 3D cube you move 90 degrees away from every point on the cube. Not sure how useful that is.

[Arthur d. S. Jr.]

But if the fourth axis is just time, why are shapes like the tesseract so crazy, if the fourth variable is just the time the object is?

 

[Phi for All]

But if the fourth axis is just time, why are shapes like the tesseract so crazy, if the fourth variable is just the time the object is?

 

If you're using the tesseract as a 4D example, then the fourth axis is a spatial one. Time is the temporal dimension.

[StringJunky]

 

If you're using the tesseract as a 4D example, then the fourth axis is a spatial one. Time is the temporal dimension.

So, it is 5D?

[Phi for All]

So, it is 5D?

 

I'm never using the kind of math where dimensions outside normal spacetime become relevant, so when someone talks about how many Ds a perspective has, I always assume they're talking about spatial, plus one temporal. This might not be best practice.

[Delta1212]

So, it is 5D?

I think it would be less misleading to describe it as 4+1D

[StringJunky]

I think it would be less misleading to describe it as 4+1D

Can you elaborate?

 

edit: As in 4 spatial + 1 temporal?

 

Does that tesseract analog with the teapot example I mentioned, as the inside becomes the outside and vice versa?

Edited November 30, 2016 by StringJunky

[mathematic]

It is important to distinguish between dimension as a mathematical concept (any number) and dimension in relativity theory (3 spatial plus time).

[Airbrush]

"One notable feature of string theory and M-theory is that these theories require extra dimensions of spacetime for their mathematical consistency. In string theory, spacetime is ten-dimensional, while in M-theory it is eleven-dimensional. In order to describe real physical phenomena using these theories, one must therefore imagine scenarios in which these extra dimensions would not be observed in experiments..."

 

So why stop at 4 spatial dimensions? There could be any number of dimensions.

 

https://en.wikipedia.org/wiki/M-theory

[studiot]

 

 

It is important to distinguish between dimension as a mathematical concept (any number) and dimension in relativity theory (3 spatial plus time).

 

 

This short statement is so profound - It carries vitally important information and is the key to resolving many of the misunderstandings on this subject. +1

 

It does beg the question as to why they are different or need distinguishing.

Edited November 30, 2016 by studiot

[StringJunky]

 

 

 

 

 

This short statement is so profound - It carries vitally important information and is the key to resolving many of the misunderstandings on this subject. +1

 

It does beg the question as to why they are different or need distinguishing.

 

Is it because non-physics people always associate dimensions in science as spatial ones rather than other parameters as well?

Edited November 30, 2016 by StringJunky

[Mordred]

or additional degrees of freedom?

 

example Kaluzu-Klien 3 spatial degrees of freedom. One for time, one for electromagnetism=5d. {U(1) guage} The U(1) guage reduces down to one degree of freedom.

 

String theory is similar in that the additional dimensions is describing additional degrees of freedom due to weak, strong and electromagnetic fields. Though it is not strictly additive with guage fixing and can get incredibly complicated with guage fixing lol

Edited November 30, 2016 by Mordred

[StringJunky]

or additional degrees of freedom?

 

example Kaluzu-Klien 3 spatial degrees of freedom. One for time, one for electromagnetism=5d. {U(1) guage} The U(1) guage reduces down to one degree of freedom.

 

String theory is similar in that the additional dimensions is describing additional degrees of freedom due to weak, strong and electromagnetic fields. Though it is not strictly additive with guage fixing and can get incredibly complicated with guage fixing lol

Brain melts... Can you elaborate what degrees of freedom means... for a thicko.

Edited November 30, 2016 by StringJunky

[Mordred]

Good question though not one easily answered.

 

In physics a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all dimensions of a system is known as a phase space, and degrees of freedom are sometimes referred to as its dimensions.

 

However this definition may be easier.

 

" the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinitesimal object on the plane might have additional degrees of freedoms related to its orientation."

 

As you can see here there is differences in definition depending on the application.

 

https://en.m.wikipedia.org/wiki/Degrees_of_freedom

 

So to describe a point in 3d I need three values x,y,z. In 4d with time as a vector I now require 4 t,x,y,z. When you spin that object you add another degree of freedom. There is some examples and a decent coverage here.

 

https://en.m.wikipedia.org/wiki/Degrees_of_freedom_(physics_and_chemistry)

 

For example lets deal with a 3d pointlike object under strictly change in position.

 

As each movement must be independant to count as a degree of freedom. The object can change x without changing y or z. Same with y or z.

 

So x,y and z are independant of each other. They are each examples of a degree of freedom. Time is also independant it can change without a change in spatial location. If you add a dynamic that is independant of the x,y,z,t degrees of freedom such as photon polarization you add additional degrees of freedom.

 

 

PS I'm also hoping Studiot will provide some good mechanical degrees of freedom examples. Hes more practiced on the engineering applications. If not I can use some robotic arm examples lol.

 

So take this object. First ask how how many degrees of freedom is needed to fully desribe this object.

 

To start it has three spatial independant coordinates. x,y,z. If you treat it with time being independant (not dependant) then thats 4.

 

Now the rotations of the total object (both boxes) has been reduced to its independent variables. Yet the inner box can move independant of the outer box. (or at least appears to move independant) lets assume it can.

 

Those independant movements adds additional degrees of freedom. Just watching the object and including time this can be described as a 5 dimensional object. (though it could have more degrees of freedom than I perceive)

Edited November 30, 2016 by Mordred

[studiot]

Brain melts... Can you elaborate what degrees of freedom means... for a thicko.

 

Mordred has the right idea and I'm sure we have lots of agreement on this subject. You can trade constraints for dimensions to some extent.

 

For instance two dimensions, x and y, with no constraints, give you access to an entire plane.

 

Adding the constraint y = x² restricts you to a particular line.

 

However it actually is very complicated.

 

I had started to prepare a development of the subject but since discussion is live here is what I have done so far but please remember:

 

It is not yet finished and the difficult bits are yet to come.

 

 

 

Mathematically dimensions refer to the coordinate axes used to measure or describe something.

Each axis is mathematically just an unrestricted list of numbers as mathematic says.

 

 

This set up can be used in several ways.

 

1. All the axes may carry the same weight or meaning, for instance a coordinate system describing length, breadth and height. This is pretty general and the basic system contains no restrictions. It acts as a self-contained entity with no reference to any other subject. Further each axis may be considered as representing a completely independent variable. We can use it to measure the size (or extents) of say a cube or other object wherever that object may be placed in relation to the axes.

   Mathematicians call this a 3 dimensional coordinate system and physicists would call it a 3 spatial dimensional system.

   We would say that the cube etc is embedded in this 3 dimensional space.

2. We can restrict generality slightly to use coordinate system (1) for location or position. Coordinate system (2) is still 3 dimensional but yields slight more information, because of the restriction. We are interpreting the position of ‘points’ in this space.

3. We can extend coordinate systems (1) or (2) by simply adding another axis of equal weight or meaning.

   This creates a simple 4 dimensional space that scientific observations suggest doesn’t physically exist in out material universe, but has mathematical existence for theoretical purposes.

4. We can also extend coordinate system (2) by introducing a fourth axis with a different weight or meaning. For instance we can introduce a temperature at every point. In this view we can regard temperature as another variable axis that intersects our 3 dimensional position space. At any point all temperatures (T) are available but only one is ‘true’. So if the temperature somewhere is say 10o then the point (x,y,z,10) is regarded by the physicists as having physical reality or validity and the point (x,y,z, 20) is not. A mathematician, on the other hand regards the whole panoply of points (x,y,z,T) as being equally valid.

5. Edit I hope Mordred can see where this is heading, but an aside on trading dimensions and constraints.

Consider the standard parabola y² = 4ax.

 

This needs 2 dimensions to fit into, although it does not fill the whole two dimensions.

It has one constraint.

 

If we add another constraint (so we have 2) we can create it as a one dimensional entity:

 

Introduce the 'parameter', t (which is one dimensional) and the constraints:_

 

x = at²

y = 2at

 

we now have The parabola described by a single dimension.

Edited November 30, 2016 by studiot

[Mordred]

I like what you have so far. Yes I can see the direction your heading on 5. Just so I don't give it away but looks likes your heading to f=p-c+2 (after constraints are applied). Good approach if I'm correct.

Edited December 1, 2016 by Mordred

[studiot]

I like what you have so far. Yes I can see the direction your heading on 5. Just so I don't give it away but looks likes your heading to f=p-c+2 (after constraints are applied). Good approach if I'm correct.

 

I was hoping that mathematic would expand on his (her?) statement. I don't want to steal someone else's thunder.

 

Anyway I was going much further than that.

 

The difficulty is which way to go next since developing (1),(2),(3) and (4) directly leads on to a very startling idea that we need more than one coordinate system and coordinate space and then onto explaining tensors.

The other way is to introduce local and global coordinate systems.

 

The two tie together in the end.

 

There are also some (mathematical 'funnies' on the way such as the non existence of the y axis on log graph paper and the non existence of both axes on log-log graph paper)

[Mordred]

Gotcha but I too don't want to distract this thread from the OP lol. Sounds like a good paper though.

 

For the benefict of others the equation I gave is Gibb's phase rule.

 

https://en.m.wikipedia.org/wiki/Phase_rule

 

Which gives the number of degrees of freedom (well according to the understanding of particles at the time period lol) the general principle still applies.

 

Today its far more accurate to use the Bose-Einstein, Fermi-Dirac or Maxwell Boltzmann statistics. These correlate the quantum numbers as additional degrees of freedom.

 

Gibbs laws are good stepping stones though. Its still accurate on everyday gases. The difference's comes into play when dealing with elementary particles.

Edited December 1, 2016 by Mordred

[studiot]

Mordred post#20

Gotcha but I too don't want to distract this thread from the OP lol. Sounds like a good paper though.

 

Well it seems that the OP has lost inerest, and I'm not sure anyone else is interested either.

 

If fact my posts were not taking the thread off topic but were introducing some necessary background to the observation

 

Which fourth dimension?

 

Don't forget that the local and global coordinate systems may contain different axes and may not match directly if they do.

 

Moreover scalars such as temperature that I placed at position (x,y,z) don't take up any xyz space.

 

But place a normal a tangent, a vector or a tensor at (x,y,z) then you need a set of local axes as well as the global ones.

 

The mathematical name for the correspondence function is a chart.

 

But of course the local axes at one point may not match the local axes at another so we also need the correspondence between these and the term here is a connection.

 

Heavy stuff, but worse is to come.

 

The above assumes isotropic or linear axes, but the axes themselves my not be linear.

The comment about log graph paper was intended as a precursor to a gently introduction to this matter.

 

[geordief]

 

 

This is ,I guess a mathematical object. Are there any physical phenomena it can be used to predict.?

 

Like others,I see it as representing 4 dimensions plus time.

 

It is simply a mathematical curiosity or can it model any physical and observable process that occurs?

 

 

Perhaps the idea is that it may model some processes that call for a larger number of dimensions in some quantum gravity theories?

 

Correct me if I am wrong but this (tesseract) is not something that can be actually manufactured physically as the "physical object" would collide with and pass through itself.

Edited December 2, 2016 by geordief

[36grit]

What I've got it's an axis (let's call it "w") that together with the three-dimensional axis (x,y,z), forms a fourth dimension, on which all the four axis (x,y,z,w) are 90° from each other. We cannot see it this way because our sense of space and dimension is tridimensional, where only 3 axis can be 90 degrees from each other; like a two dimensional character from a game, it cannot see depth, and for him/her there is only x and y, in a 90° angle and it isn't possible to imagine a third axis in 90 degrees to it. In the same way, we, three-dimensional oriented beings, can not imagine another vector with ninety degrees other then x, y and z. But the problem is: some say the fourth dimension is time, some say it's not, Is it or not? And if it is, why? what does time have to do with x,y,z, coordinates or dimensions?

 

 

Thanks;

Arthur.

I think about it like this, distance is relative, and x, y, and z, all begin and end as points in time. Measure a yard stick, It's three feet, stand a block away and measure it again, it's quarter inch from your personal perspective in relative space time. Simple.

[StringJunky]

I think about it like this, distance is relative, and x, y, and z, all begin and end as points in time. Measure a yard stick, It's three feet, stand a block away and measure it again, it's quarter inch from your personal perspective in relative space time. Simple.

You are talking out of a dark place.

[AbstractDreamer]

Measure a yard stick, It's three feet

 

How are you measuring the yard stick?

 

 

stand a block away and measure it again

 

Its now a block away.

it's quarter inch from your personal perspective in relative space time. Simple.

 

Close my eyes and it no longer exists, from my perspective.

Edited December 3, 2016 by AbstractDreamer