Understanding Dimensions: The Interplay of Mathematics and Philosophy

[Sam Owen]

yes I have read the wikipedia definition and followed my nose around to see how it gets interpreted in various models and theories but it doesn't say why it is that when 3 come together it forms a sphere and creates an instant of time ?

how can one dimension exist independently of others and what could possibly exist in only one except maybe a thought ?

 

[Chronos]

A dimension is basically a degree of freedom within a specified coordinate system. In other words, it is an attribute that must be quantified to uniquely identify an entity within that coordinate system. The reason we talk about 3 spatial and one time dimension is it is otherwise impossible to uniquely locate an object in this universe. Having 3 spatial dimensions is pretty obvious. The dimension called time is more subtle. It would not be necessary if everything in the universe was rigidly fixed in place and never moved. But stuff does move in our perceptual universe and the only way to uniquely describe the location of any object in the universe, is to add at least one more dimension to our coordinate system.

 

[Sam Owen]

so does time exist in one dimension or even 2 if there is no space to move around in until we have 3 ?

Is it fair to say that in the multidimensional background of string nothing moves it is only the background changing shape only we can't tell as we are locked into a 3d existence so the perception of actual movment is merely an illusion ?

 

[Sam Owen]

so a dimension is basically an intangible concept within which you can assign a coordinate, an attribute of a larger tangible reality but only when you get 3 of them coming together

so we can only truly uniquely locate an object in this universe if time stood still given that everything is in motion and subject to a certain amount of predictability/unpredictability

I mean we percieve movement but is there anything really to say that it's not just the background changing shape

take the letters on the screen as you type they look like they are moving but it's only the pixels changing shape the screen hasn't moved

 

[RandallB]

Sam
I’m trying to get clear on understanding dimensions myself, but let me give it a shot and maybe Chronos can help straighten me out if I mess up.
Usually we start looking at dimensions because we are looking at Strings and were already dealing with 11 of them there. Let’s start from the other end where the is only three in a true classical sense.

Now by classical that is three dimensions only, just as we see them every day – NOT including time. That is time as NOT a dimension, but as a independent variable, separate from the geometry of space. That your are where are and time is just a measure of how long you were there and when you might move. Where you are there is no “other place” at that same location just down the road back an hour ago that still exists as a “place”. No worrying about if reality is still holding a younger copy of you there, or you just went through that “point” of time to get to this one, and something else is using that old time and place. None of that to think about in the classical because time is not a dimension just a unique measurement of our reality in Euclidian 3D space.

Worked great for Centuries – even when SR Special Relativity comes along still OK. Measurements look a bit weird at high speeds, like looking through wide angle photo lenses but with Einstein’s formulas even $$E=mc^2$$ fits in OK after we crunch the numbers. BUT very much not looking at Acceleration or Gravity!
Even if we can’t put anything physical (like ether) on it, understanding a dimension here is very simple, BUT;

We work our way into General Relativity, and things get a bit harder to describe. GR needs some help to picture what is going on - and WARPING space and time was the key. I always thought - Just make time a 4th dimension and warp away. But that won’t work – whose time to we warp against? The time at Earth reference, or the galaxy speeding away from us? To pick one is to pick a reference frame and we know that won’t work.
So here is where I get a bit confused.
Does this mean of the 4 dimensions you don’t get three that still match up DIRECTLY with what we see as our local xyz realty? Maybe all 4 of them (whatever they are) are required to create that “xyz” image for us and within the four it can warp. Creating different kinds of “xyz” with different time frames at different places.
Thus the four together are required as a “background” to build our reality. With no one of “them” directly linked to a measurement we can make.

I expect this is where the background can be “independent” vs. “dependent”.
I don’t really understand that issue to say, so I’ll let someone else speak to that.

Once we leave the classical 3D, to create 4D adding more dimensions should be OK - if it can be shown to work, how, and why, for however many. That hasn’t happened yet.
I’m sure basic QM (before Strings) has to be considered “non-classical” due to uncertainty - - but I’m not sure basic QM is non-classical in its demands on more than 3 Dimensions as well. Maybe someone can address that issue too.

Hope that helps or at least give you something to think about.

 

[mccrone]

Another way of thinking of time vs space is that - as Chronos says - a dimension is a degree of freedom. Physical reality has two particularly obvious kinds of freedom - the freedom to be at a location and the freedom to be moving. These are kind of contradictory freedoms as if you are moving, you aren't located, and vice versa.

Physics is about modelling reality. A simple way to model this ability of reality to contain both position and momentum is to dichotomise measurements towards two kinds of dimension (or ways of measuring). So you basically bin all the observed location in three spatial dimensions and all the observed change in location (ie: motion) in the temporal dimension.

Newton and then relativity showed that motion is as basic as location, so time is not less real as a dimension. But difficulties of interpretation arise because location (brute existence) is still taken as the unproblematic part of the physical story and then change is divided into inertia change and accelerative change. Then eventually with thermodynamics, it becomes irreversible change.

So the temporal dimension - the "fixed background" against which various change measurements are made - is accepted as having a hierarchical complexity. But for some reason (actually, because it makes the calculations nicely simple for most modelling purposes) the spatial dimensions are treated as just a string of discrete locations. Or is that a continuous line?

Oh don't bother me with your pointless metaphysics! Who cares what Zeno was really on about.

 

[εllipse]

We work our way into General Relativity, and things get a bit harder to describe. GR needs some help to picture what is going on - and WARPING space and time was the key. I always thought - Just make time a 4th dimension and warp away. But that won’t work – whose time to we warp against? The time at Earth reference, or the galaxy speeding away from us? To pick one is to pick a reference frame and we know that won’t work.
So here is where I get a bit confused.

It is true that space and time are relative, but spacetime is absolute. Different observers will have different views of how far apart distances are separated in space and how long events are separated in time, but everyone will agree on the "distance" between them in spacetime. This is what's meant by the equation $$x'^2+y'^2+z'^2-c^2t'^2=x^2+y^2+z^2-c^2t^2$$.

Simple Derivation of the Lorentz Transformation (note equation 11a)
Minkowski’s Four-Dimensional Space

 

[Chronos]

Aye, but here is the rub. Everyone will not agree how much time has elapsed unless they are in comoving reference frames.

 

[RandallB]

Aye, but here is the rub. Everyone will not agree how much time has elapsed unless they are in comoving reference frames.

Exactly, that’s why I’m having so much trouble understanding a 4D interpretations that at the start assumes that 3 of those 4D's match up directly with what we understand as "xyz" spatial space. With all the weirdness piled up into just the one dimension of time, it doesn’t take long to run into irresolvable infinities. Einstein with GR and QM with quantum theories have never been able to resolve those infinities. It’s just to complex to handle in one of the four dimensions. But maybe mathematically if the strange complex action was spread over all four dimensions it could be worked out.

So I'm assuming that’s the hope of the "indeterminate background" approach. By unhooking any direct measure from our “xyz” to any group of the 4 Dimensions (Or more D’s if they really are needed), that an way can be found to make sense of it all.

A good analogy of what indeterminate and determinate ‘really’ mean – I don’t know.
But it makes sense, that if we are willing to leave the Classical to go to 4 dimensions, why should we insist on holding on to a classic link of our “xyz” directly matching up to three of the dimensions.
Which means that Einstein did depart from the Classical with warped space-time requiring 4D, but I’m not sure how he would define his view between indeterminate and determinate.

 

[mccrone]

The infinities arise at the limits of the position~momentum dichotomy. So as you try to locate a mass particle to a point, or even a scrap of "vacuum", you get infinite (or rather Planck-scale) quantum jitter. Likewise if you try to move a mass particle at the speed of light, you get an infinite (or again Planck-scale?) swelling in size, slowing in time.

The infinities are evidence of there being a position~momentum dichotomisation in which two limits are being asymptotically approached.

The comoving observers issue is just about the fact that "simultaneous" observers would be at rest with each other - so all the accounting done in terms of location, momentum factored out of the story for simplicity. Then with relativistic spacetime, a formula for re-introducing the motion as required.

The interesting question is why did it seem natural to set the three spatial dimensions as the simple ones, and time be complex?

Location = stable existence. For something to be located, it is also implied that it will linger long enough to matter (unlike a virtual particle say). So it may also exist in time, but this existence is inertial. The particle travels freely without need of a force to keep it moving forward in time at a "constant" rate.

So perhaps there is more complexity in the idea of spatial location after all. It is assuming inertial existence (or in the case of the void, inertial non-existence).

Then there is also the implied property of locality. It takes time for effects to propagate across a space. So two located objects are isolated until they have had time to interact.

Again, the property of "being located" is more complex than first meets the eye. Once you step back to a spacetime block perspective where position and momentum are values to be assigned in yo-yo, dichotomistic, fashion.

But still, the Newtonian view was to make things simple. Start with existence (locateness) and then introduce motion. This seems obvious in a world of mass where an inertia of existence is something that can be taken for granted (for protons and electrons at least) but momentum is variable (between the asymptotic limits of absolute rest and light speed).

The reverse case would be to take a motion as basic, primal, and existence as variable between limits.

This might be the more natural reference frame for a pre-Higgs realm where there is no mass and only a flood of massless particles. Now you could be certain that all particles travel at the same speed - the speed of light. So at least one of the complex temporal variables drops out of the picture. Though you still have variable direction for the particles. Perhaps (given the heat) you have less stability of existence?

Of course now you also have to deal with the idea that at light speed, time is frozen (for the particle concerned).

Excuse my rambling, but you can appreciate the basic point. Measuring anything demands that one end be pinned down as the background and then the wriggling of the other end can be observed. Choice of whether location or motion is treated as the fixed ground it just that, a choice. But they are the pair of alternatives that do extremitise the situation. They are a natural fundamental dichotomy.

Then we have the notion of a dimension arising as the description of these natural limit states. A simple model would treat the emergent results as "a degree of freedom". A complex model would see that the freedoms arise within a context of constraint.

So a simple model would treat reality as atomistic. An atom exists at a location freely. It's persistence is not being caused (so its existence is inertial and involves no "accelerations"). And the same is assumed for the void. It is a collection of locations that exist freely with no need to be caused as empty places. The complexity of the further property of locality (the fact that any two atoms are spatiotemporally separated and so it takes time for effects to propagate) is only implied in the dimensional description.

But the bigger picture of dimensionality would have to take into consideration that degrees of freedom arise within a context of constraints. The ancient atomists could assume that atoms and void were eternal and uncreated. But modern cosmology demands some story on how existence (of both atoms and "voids") came to be.

Oh, one further obvious point. The universe expands "freely". So that is another way Newtonian static spatial dimensions actually have a more complex dynamism. It would be interesting to have a model of dimensionality that incorporates this complex property - and this is where fractal and scale-free network approaches may have promise.

Atoms - lumps of mass - are interesting because they can travel (change location) at a range of speeds between rest and lightspeed. But even the void moves at all rates between rest and lightspeed (and superluminal once over the event horizon). The difference with the void is that the rate of motion/expansion is a single coherent one - the Hubble flow.

So dig beneath the apparent simplicity of the spatial dimensions and you discover plenty of complexity. It is a particular modelling choice to make location the simple idea and bin all the complexity in the notion of motion (for both atoms and voids).

 

[Berislav]

but it doesn't say why it is that when 3 come together it forms a sphere and creates an instant of time ?

I'm not sure what you're referring to, but I will try to elaborate it, nonetheless.
Since you're posting in a forum dedicated to quantum gravity I think that you're basically talking about space-like hypersurfaces, which are useful in quantum gravity. One can divide spacetime into these surfaces. In the most common and well-known space, the Schwarzschild one, we define a space-like hypersurface to be orthogonal to a time-like Killing vector field (which, in a way, simulates time), as such it is a function of time and can represent space at some moment in time. In that geometry we use spheres to model the space, which for mathematical reasons are called 2-spheres, because we want spacetime to be spherically symmetrical.

 

[Chronos]

It's much simpler than that, I think. Just rotate the coordinates. I side with Berislav on that count. You don't need Euclidian concepts to create a coordinate system that makes sense. [Berislav explains this very well despite being victimized by english as a second language]. He has my attention. I think he knows what he is talking about. I also respect what mccrone has to say. I think he too is sharp.