Multi-dimensional cosmology question

[Dyothelite]

Again, I apologize for my novice level of physics but here goes:

 

For the last 14 years I have been struggling with a multi-dimensional view of cosmology that would help describe time and gravity and the universe in a geometric model. This actually carries into the books I've written on Qabalah.

 

say you take a one-imensional axis, if you square it you get a two-dimensional plane (x2) then if you square the second axis you get three dimensional space (x3), then if you square the third axis as in a linear graph of velocity as distance by time you get 4D (x4).

 

now lets say then you square the fourth axis, you then get acceleration as a function of T2 (x4 * x). Where a fourth dimensional graph of 3D distance by time is linear, acceleration is 3D by time squared, which implies adding a theoretical spatial dimension.

 

If this is correct I was then thinking, isn't it fair to call acceleration (more importantly) gravitational acceleration a five dimensional concept?

 

My idea then is if it is fair to call acceleration a 5D concept and the natural state of objects rests in 5D, what would be 6D? Well I was thinking that in order to experience 6D you need to see a change in gravitaional acceleration, this would happen as you add mass to a black hole. The 5D constant would change. and if you add mass exponentially to a blackhole the 5D would change with acceleration making a 7D, possibbly like the collapse of the universe.

 

Just some ideas I'd apreciate honest feedback, I know this is a little sketchy but I'm working on it.

[Qfwfq]

isn't it fair to call acceleration (more importantly) gravitational acceleration a five dimensional concept?

Frankly I don't see why.

[CraigD]

If this is correct I was then thinking, isn't it fair to call acceleration (more importantly) gravitational acceleration a five dimensional concept?

I don’t think so.

 

The mathematical concept of a dimension includes the quality of independence from other dimensions, what’s sometimes called a “degree of freedom”. A coordinate space that’s useful in describing physical reality need to have at least 1 (usually has at least 3) spatial dimension, and 1 temporal dimension. Once you have these, concepts such as velocity are dependent on the values of the coordinates of a point in this space, eg: speed = change in spatial dimension / change in temporal dimension, acceleration = change in speed.

 

A value that can be represented in terms of existing dimensions can’t meaningfully be a dimension.

 

Though it’s common to use the word “dimension” in a wide variety of sentences (eg: “the dimension of love”, “visitors from another dimension”), mathematically, this is fuzzy thinking.

[Dyothelite]

I don’t think so.

 

The mathematical concept of a dimension includes the quality of independence from other dimensions, what’s sometimes called a “degree of freedom”. A coordinate space that’s useful in describing physical reality need to have at least 1 (usually has at least 3) spatial dimension, and 1 temporal dimension. Once you have these, concepts such as velocity are dependent on the values of the coordinates of a point in this space, eg: speed = change in spatial dimension / change in temporal dimension, acceleration = change in speed.

 

A value that can be represented in terms of existing dimensions can’t meaningfully be a dimension.

 

Though it’s common to use the word “dimension” in a wide variety of sentences (eg: “the dimension of love”, “visitors from another dimension”), mathematically, this is fuzzy thinking.

 

How bout this though.... say you graph something with 0 velocity (slope = 0/time) as 3D by time, then you graph a moving object (positive slope). both are a linear graph with a specific slope which defines the velocity. Now if you graph acceleration you see a curve in the linear graph as you square the time coordinates. If it is fair to say that the linear graph of velocity is a 4D concept, isn't the curvature of the line and the squaring of the time coordinates suggestive of 5D?

[CraigD]

If it is fair to say that the linear graph of velocity is a 4D concept, isn't the curvature of the line and the squaring of the time coordinates suggestive of 5D?

No.

 

In order to represent a point in a coordinate space in such a way that the concept of “change in position” exists, it must have at least one position coordinate (which can also be called a “dimension”) and exactly one time coordinate. The characteristics of physical reality that we observe, such as 2 straight lines being either parallel, intersecting, or neither, lead us to use 3 position coordinates rather than 1 – though when trying to explain things such as the behavior of fundamental particles as vibrating strings, additional position coordinates are often required. However, it makes no sense to create a coordinate for change in position (velocity), or change in velocity (acceleration), or change in acceleration, etc, because these can be calculated from several points of the same body’s “world line”. This is more than a matter of just keeping the coordinate system as simple as possible – we can’t include these other, “dependent” coordinates, without placing limits on the possible behavior of bodies that disagree with observations of physical reality, or allowing any observed motion that can’t be represented with this “extended” coordinate system to generate additional “dimensions” as needed, an impractical and un-useful approach.

[Dyothelite]

Yeah I've been kinda worried that anything can be represented using the coordinates that already exist.

 

How then do we describe a curvature of space-time? Does it curve within itself? Or can we suggest it curves into and within a higher dimension? If space-time is a set framework, how then can it be curved if not relative to another coordinate? When you curve a line you create 2D and add a dimension.

[Jay-qu]

You are thinking to much about conventional curving. Its more prudent to say that it is warped. You could describe gravity as each point having an amount of warping and objects tend to move in the direction of greater warping.

[Qfwfq]

Actually it is called curvature in the lingo of differential geometry. I suppose Dyothelite needs an illustration but I would tend to talk about stretch and not warp.

 

What you have in mind Dyothelite is called embedding an n-dimensional manifold into a flat one of more than n dimensions. This can be helpful for understanding but isn't essential to the given manifold itself. There is also an arbitrarity. Consider a piece of paper, which is a portion of flat 2-D space, and roll it up without any stretching; it is essentially the same manifold as before, with the exact same geometry, only the embedding into 3-D space has changed.

 

Now consider a sheet of rubber stretched taut like a trampoline; each point trivially has coordinates once you choose a pair of cartesian axes. Now push a convex surface into it, you have a manifold with non-zero curvature. You may however associate each point with the same (x, y) pair as it had before, this is just one possible coordinate mapping. The distances are however not the same as before. This means that the metric is no longer the usual one:

 

[math]d^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2[/math]

 

Now call this state of the rubber sheet A and let it return flat, or even stretch it any other way, but consider the distance between any pair of points to be what is was on A. This choice of metric is what defines the manifold and its curvaure, regardless of any embedding into a space with more dimensions.

 

To be precise, some maniflds can't be described with a single coordinate map; a set of maps that can describe the whole thing is called an atlas. For instance the surface of a sphere or an ellipsoid needs two maps, each of which can map, at the most, all points except one to the cartesian plane.

[Dyothelite]

Ok but if we see the space-time fabric as the 2D trampoline surface... it is being warped in a third spatial dimension. This suggests analogously that space-time must also warp in a hyper spatial dimension than the space-time fabric itself.

[Jay-qu]

Ok but if we see the space-time fabric as the 2D trampoline surface... it is being warped in a third spatial dimension. This suggests analogously that space-time must also warp in a hyper spatial dimension than the space-time fabric itself.

"Matter tells spacetime how to curve.

Curved spacetime tells matter how to move." John Archibald Wheeler

 

At each point the matter is 'told' how to move, it is not falling under gravity through a curve like in the analogy. It does not require another dimension to curve into.

[Dyothelite]

Ok but if 4D space-time without matter is in equilibrium and then with matter it curves, what dimension is it curving in? 4D space-time by itself isn't curved, so which dimension does it curve in when matter is present?

[Jay-qu]

:) AFAIK relativity does not specify a dimension that space curves into, because it is not needed. Try re-reading the above few posts again.

[Qfwfq]

Ok but if we see the space-time fabric as the 2D trampoline surface... it is being warped in a third spatial dimension.

See the second last paragraph.

[CraigD]

Ok but if 4D space-time without matter is in equilibrium and then with matter it curves, what dimension is it curving in? 4D space-time by itself isn't curved, so which dimension does it curve in when matter is present?

I think it’s helpful to consider the coordinate space of the surface of a sphere.

 

Any point can be defined by 2 coordinates, such as latitude and longitude. It’s a bit complicated to calculate the distance between 2 points, and the poles are weird points where longitude is meaningless.

 

It’s helpful and intuitive to visualize it in 3 dimensions/coordinates (they follow the nice, simple formula [math]x^2+y^2+z^2=r^2[/math]), but this doesn’t make the surface of a sphere 3-dimensional, because you’re not free to chose any value for any of the coordinates within a defined range. For example, given [math]r=1[/math], [math]x=\frac12[/math], [math]z=\frac13[/math], [math]z[/math] is limited to 2 possible values, [math]\frac{\sqrt{23}}6[/math] or [math]-\frac{\sqrt{23}}6[/math].

 

In the same way, it’s helpful to visualize the 3-spatial-dimensional universe described formally by General Relativity with the 3 coordinates reduced (for conceptualizing purposes only) to 2, and an additional coordinate to define the acceleration due to gravity of at any point – the famous “rubber sheet” illustration. The space is still, however, 3 dimensional, because you’re not free to choose any value within a defined range for the additional coordinate – it’s limited to a single possible values, that can be calculated exactly from the position and mass of every point particle in it (with equations much less simple than [math]x^2+y^2+z^2=r^2[/math] ! :( ) .

 

It’s reasonable, though, to think of 3 space + 1 time points with mass and gravity as having a different independent 4 coordinates – 3 space, 1 mass, and time as a dependent coordinate. A collection of 3 space + 1 mass points at an arbitrary time=0 can, in principle, be used to calculate the coordinates of all the points for time=anything. Practically, there’s no way yet known to perform this calculation exactly with a finite number of arithmetic operations – see the n-body problem. Although this problem is usually associated with classical gravity, not GR gravity, I believe it’s even harder using GR.

 

Also note that all these “classical-like” approaches to physics, including Relativity as developed by Einstein, though very useful, are almost certain to be unable, ultimately, to explain reality. What might be able to is a gigantic subject, way off topic for this thread.