How Does Dimension Influence Direction and Freedom in Physics?

[Antonio Lao]

All Theorists,

I have been searching for the complete understanding of "dimension" since the time when I fell down the stairs as a baby. There never was any difficulty when I walk or crawl around in two dimensions but for three dimensions, there were a lot of pains (I found out the hard way). Now I know it's the force of gravity that makes anyone with mass so miserable.

My ability to walk or crawl around in two dimensions contains an infinite number of choices for a direction that I will eventually decided to take. But in three dimensions, I have no such freedom, it all fall downward to the center of the earth. (True or not) Newton found this out the less painful way by just being hit on the head by the falling apple. Nevertheless, Newton invented his law of universal gravitation. But he never really explain what mass is. He assumed it. What I wanted to do is to understand mass completely and its relationship to the entire universe. Now I know there is such a thing as a photon, which has no mass but only energy. The electromagnetic equations of Maxwell do not have a mass term. And because of this EM waves can go anywhere.

Then Einstein came up with rest mass and energy equivalence. What he did is add the time dimension to the three space dimensions. Although a photon can go anywhere and anywhen in three dimensions of space, it cannot go anywhere (omnipresence) in four dimensions of space-time. In general relativity, it can be trapped by a black hole. So photon can have infinite directions to move in 3-space, it can only have two in 4-dim space-time, to fall into the black hole or to get away from it.

From these brief descriptive analyses, we can make the following postulates:

1. The one dimension of time has two directions.
2. The one dimension of space has two directions.
3. The two dimensions of space has infinite directions (infinite freedom of mass).
4. The three dimensions of space have double infinite directions (doubly infinite freedoms of photon).

Antonio

 

[Antonio Lao]

All Theorists,

If the concept of "direction" is correlated to the concept of dimension, the followings can be asserted:

1. The inertial force of Newton's 2nd law of motion is a two dimensional force (this assertion is 100% true only for point-mass particles).

2. The attractive force of Newton's law of universal gravitation is a three dimensional force (this assertion is 100% true only for point-mass particles).

The dimensional forces of electromagnetism, nuclear strong force, and nuclear weak force are still under investigation at this time.

In Einstein's general theory of relativity, he made the principle of equivalence, which means that a 2-dim force is equal to a 3-dim force.
This principle indirectly implies that the geometry of space-time is closed (boundless but finite as that of a spherical surface).

Antonio

 

[Antonio Lao]

All Theorist,

The transformation of dimensional forces:

For a scalar field resulting from zero forces, the mapping is

$$f:R^n \rightarrow R$$, mass decrease to zero.

For a vector field staying in the same n-space, the mapping is

$$f:R^n \rightarrow R^n$$, mass stays the same.

By the same logic, dimensional forces should be allowed the following mappings:

$$f:R^n \rightarrow R^{n-1}$$, mass decreases

and

$$f:R^n \rightarrow R^{n+1}$$, mass increases.

Antonio

 

[Antonio Lao]

To all theorists,

By the logic of dimensional forces:

1. Force of gravity with quantum of a graviton (mass is zero) is mapped by

$$f:R^n \rightarrow R^{n-m}$$

2. Force of electromagnetism with quantum of a photon (mass is zero) is mapped by

$$f:R^n \rightarrow R^{n-m}$$

3. Force of strong nuclear force with quantum of a gluon (mass is zero) is mapped by

$$f:R^n \rightarrow R^{n-m}$$

4. Force of weak nuclear force with quantums of $$W\pm$$
and $$Z^0$$ (masses >> 0) is mapped by

$$f:R^n \rightarrow R^{n+m}$$

This last dimensional force is the force implied in the Higgs mechanism.

Note: The values of n's and m's are not specified.

Conclusion: Increasing the dimension of a force give mass. Decreasing the dimension give zero for mass of the force’s quanta.

Antonio

 

[Antonio Lao]

To all theorists,

A. Zee's book "Eisntein's Universe, Gravity at Work and Play" explains in layman terms an excellent description of why the gravitational force is restricted or effective only on a defined spherical surface. This description appeared in the Prologue: The Apple and the Moon

Antonio

 

[matt grime]

Originally posted by Antonio Lao
All Theorists,

If the concept of "direction" is correlated to the concept of dimension, the followings can be asserted:

And if it isn't?

 

[Antonio Lao]

matt grime,

When Newton formulated his law of universal gravitation and the 2nd law of motion in the 17th century, the concept of vector still wasn't formalized in mathematics.

The concept of "direction" is a property of a vector, the other property is "magnitude or scalar" of a vector. The concept of a vector originated in the middle of the 19th century (about 200 years later) when Sir William Rowan Hamilton invented quanternions. A quaternion is made up of a vector and a scalar. Later, the math of quaternions were divided into a branch of math called vector analysis.

Traditionally, the definition of coordinates in a coordinate system, for example the Cartesian coordinate system, is the same as that of dimension. The Cartesian system has three coordinates (x,y,z), this means that it has three dimensions. In vector analysis, each of these coordinates is associated with a unit vector. It is this association of unit vector to a dimension that started me to investigate further.

Basically, I am redefining the concept of coordinates and unit vectors. In one dimension, there are two unit vectors (two opposite directions). In two dimensions (not two coordinates but more of a surface or area), there are an infinite number of unit vectors (half of this infinite units vectors have the other half as "opposites." In three dimensions (not three coordinates but more of a volume or a closed surface), there are a doubly infinite ( infinity multiply by infinity) of unit vectors. Half of these are "opposites" to the other half.

The correlations that I came up with are the followings:

1. one dimension - two directions.
2. two dimension - infinite directions.
3. three dimension - doubly infinite directions.

The traditional definition of dimension and direction is one to one (homeomorphism).
My new definition is one to many (polymorphism).

If the concept of "direction" is not correlated to the concept of dimension then we are back to square one of the traditional formulations.

Antonio

 

[Pergatory]

My ability to walk or crawl around in two dimensions contains an infinite number of choices for a direction that I will eventually decided to take. But in three dimensions, I have no such freedom, it all fall downward to the center of the earth.

So photon can have infinite directions to move in 3-space, it can only have two in 4-dim space-time, to fall into the black hole or to get away from it.

I don't understand your assertion that possible vectors are eliminated as the number of dimensions increases. It is, in fact, the exact opposite. Please explain in more detail, thanks!

 

[Antonio Lao]

Pergatory,

The "dimension" here is mapped to geometric figures.

One dimension is mapped to lines. Two dimensions mapped to surface. Three dimensions mapped to volume.

In a line, there are two directions of motion for a point on the line.
In a surface, there are infinitely many directions for a point to move. In a volume, there are doubly infinite directions to move.

A point can move (that is accelerate and not constant velocity) only if there is a force. But now we create a force, on the one dimensional line, the attractive force can pull in one direction, while the repulsive force can push in the other direction.

In a two dimensional surface, these forces can push or pull in infinite directions. Half of these directions are the exact opposite of the other half. So in a sense, equal magnitudes forces canceled each other. The same analogy for the doubly infinite three dimensional forces. The key to understand dimensional forces is the concept of mass. If two masses are exactly equal, one dimensional forces are created. If there are infinite number of equal masses, two dimensional forces are created. If one mass is infinitely larger than another mass, three dimensional forces are created. Three dimensional forces are derived form outer product of vectors, while two dimensional forces are derived from inner product of vectors. One dimensional forces are always zero because of the principle of directional invariance.

Newton's inertial force in his 2nd law of motion is a two dimensional force on a surface, while his gravitational force is a three dimensional attractive force. These forces can only be applied to point-mass. For extended mass, there are four fundamental forces,
1. gravity
2. EM force
3. Strong force
4. Weak force
When we combined time to three space dimensions, we create an event.
An event is a world-point. And when a world-point move because of a space-time (4-dim force or tensor force), it create a world-line, which is really one dimension (map of a line is always one dimensional). The photon travels along a world-line such that the space-parts is equal to the time-part. In other words, the space-time interval of the photon is always zero. This implies that a photon cannot age but it cannot be everywhere. It has to move from here to there at the speed of light in vacuum (300,000 km/s).

Map of dimension to direction:

1-dim maps to 2 directions.
2-dim maps to $$\infty$$ directions.
3-dim maps to $$\infty^2$$ directions.
4-dim maps to 2 directions.
5-dim maps to $$\infty$$ directions.
6-dim maps to $$\infty^2$$ directions.
7-dim maps to 2 directions.
8-dim maps to $$\infty$$ directions.
9-dim maps to $$\infty^2$$ directions.

Antonio

 

[Pergatory]

I'm sure I'm missing something by my complete lack of understanding of most of what you've said in this thread so apologies if I'm asking you to explain something you already have. Either way, perhaps we can make it more 'concrete.' :)

Originally posted by Antonio Lao
Pergatory,

The "dimension" here is mapped to geometric figures.

One dimension is mapped to lines. Two dimensions mapped to surface. Three dimensions mapped to volume.

In a line, there are two directions of motion for a point on the line.
In a surface, there are infinitely many directions for a point to move. In a volume, there are doubly infinite directions to move.

A point can move (that is accelerate and not constant velocity) only if there is a force. But now we create a force, on the one dimensional line, the attractive force can pull in one direction, while the repulsive force can push in the other direction.

In a two dimensional surface, these forces can push or pull in infinite directions. Half of these directions are the exact opposite of the other half. So in a sense, equal magnitudes forces canceled each other. The same analogy for the doubly infinite three dimensional forces.

I'm with you so far! But next is where you throw the proverbial 'curve'

Originally posted by Antonio Lao
The key to understand dimensional forces is the concept of mass. If two masses are exactly equal, one dimensional forces are created. If there are infinite number of equal masses, two dimensional forces are created. If one mass is infinitely larger than another mass, three dimensional forces are created. Three dimensional forces are derived form outer product of vectors, while two dimensional forces are derived from inner product of vectors. One dimensional forces are always zero because of the principle of directional invariance.

How can a force be one-dimensional or two-dimensional? I mean, aside from requiring only a certain number of dimensions to describe its vector (straight line = 1-d, wave = 2-d, field = 3-d), doesn't it still exist in the whole of space-time? Aren't the other dimensions still required to describe its vector in relation to other 4-dimensional bodies?

 

[matt grime]

Wow, patronized by an evident crank who cannot define quaternions correctly, nor even use infinity in any mathematical way.

 

[Antonio Lao]

Pergatory,

If we make the distinction between dimension and direction then we can defined dimensional forces.

The resultant vector (which became a scalar hence its directional property is destroyed) of a dot product of two vectors remains on a "surface" where all the other vectors live. This surface is what gives meaning to two dimensions. On this same surface, vector addition can also be defined. if we called this vector as a force, we can define a 2-dim force.

The resultant vector of a cross product of two vectors is perpendicular to the "surface" into the third dimension. This is a vector not found on the surface. If this vector is a force, it is defined as a 3-dim force.

A 1-dim force is the same as a scalar, it has magnitude but no direction. all the infinite number of 1-dim forces comprise a scalar field. All forces are zeros. A vector is transformed into a scalar.

A 4-dim force (space-time tensor force) is really a 1-dim force at a higher level of existence. It is built up by lower levels 1-dim, 2-dim, 3-dim forces.

When we try to define mass in terms of dimension and direction, it can be shown that mass are 1-dim forces.

Almost all forces in nature are known to be conservative forces. That is to say that some properties of the forces remain the same after interactions.

2 vectors transform into a scalar by dot product gives 1-dim force.
vector addition of 2 co-surface vectors stays on the same surface.
2 co-surface vectors transform into 3-dim vector by cross product gives a 3-dim force.

Antonio

 

[Antonio Lao]

matt grime,

I never profess to be an authority in mathematics. I am just using math. The properties of quaternion that I am using is scalar and vector. Direction can only be defined for vector, not for scalar. I am not sure whether direction can be defined for a tensor? To me tensors are matrices. So the concept of direction is implied in a matrix.

You are welcome to give the correct definitions for quaternion, vector, scalar, matrix or tensor for my benefit and for Pergatory as well.

Thanks, if you do.

Antonio

 

[matt grime]

The quaterions are a 4 dimensional real vector space. They are also a division algebra.

you proabably think it is a vector and a scalar because they are written

x+yi+zj+wk

all the x,y,z,w a scalars, but you haven't noticed that there is an implicit basis vector the x is multiplying, often denoted 1.


Your insistence on direction for vectors is touching but highly misplaced. It is not an intrinsic part of a vector. It is a myth propagated by bad teaching that a vector is an object with length and direction. A vector is an ordered n-tuple of entries from some field. That it is used to model space, where we can define distance and a relative notion of direction that tallies with our observations, is handy. But there are many vector spaces where this makes no sense, although we do formally declare things to be orthogonal wrt to some inner product.

 

[matt grime]

forgot to address your misuse of infinity.

all the quantities you label as powers of infinity (finite powers) have the same cardinality, that of the continuum

 

[Antonio Lao]

matt grime,

Thanks for your free knowledge about quaternions, vectors, and math in general. Normally, it takes 10 years or more of higher schooling before one can acquired these knowledge that you just gave. And even then, as you said, still the knowledge learned can be misplaced.

Can you say with confidence and conviction that the physical concept of absolute direction (instead of relative notion) can or cannot be resolved using math? Among abstract vector spaces, there are tangent spaces and norm spaces. Are norm spaces perpendicular to tangent spaces? By definition? By deeper abstraction? How far can we keep on covering our ignorance with further abstractions and more abstractions?

If direction is really a physical concept and current math cannot describe it then what can one do to resolve one sense for all the idea of directions (concept of motion in physics): the absolute direction of force, the absolute direction of acceleration, the absolute direction of velocity, the absolute direction of momentum, the absolute direction of angular momentum, the absolute direction of the flow of time? Absolute in these sense means conservation (something don't change. I called it the principle of directional invariance).

The bottom line of what I'm trying to do is to get to the bottom of defining the concept of mass using the concept of direction. Mass is a scalar in physics. By its nature, mass has no direction. But my idea is that mass is derived from a quantization of an infinitesimal direction of one dimensional space. This region of space is many factors less than the Planck length. It's almost a mathematical point (actually, it's a region of two points in space), while it still can be defined as a physical system of two points because of the existence of an infinitesimal metric between these two points.

Antonio

 

[matt grime]

It doesn't take many years of schooling. A quick search on google will give you more information abuot quaternions than you can learn.

Like many things in this theory development forum your initial question is ill-posed to a mathematician: I don't know what you mean by it.

Norm spaces are not perpendicular to tangent spaces. Firstly that makes little sense (what does it mean for class A of objects to be orthogonal to class B of objects?).

A norm space (by which i presume normed) is one with a norm, A tangent space requires some manifold.

Dunno why you're going on about 'our' ignorance. these objects are all well understood.

 

[Antonio Lao]

matt grime,

I'm from the old school. A time when fiber optics and internet are yet to be born. My foundation in math is worse than that of a starving artist trying to sell his/her first series of paintings. I never have the good fortune of a proper schooling. I could also say the same about my education in physics.

What I wanted in life is to do physics. When I graduated in 1973, I can't find works in physics. I had to take any jobs that I can find. In my spare time, I'm self studying math and physics all over again and again. But without proper guidance from a good physics/math teacher, the progress is that of the pace of a turtle, and sometime, I got the feeling of actually going backward.

What I mean by norm is the normal vector. This is orthogonal to the tangent vector. All the normal vectors taken together comprise a normal space and same with tangent vectors to tangent space.

The inner product of orthogonal vectors is always zero. This is math! But from a physical point of view, my question is what happened to the vectors? If these vectors are forces, what ever happen to them? They can't just vanish by doing inner product? Maybe its the geometry that I am not clear or ignorant about. The vector additions of orthogonal vectors give a larger vector (this is math again). But the logarithm of multiplication is just the same as addition. I am not a mathematician. So I could never answer my own questions. It might be obvious to you but not to me.

Antonio

 

[matt grime]

If you're simply thinking about how can these dot products of forces vanish then the answer is simple.

suppose you want to find work done, then its F.d, force dot displacement, and this can be zero because you are omitting to remember you are finding the work done in *the direction of d*. Dotting finds compents in some direction. For instance, a particle confined to moving on a sphere centred on the origin will always have velocity orthogonal to displacement, nothing special really, you're just saying it isn't leaving the surface of the sphere.

 

[Antonio Lao]

matt grime,

Thanks for your help in my shortcomings in math. Now I can try to answer Pergatory's questions.

Pergatory,

1-dim force - a force that confines a point-mass particle in a line.
2-dim force - a force that confines a point-mass particle in a surface.
3-dim force - a force that confines a point-mass particle in a volume.
4-dim force - a force that confines a point-mass particle in space-time.

Antonio

 

[matt grime]

A force is a vector quantity, it doesn't have a dimension.

 

[Antonio Lao]

matt grime,

Thanks again for your certitude. But I still have one question for you. Can you please define "dimension" for me? Thanks.

to all other theorists,

There is nothing more that I can add to this thread until I can finish square 11. I decided to go back to square 1 to do more reading and research the way I know how.

Antonio