While reading a book by Stephen Hawkins, I came across this sentence,
which roughly said "Earth travels linearly in 4D, whereas, it appears
to travel in curves (circular) in 3D"
I wasn't immediately interested in drawing up equations and proving
the above, but, needed a small model in which I could visualize 4D in
a rough way.

To start, I have two assumptions/observations about 4D:
1. Every Point in 3D space has a point in 4thD which is at
infinitesimally small distance.
2. As 3D can be visualized "in prespective" in 2D (i.e. on Paper), 4D
can be viwed "in prespective" in 3D.

For which, I propose a simple method to imagine/visualize a 4D Space:
1. Draw a 3D co-ordinate system on paper "in prespective"
2. Imagine a stack of such papers w/ 3d "prespective" coordinates,
and, there we have a simple 4D model.

Please note:
1. For every point that can be mapped on a "prespective" 3D
diagram on paper, there is a point in the 4th D which is at a
infinitesimal distance (i.e. on papers immediately above/below the
paper in question)
2. The "prespective" view maps more than one 3D point on a 2D paper,
and hence, I am not in a position to comment the loopholes that may be
introduced due to such a 4D visualization, when mathematics is imposed
on this simple 4D model.

Observation:
I made the following observation using my simple 4D model. I
am not sure about the validity of what I state in certain parts below:
1. When viewed "in prespective" a (3D) Sphere appears to be a (2D)
disc.
Hence, I infer,
2. When viwed "in prespective" a 4D counterpart of Sphere, which has
no restriction on the 4th Dimension, will roughly appear as a (3D)
Cylinder.

Sincerely,
Araivndh Krishnamoorthy

Aravindh K wrote:
While reading a book by Stephen Hawkins, I came across this sentence,
which roughly said "Earth travels linearly in 4D, whereas, it appears
to travel in curves (circular) in 3D"

He means linearly in 4D spaceTIME. And by "linearly" he really means
"along a geodesic".

Earth orbits the sun in a plane (neglecting the other planets'
influences), so let us ignore the spatial dimension normal to that
plane, and imagine the reduced 3d spacetime projected onto a 3d space.
In this space, earth moves in the X-Y plane, and the Z axis is time. The
path of the earth is an elliptical helix with axis along Z; the period
of the helix is one year, and in that year the earth travels one
light-year along Z and around an ellipse about 8 light-minutes across in
X and Y. So this path deviates from a straight line by approximately 8
minutes / 1 year ~ 10^-5.

Implicitly I am using "background" Minkowski coordinates.
That's OK because gravitation is so weak here.


Tom Roberts

A Method to Visualize 4th Dimension

You can best visualize the 4th dimension by imagining a rubber-band, red on the
inside and green on the outside. If you, a 3-D being, "flipped" this
rubber-band inside out a 2-D being, located outside of it, would only see the
rubber band go from green to red. i.e., the 2-D being would have to cut the
rubber band at some point and bring one end back around to the other to do this
whereas you, being 3-D, would not.

From this same perspective, a 4-D being could turn a basketball, or any hollow
2-D sphere, inside out without cutting it, whereas you or I , being we are only
3-D beings, would find it neceaasry to cut it first in order to turn it inside
out.

all the best,

Jeff Lee CENTER for REALITY PHYSIC

Aravindh K wrote:

While reading a book by Stephen Hawkins

The man's name is Hawking, not Hawkins. If you were reading
the book, couldn't you at least checked the author's name
before you posted this?

, I came across this sentence,
which roughly said "Earth travels linearly in 4D, whereas, it appears
to travel in curves (circular) in 3D"
I wasn't immediately interested in drawing up equations and proving
the above, but, needed a small model in which I could visualize 4D in
a rough way.

To start, I have two assumptions/observations about 4D:
1. Every Point in 3D space has a point in 4thD which is at
infinitesimally small distance.
2. As 3D can be visualized "in prespective" in 2D (i.e. on Paper), 4D
can be viwed "in prespective" in 3D.


That isn't exactly new.

For which, I propose a simple method to imagine/visualize a 4D Space:
1. Draw a 3D co-ordinate system on paper "in prespective"
2. Imagine a stack of such papers w/ 3d "prespective" coordinates,
and, there we have a simple 4D model.

Please note:
1. For every point that can be mapped on a "prespective" 3D
diagram on paper, there is a point in the 4th D which is at a
infinitesimal distance (i.e. on papers immediately above/below the
paper in question)
2. The "prespective" view maps more than one 3D point on a 2D paper,
and hence, I am not in a position to comment the loopholes that may be
introduced due to such a 4D visualization, when mathematics is imposed
on this simple 4D model.

Observation:
I made the following observation using my simple 4D model. I
am not sure about the validity of what I state in certain parts below:
1. When viewed "in prespective" a (3D) Sphere appears to be a (2D)
disc.
Hence, I infer,
2. When viwed "in prespective" a 4D counterpart of Sphere, which has
no restriction on the 4th Dimension, will roughly appear as a (3D)
Cylinder.


Sorry, but you're reinventing the wheel.

John Anderson

On Tue, 14 Sep 2004 15:24:00 GMT, Tom Roberts wrote:

Aravindh K wrote:
While reading a book by Stephen Hawkins, I came across this sentence,
which roughly said "Earth travels linearly in 4D, whereas, it appears
to travel in curves (circular) in 3D"

He means linearly in 4D spaceTIME. And by "linearly" he really means
"along a geodesic".

Earth orbits the sun in a plane (neglecting the other planets'
influences), so let us ignore the spatial dimension normal to that
plane, and imagine the reduced 3d spacetime projected onto a 3d space.
In this space, earth moves in the X-Y plane, and the Z axis is time. The
path of the earth is an elliptical helix with axis along Z; the period
of the helix is one year, and in that year the earth travels one
light-year along Z and around an ellipse about 8 light-minutes across in
X and Y. So this path deviates from a straight line by approximately 8
minutes / 1 year ~ 10^-5.

I find this quite extraordinary...coming from such a noted SRian.

Firstly, in your 2D +T representation, the Earth doesn't MOVE anywhere.

The helix is static. The +ve direction of the Z axis defines what we call
'increasing time'. The gradient of the helix wrt Z, IS velocity.

In one year, the Earth doesn't travel 1 LY along Z at all. Nothing travels
anywhere in your diagram.

Distances such as LYs are represented as projections in the X, Y plane, not
the Z direction.

Tom I am very surprised at you.





Implicitly I am using "background" Minkowski coordinates.
That's OK because gravitation is so weak here.


Tom Roberts


HW.

www.users.bigpond.com

Henri Wilson wrote:
On Tue, 14 Sep 2004 15:24:00 GMT, Tom Roberts wrote:
Earth orbits the sun in a plane (neglecting the other planets'
influences), so let us ignore the spatial dimension normal to that
plane, and imagine the reduced 3d spacetime projected onto a 3d space.
In this space, earth moves in the X-Y plane, and the Z axis is time. The
path of the earth is an elliptical helix with axis along Z; the period
of the helix is one year, and in that year the earth travels one
light-year along Z and around an ellipse about 8 light-minutes across in
X and Y. So this path deviates from a straight line by approximately 8
minutes / 1 year ~ 10^-5.

Firstly, in your 2D +T representation, the Earth doesn't MOVE anywhere.

You have to distinguish between time IN THE MODEL, and time to us
analysts. IN THE MODEL the earth does indeed move; to us analysts the
entire model is static.

At time t=0 the earth is at one point on the helix, and at t=1 year it
is 1 light-year in the +Z direction. As I have expressed this in X,Y,Z,
it is indeed a light year along Z, and 1 year in time.


The helix is static.

Sure. To us analysts. IN THE MODEL the earth is simply located at a
specific point, determined by the t coordinate. To an observer in the
model there is no helix, there is simply an earth that is moving in the
X-Y plane.


The +ve direction of the Z axis defines what we call
'increasing time'. The gradient of the helix wrt Z, IS velocity.

Sure. Except for units.


In one year, the Earth doesn't travel 1 LY along Z at all.

Sure it does. This is one year IN THE MODEL, not one year to the analyst
(you and me).


Nothing travels
anywhere in your diagram.

Sure the earth does. At t=0 it is located at one point in X,Y,Z, and at
t=1 year it is at a different place, one lightyear away along Z. This is
traveling IN THE MODEL. We analysts have a completely different time,
and to us the model is indeed static, and we can consider any value of t
IN THE MODEL (we have no control over our own time, of course).


Distances such as LYs are represented as projections in the X, Y plane, not
the Z direction.

They also apply in the Z direction, because of the way the model was set up.


Tom Roberts