wrote:
Hi Tim and Ken,

I am glad you dig into this subject. Let me give you some history of
my thinking, which would be hard to present in my initial post.

1. The linear scales of Lorentz spacetime can be defined in the
following way:

- Choose one light ray t in x direction. Let it travel from a to b.
Define the time it travels from a to b be 1 second and the distance ab
= 1 light-second. During this time, ray x travels from b to c, thus
define distance bc = 1 light-second. Define the time duration ray t
travels from b to c as the next second. Define the distance cd ray x
travels during the second second as cd = next light second, ... Thus
all the scales of x dimension and time dimension are defined.
- Run another ray y in y direction, mark the distance scale on y
dimension as the first light second, second light-second, third
light-second, ... during the first second, second second, third second,
...
- Do the same thing on z dimension. Thus the scales of time dimension
and all 3 spatial dimensions of this Lorentz spacetime are defined.
- Definition of simultaneity: Run two light rays from o in opposite
directions and let them be reflected back at two mirrors at p and q, if
the two reflected rays reach back at o at the same time, define the
time they reach p and q as simultaneous.
- Maxwell theory asserts any other light ray will be measured as of the
same speed as these 4 light rays.

However, if there are no magnetic fields running from one dimension to
another, or if the magnetic field line on xy-plane running from the 2nd
light-second on x-axis doesn not run through the 2nd light-second, but
through the 3rd light-second on the y-axis, the above definition would
still be meaningless. It is the whole Maxwell EM theory as an
integral, including light propagation and the electric and magnetic
fields running between different dimensioins, which guanrantees the
above definition of linear scales to make sense. From the fact that
all lights travel at the same speed in all directions, we can tell
there are guaranteed magnetic (and electric) fields running between
different dimensions. (Otherwise, lights would not travel at the same
speed in different dimensions).

Now, from the fact that plane angle scales on different planes are
equivalent to each other, we can also tell there are "guaranteed"
(solid angle) fields running from one plane (with its plane angle
scale) to another plane to ensure their equivalence. Otherwise, plane
angle scales cannot be equivalent to each other on different planes (as
manifested in numerous physics facts).

2 What we anticipate is a field running from one plane, with plane
angle scale, to another plane. Initially I thought it is something
like a cone shrinking from a plane (say xp-plane) to a needle, then
expand from the needle back to another plane (say zt-plane). But for
others which rotate from xy-plane to xz-plane, it becomes harder to
visualize from a Cartesian coordinates. It becomes more like a bent
cylinder coinnecting one plane angle scale to another plane angle
scale, while the axis and the fixed singular point become unnecessary.
To serve our purpose of comparing different plane angles, a cylinder is
essentially the same and topologically equivalent to a cone, even
though it is actually not a cone as visualized in a Cartesian
coordinates. That's why I continue to call it "solid angles".

3. Since particle classifications deal only with solid angular momentum
and plane angular momentum, without involvement of linear momentum, we
really don't need to bother the Cartesian coordinates. In fact, this
is the only way "my solid angle" can be defined. It is simply the
plane angle of a 6-d space, with each axis representing a plane angle
scale of the 3+1 Lorentz spacetime. (For the more natural 4+1
spacetime, it is a 10-d space representing the 10 palne angle scales).
We have to abandon Cartesian coordinates here, which is not wrong
because particle classifications don't bother linear momentum and, as
you can see, parity is also violated. On the other hand, when you
insist on a Cartesian view, such as in particle interactions (where
linear momentum is conserved), you will see conservation of solid
angular momentum (e.g. iso-spin, strangeness, etc.) is not observed.
So, there is an inherent incompatibility between the 6-d (or 10-d)
solid angle space and the Cartesian space.

Hope this calarifies a little bit more of this hard-to-convey concept.


Qchiang

I don't understand plane angle scales so much of the above context is
lost on me. An angle between points in a higher dimensional space is
just the same as it is in 3D or even 2D. The solid angle in higher
dimensions must be somewhat similar. An axis off of which an angle is
swept in every degree of freedom would be a loose definition. Anyhow
I'm down at too simple a level here. Thanks for the brain workout.
-Tim

wrote:
Hi Ken,

I lived in New York city for 20 years, but just moved to Vancouver this
summer. From your email address, it seems you are also in Canada.
Nice to be neighbors.
Qchiang

Small world. Got kids in Vancouver, daughter's a geo-eng helping
fix Seymour Dam. I'm over in the Okanagan right now just hangin
out.

As mentioned earlier, solid angle (and an axis) is only a rough
concept, without concrete definition, in Cartesian coordinates. The
only definition is on a 6-d or 10-d space, whose axes are plane angles
(of Cartesian coordinates) and whose plane angles are solid angles (of
Cartesian coordinates). But mathematically it can be more easily
understood from the equations in section 2.

Qchiang

Hi Ken,

Really a small world. If you ever stop at Vancouver, let's have a
coffee and chat. I'll send some details to your private Email. If
you don't receive it, pls contact me at . Regards,


Qchiang

wrote:
Hi Ken,

Really a small world. If you ever stop at Vancouver, let's have a
coffee and chat. I'll send some details to your private Email. If
you don't receive it, pls contact me at . Regards,
Qchiang

Ok will do, my email address above is bogus, I'm at

dynamiXs (at) uniserve.com

but X=c.

My kids have facilities in West and North Vancouver.
What I'd enjoy is using big sketch paper so you can
draw the 6D rotations, like a brief lecture.
Regards
Ken