Re: More Trivialities

On Tue, 16 Jan 2007 18:07:34 -0800, Barry wrote:

sal wrote:
Barry wrote:

In the 8-space, the metric is

(8^2 - 4^2)^0.5 - (8^2 - 4^2)^0.5 = 0

I don't understand what you would use this for. Certainly your metric
didn't enter into this problem at all -- you just used the ordinary
Minkowski metric, above.


You asked for a calculation of proper time. So I only needed to deal with
the Proper Time space.

Any event in R^4 has coordinate in R^8 of the form (t,x,y,z,t,x,y,z).


But _every_ event can be assigned coordinates in R^4. So, _any_ event
must be represented in R^8 by coordinates of the form

(t,x,y,z,t,x,y,z)

Yes, it's trivial.

and I don't understand what the second copy of (t,x,y,z) is buying you.


Lebensraum.

Take two space-like separated events A (a, b, c, d) and B (e, f, g, h) in
Minkowski space. They can't be connected causally.

In 8-space, they are (a, b, c, d; a, b, c, d) and (e, f, g, h; e, f, g,
h) and are null separated. They can be connected causally in 8-space.
Every event can, in principle, be causally connected to any other event.

I don't see how that corresponds to the usual definition of
"causal connection". A "causal connection" between event A and event B
conventionally means events A and B are both on the worldline of one
observer, and the observer's proper time is smaller at A than at B. That
observer would say event A occurred "before" event B.

You've brought in a null metric here, in which the distance between any
two points in 4-space -- after mapping them to points in 8-space -- is
zero. But when I asked you to find the proper time between two events
(in the first half of the "twins problem") you used the Minkowski
metric, of course, which is perfectly reasonable -- but it points up the
fact that the proper time is unrelated to the distance using your metric,
and the fact that your metric reads zero as the distance between the two
events doesn't seem to indicate the presence of a "causal connection" in
the usual sense of the term.

If you care to redefine the term "causal" then you can certainly say
they're causally connected, of course.


It reconciles SR with observation - by itself, that's enough.

Observations are done in 4-space where observation of causality
corresponds with finding two events on the same worldline, one at smaller
proper time than the other. I don't see how introducing a second metric
-- which does not measure proper time -- helps with that, or indeed has
any effect on it at all.


In particular, the causal connection has no preferred direction. A and B
are causally connected, but fundamentally it makes no sense to ask if A
caused B or if B caused A.

Such questions are frame dependant.

Look at Feynman diagrams, they can be oriented in any direction and
still make sense.

So, could you show a worked example (with numbers) in which you make
use of the full R^8 to obtain an answer to a problem?


Above we dealt with the 4-temporal space.

I'll just give a brief picture of some issues in 4-space (not
4-spacetime) to show how I'm thinking.


Consider a point A at the spatial origin (0i, 0, 0, 0) of a 4-space.
It's *directly* connected via a Minkowski-like metric to an array of
points that's arranged in the form of a light cone.

(envision a typical Minkowski diagram, but in this case representing a
spatial space).

Consider a second point B at (0i, 5, 0, 0) on the x axis.

In the real subspace ( the last three coordinates) A is five units of
distance away from B.

Keeping A still, let's virtually displace B to the point (i, 5, 0, 0).
Now, directly to the left of B there's a slice of the light cone of
points that's directly connected to A. - forming a sphere.

Let's imagine that a red particle is located at A and a blue particle at
B. Keeping A still, let's virtually slide B up along the w axis from
(0i, 5, 0, 0) to (i, 5, 0, 0)

Tracing a "line" from B to its left, A spreads out from a red point
particle located at (0i, 0, 0, 0) to a red particle sphere centred at
(i, 0, 0, 0) with a radius of one unit.

In B's frame, a red point particle has become a red sphere.

Let's pretend that the particle is an electron E. At some events during
this process, the electon will interact iwith its surroundings. At each
such event, the electron becomes localized at that event and then
commences spreading out again.

It follows that Electron E will always remain inside some sphere whose
surface is expanding at c relative to its original position.

An observer at B searching for E, will not observe these interactions,
but can only search for E within that sphere.

Should E be contained within an atom by an unknown number of
interactions, then E's position must be indeterminate until it is at an
event which is actully observed by B - at which point it's range of
possible locations will collapse to that event.

Sure. But you did this all in 4-space, without using the other 4
dimensions, and without using the R^8 metric for anything. So, I still
don't understand what you actually use the extra dimensions for.

When do you ever deal with events which have _no_ 4-space coordinates --
events which fall outside the 4-dimensional subspace which is the
embedding of Minkowski space in R^8?




Barry

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