(Ken S. Tucker) wrote in message . com...
(Gregory L. Hansen) wrote in message ...

Hi Greg

Here's another one to ponder. In special relativity make the
transformation to rotation coordinates,

t' = t

x' = x cos wt - y sin wt

y' = x sin wt + y cos wt

z' = z

From that you can find a metric, connections, and geodesic equations. And
the geodesic equations will have terms interpretable as centrifugal and
Coriolis forces, which can be generalized in vector notation. Those
inertial terms come from g_00 and g_0i=g_i0.

I wanted to follow a similar program in Newtonian mechanics, since it's
the same change of coordinates, it's still just geometry. But Newtonian
spaces don't have g_00 or g_0i because time isn't a component of Newtonian
vectors.

I think your tying one foot behind your back and entering an
ass kicking contest.

In Newtons Geometry you can still have 4D graphs, except
that these would be cartesian, so
g11 = g22 = g33 = g44 =1 and covariant and contravariant
components are equal.

So that wonderful derivation doesn't seem available in Newtonian
mechanics, and the centrifugal and Coriolis terms have to be pounded
together by other means.
The geodesic equations tell you what a straight line is, I'd have figured
it would tell you what a straight line is even in the Newtonian rotating
frame. I really don't understand what went wrong there.

I don't see what's wrong, if you permit Cartesian 4D.

I was going to give a kneejerk response, then went away to think about
it. Galilean relativity doesn't have four-vectors, it has ordered
pairs of three-vectors and time, which are separately conserved under
transformations. A Cartesian metric clearly can't work in Galilean
relativity. E.g. in special relativity, for an observer at rest in
the lower-case frame observing something in the capital frame,

c^2 dT^2 = c^2 dt^2 - dx^2

dT^2 = dt^2 (1 - (dx/dt)^2/c^2)

dt = dT / sqrt(1 - v^2/c^2)

Time dilation. In a Cartesian metric,

k^2 dT^2 = k^2 dt^2 - dx^2

dt = dT / sqrt(1 + v^2/k^2)

Time contraction.

But consider k=0,

ds^2 = 0*dt^2 + dx^2 + dy^2 + dz^2

When you work the metric in special relativity for a rotating frame
you get

ds^2 = (-c^2 + w^2(X^2+Y^2)) dT^2 + dX^2 + dY^2 + dZ^2

+ 2 Y w dX dT - 2 X w dY dT

The only contribution from the dt^2 term in the metric is the -c^2 in
the first term. If you work the problem as before for the Galilean
metric I gave, you get exactly the same thing, but without the -c^2.
So the 0*dt^2 is acting as a place holder to say that something could
be there, even if there isn't anything there right now.

And then when you consider Galilean time intervals you need to
consider the separate metric

(ds_t)^2 = dt^2 + 0*(dx^2 + dy^2 + dz^2)

which I find kind of ugly. Besides the fact that the four-dimensional
version is clearly derived from another theory that's not Newtonian
mechanics from a guy that already knows what kind of answer he wants
to get.

The usual way of going to Newtonian mechanics is to let c-inf, or the
ratio v/c-0. Which works fine in the time,

dT^2 - (dX^2 + dY^2 + dZ^2)/c^2 = dt^2 - (dx^2 + dy^2 + dz^2)/c^2

- dT^2 = dt^2 as c-inf

But what can we say about the spatial part? We can ignore the spatial
part when the time part is much larger, but we can't ignore the time
part when the time part is much larger.

Maybe I'll try to understand chapters 4 and 5 of Goldstein in the
language of differential geometry. But I wanted to get a reply off to
you because I never know in advance how long my attention span will
hold out.