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Invariant Galilean Transformations On All Laws
Section - 3. The Principle of Relativity and Transformation

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If a law is different over there than it is here,
it is not one law, but at least two, and leaves us
in doubt about any third location. This is the 
Principle of Relativity:  a natural law must be the 
same relative to any location at which a given event 
may be perceived or measured, and whether or not the
observer is moving.

The idea of location translates to a coordinate
system, largely because any object in motion could
be considered as having a coordinate system origin
moving with it. If you perceive me moving relative
to you - who have your own coordinate system - will
your measurements of my position and velocity fit
the same laws my own, different measurements fit?

If a law has the same form in both cases it is
called covariant. If it is identical in form, var-
ables, and output values, it is called invariant.

What we're asking is that if the x-coordinate, x, 
on one coordinate axis works in an equation, does 
the coordinate, x', on some other, parallel axis 
work?  Speaking in terms of the axis on which x is
the coordinate, x' is the 'transformed' coordinate.

The situation is complicated because we're talking
about coordinates - locations -  but in most mean-
ingful laws/equations, it is lengths/distances (and 
time intervals) the equations are about, and x coord-
inates that represent good, ratio scale measures of 
distances are only interval scale measures on the x' 
axis. [See Table of Contents for discussion of scales.]

So, if we have an x-coordinate in one system, then
we can call the x' value that corresponds to the same
point/location the transform of x.

In particular, the Principle of Relativity is embodied
in the form of the Galilean transformation, which
relates the original x, y, z, t to x', y', z', t' by
the transform equations x'=x-vt, y'=y, z'=z, t'=t in
the simplified case where attention is focused only
on transforming the x-axis, and not y and z. In the
case of Special Relativity, the x' transform is the
same except that x' is then divided by sqrt(1-(v/c)^2),
and t'=(t-xv/cc)/sqrt(1-(v/c)^2). In either case, v
is the relative velocity of the coordinate systems;
if there is already a v in the equations being trans-
formed use u or some other variable name.

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Top Document: Invariant Galilean Transformations On All Laws
Previous Document: 2. Table of Contents
Next Document: 4. The Encyclopedia Brittanica Incompetency.

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Last Update March 27 2014 @ 02:12 PM