Search the FAQ Archives

3 - A - B - C - D - E - F - G - H - I - J - K - L - M
N - O - P - Q - R - S - T - U - V - W - X - Y - Z

Invariant Galilean Transformations On All LawsSection - 4. The Encyclopedia Brittanica Incompetency.

( Single Page )
[ Usenet FAQs | Web FAQs | Documents | RFC Index | Property taxes ]

Top Document: Invariant Galilean Transformations On All Laws
Previous Document: 3. The Principle of Relativity and Transformation
Next Document: 5. Transformations on Generalized Coordinate Laws
```One example of the traditional fallacious idea
that an equation is not invariant under the galilean
transformation comes from the Encyclopedia Brittanica:

"Before Einstein's special theory of relativity
was published in 1905, it was usually assumed
that the time coordinates measured in all inertial
frames were identical and equal to an 'absolute
time'.  Thus,

t = t'.              (97)

"The position coordinates x and x' were then
assumed to be related by

x' = x - vt.         (98)

"The two formulas (97) and (98) are called a
Galilean transformation. The laws of nonrelativ-
istic mechanics take the same form in all frames
related by Galilean transformations.  This is the
restricted,  or Galilean, principle of relativity.

"The position of a light wave front speeding from
the origin at time zero should satisfy

x^2 - (ct)^2 = 0          (99)

in the frame (t,x) and

(x')^2 - (ct')^2 = 0       (100)

in the frame (t',x'). Formula (100) does not
transform into formula (99) using the transform-
ations (97) and (98),  however."
.................................................

Besides the trivially correct statement of what the
Galilean 'transform' equations are, there is exactly
one thing they got right.

I.   Eq-100 is indeed the correct basis for discussing
the question of invariance, given that eq-99 is
the correct 'stationary' (observer S) equation.
[Let observer M be the 'moving'system observer.]

In particular, eq-100 is of exactly the same
form [the square of argument one minus the square
of argument two equals zero (argument three).]

II.  It is nonsense to say eq-99 should be derivable from
eq-100; for one thing, the transforms are TO x' and
t' from x and t, not the other way around, and the
idea that either observer's equation should contain
within itself the terms to simplify or rearrange to
get to the other is ridiculous. As the transform
equations say, the relationship of t', x' to t, x
is based on the relative velocity between the two
systems, but neither the original (eq-99) equation
nor the M observer equation is about a relationship
between coordinate systems or observers. One might
as well expect the two equations to contain banana
export/import data; there is no relevancy. The
'transform' equations are the relationships between
x' and x, t' and t and have nothing to do with what
one equation or the other ought to 'say'.  The
equations' content is the rate at which light emitted
along the x-axes moves.

III. Most remarkable, the True Believer SR crackpots who
most despise the consequences of measurement theory
(demonstrable fact) contained in this document are
those who want to argue against our saying the Britt-
anica got eq-100 right;

They insist that the correct equation is derived
directly from x'=x-vt and t'=t. Solve for x=x'+vt
and replace t with t', then substitute the result
in eq-99: (x'+vt')^2 - (ct')^2 = 0.

Besides the fact that this results in an equation
with arguments exactly equal to eq-99, they will
insist the transform is not invariant.

IV.  A major justification they have for their idea of
the correct M system equation on which to base the
the discussion of invariance, is that the variables
are M system variables, never mind the fact that
the arguments are S system values.

That argument of theirs is arrant nonsense. The
velocity v that S sees for the M system relative
to herself is the negative of what the M system
sees for the S system relative to himself.

In other words, x'+vt' is a mixed frame expression
and it is x'+(-v)t' that would be strictly M frame
notation, and that equation is far off base. [Work
it out for yourself, but make sure you try out an
S frame negative v so as not to mislead yourself.]

V.   In I. we said: "given that eq-99 is the correct
'stationary' equation. Let's look at it closely:

x^2 - (ct)^2 = 0          (99)

This whole matter is supposed to be about coordinate
transforms. Is that what t is, just a coordinate?

No. It isn't, in general.  Suppose you and I are both modelling
the same light event and you are using EST and I'm using PST.
'Just a time coordinate' is just a clock reading amd your t clock
reading says the light has been moving three hours longer
than my clock reading says. Well, that's what the idea that
t is a coordinate means.

Eq-99 works if and only if t is a time interval, and in
particular the elapsed time since the light was emitted.
Thus, that equation works only if we understand just
what t is, an elapsed time, with emissioon at t=0.

However, we don't have to 'understand' anything if we use
a more intelligent and insightful form of the equation:

(x)^2 - [ c(t-t.e) ]^2 = 0,

where t.e is anyone's clock reading at the time of light
emission, and t is any subsequent time on the same clock.

Similarly, x is not just a coordinate, but a distance
since emission.

(x-x.e)^2 - [ c(t-t.e) ]^2 = 0        (99a)

VI.  In the spirit of 'there is exactly one thing
they got right', the correct M system version
of eq-99a is eq-100a:

(x'-x.e')^2 - [ c(t'-t.e') ]^2 = 0   (100a)

Every observer in the universe can derive their
eq-100a from eq-99a and vice versa, not to mention to and
from every other observer's eq-99a.

Now, THAT's invariance. [You do realize that every
eq-100a reduces to eq-99a, when you back substitute
from the transforms, right? t.e'=t.e, x.e'=x.e-vt.]

```

User Contributions:

Top Document: Invariant Galilean Transformations On All Laws
Previous Document: 3. The Principle of Relativity and Transformation
Next Document: 5. Transformations on Generalized Coordinate Laws

Single Page

[ Usenet FAQs | Web FAQs | Documents | RFC Index ]

Send corrections/additions to the FAQ Maintainer:
Thnktank@concentric.net (Eleaticus)

Last Update March 27 2014 @ 02:12 PM