Re: The Essential History of Special Relativity

On Jun 15, 8:47 am, "Juan R." González-Álvarez
<juanREM...@xxxxxxxxxxxxxxxxxxxx> wrote:
EdwardGreenwrote on Sat, 14 Jun 2008 07:52:58
-0700:

On Jun 14, 6:06 am, "Juan R." González-Álvarez
<juanREM...@xxxxxxxxxxxxxxxxxxxx> wrote:

Are you noticing the difference between
invariant and covariant laws?

Please provide brief abstract definitions.
Thanks.

Covariance:

{T^(a...)_(b...); g_ab; f(g_ab)} -->
{t^(m...)_(n...); y_mn; h(y_mn)}

Invariance:

{T^(a...)_(b...); g_ab; f(g_ab)} --> {phi
T^(a...)_(b...); g_ab; f(g_ab)}

with phi an isometry.

Some authors call "general covariance" to the
former and "special
covariance" to latter.

Hmm... I got exactly what I asked for! Too brief and
too abstract for
me. Sorry to waste your time.

If you care to throw some good money after bad
though, how about a
brief gloss, in words, what those extremely compact
formulae have to
say for themselves? Anybody?

I don't thnk they say anything at all.

How about if we look at Einstein's own words?

In The Meaning of Relativity (Princeton U Press, 1956
pp. 9--11), he shows that linear orthogonal
transformations in the Cartesian system of coordinates
"...have an objective significance, independent of the
particular choice of the Cartesian coordinates, as can
be expressed by an invariant with respect to linear
orthogonal transformations. This is the reason that the
theory of invariants, which has to do with the laws that
govern the form of invariants, is so important for
analytical geometry." He follows with another example of
a geometrical invariant, using volume.

Einstein next shows that invariants "...are not the only
forms by means of which we can give expression to the
independence of the particular choice of the Cartesian
coordinates. Vectors and tensors are other forms of
expression." He demonstrates that the equations of
straight lines " ...have a significance which is
independent of the system of coordinates." He
continues, "[I]f the three components of a vector
vanish for one system of Cartesian coordinates, they
vanish for all systems, because the equations of
transformation are homogeneous. We can thus get the
meaning of the concept of a vector without referring to
a geometrical representation. The behaviour of the
equations of a straight line can be expressed be saying
that the equation of a straight line is co-variant with
respect to linear orthogonal transformations."

This is why I told Shubee earlier that he does not
grasp the mathematical model of special relativity--he
tries to argue that the clocks and rods of measure are
not independent of the measurement. Einstein's case
is, however, both mathematically and physically
crank-proof.

Tom
.

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