Re: Frequency and invariance, some trivialities.


bergeron wrote:
Barry wrote:
In SR, spatial separations (l) and temporal separations (t) are not
invariants They do not remain unchanged by a Lorentz transformation.

There is, however, a related invariant called proper time (p) , such
that

p^2 = l^2 - t^2

which is invariant. p can, in general, be measured by clocks (rulers if
necessary)

This equation can be rewritten as

p^2 + t^2 = l^2

In the rest frame of a rod of length l, it follows (from the
definition of a rest frame) that the coordinate time (t) difference
between the ends of the rod is zero.

Thus

p^2 = l^2

Generalizing this result, we can see that the spatial separation
between two events is equivalent to a temporal separation.

No, it isn't. A null ray (p^2 = 0, using your nomenclature),
separates
spacetime into spacelike and timelike intervals. A spacelike interval
can never be made timelike, under any Lorentz transform.

*snip*
Since invariance is so highly valued in relativity why don't we do all
our maths against against a background of proper time?

For the same reason we do not use the r^2 in r^2 = x^2 + y^2 + z^2 to
try and define x, y and z in ordinary 3-d Euclidean geometry. It
doesn't
make sense. The proper time (like the r above) is an affine parameter
of
a curve. It gives the length of the curve. There an infinite number of
curves with the same length, but that tells you nothing about any
particular curve connecting to points (unlessof course, you know the
parameterization, in which case you had to already know what you are
seeking to avoid as a starting point, making the point irrelevant).


The use of non-invariant lengths is like smoking - a bad habit.

But in SR, spacetime is 4-dimensional, so proper time must also be
3-dimensional

The proper time is a Lorentz scalar and scalars are numbers, not
3-dimensional quantities.

*snip*

So might it not be simpler to do our physics in a framework of
4-dimensional proper time.

That _is_ what physicists do, except not in the way you are trying
to go about it. What you are trying to do does not make sense.
It is equivalent to saying ``... it might be simpler to do [fill in
the blank] in terms of distances.'' Lots of distances are equal,
but equality does not imply any relationship between the phenomena
which produced the measured distances.


And since proper time is like length, it must be an entirely relational
concept. To return to the same relative point in proper time is *not*
the same as returning to the past.

In particular

E^2 = (m*c^2)^2 - (pc)^2 (where m is proper/rest mass)


The relation is E^2 = p^2 + m^2 (I am not going to bother writing
c).
If your minus sign was intentional, then what you have done is simply
shuffle the labels the quantities. In relativity, the invariant is
well-defined. It is called the mass and it has a straight-forward
connection to quantities in real experiments.

There is some similarity in what you are trying to do to what is
called a Wick rotation which results in a 4-d Euclidean system.
Your notion of an invariant energy (rigorously formulated) is
called an instanton.

You are trying to teach a very old troll. He pulls up the same ***
periodically. Don't count on him learning or understanding anything.

.

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