Frequency and invariance, some trivialities.

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.

This is an apparently trivial point, but there may be non-trivial
consequences.

We can go on to consider a sphere of radius l.

In the rest frame of the sphere, its radius is equivalent to the proper
time p.

Since p is an invariant, then p is the radius of a sphere in any
inertial frame - but l is not.

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

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 -mathematically, philosophically and physically.

It's a trivial point, but it doesn't seem to be discussed much.

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

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)

becomes

E^2 + (pc)^2 = (m*c^2)^2

and, against a background of proper time, E can be treated as an
invariant

E^2 = (m*c^2)^2

m thus becomes, like energy, a vector in proper time.

Those interested might like to investigate Newton's Second Law in such
a system.

Frequency can be treated in the same way, a vector with different
components in different coordinates..

Frequency is a vector and atomic clocks will tick at different rates
according to the relative motion of the observer in proper time.

Barry

.

Relevant Pages

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  • Re: Where is the flaw?
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  • Re: Is a clock second an interval of universal time???
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  • Re: Frequency and invariance, some trivialities.
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    (sci.physics.relativity)
  • Re: Frequency and invariance, some trivialities.
    ... invariants They do not remain unchanged by a Lorentz transformation. ... There is, however, a related invariant called proper time, such ... Since invariance is so highly valued in relativity why don't we do all ...
    (sci.physics.relativity)
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