Re: Taylor & Wheeler:Spacetime Physics, Ex,L-5 Doppler Shift

John C. Polasek wrote:

On Thu, 28 Jun 2007 10:51:31 -0400, hetware <massless@xxxxxxxxxxxx>
wrote:

John C. Polasek wrote:

On Thu, 28 Jun 2007 09:09:11 -0400, hetware <massless@xxxxxxxxxxxx>
wrote:

In exercise L-5 of Taylor and Wheeler's Spacetime Physics, they ask me
to show that the location x of the nth pulse of light from an emitter
with constant frequency satisfies the relation:

n=(f/c)(t-x)

Where time is given in meters. So WTH is c? They say that the wave
travels in the positive x direction with speed c, but c=1 unless we are
dealing with phase velocity or propagation through some material where
the speed of
light != c. Neither of these conditions are mentioned in the statement
of the problem.

If the problem were given in conventional units, n=(f/c)(t-x) fails
dimensional analysis. f[1/T], c[L/T] t[T] x[L], f[(1/T)(T/L)(T-L). The
last term (T-L), in particular doesn't make sense. Does anybody
understand what they mean here?

You must insert c m/s for ct. c=1 is bad practice. Your query attests
to that.
f/c is commonly called the wave number K with units in cycles per
meter, So the number of cycles is given by
n = K cycles/meter * distance(ct-x)meters = cycles.
John Polasek

Thanks. That looks like the proper interpretation.

Ya reckon?

Well, with the exception that I now recall that wave number is typically
given as omega/c, where omega = 2pi nu.

I'm not really sure why
the expression appears in that form. It seems reasonable that if c is to
be used anywhere, it should appear everywhere it would appear in the
expression written in conventional units.

I tend to agree that measuring time in units of distance is problematic.
I've entertained the idea of using a unit which is identical to a meter of
time, but which is not interchangeable with units of length. The problem
caused by c=1 is really a problem of mixing units of length with units of
time, and not the relative magnitude of the units when compared to say,
seconds.

I'm not sure, however, how my proposition would produce much benefit. One
reason I like using c=1 is because I find the hyperbolic trig functions to
be a nice way of expressing SR. Writing the Lorentz transformation as

x = x' cosh[theta] + t' sinh[theta]
t = x' sinh[theta] + t' cosh[theta]

doesn't preserve the dimensional information. Trying to tack it on seems
to defeat the purpose of using that notation in the first place.

--
http://www.vho.org/GB/c/DC/gcgvcole.html
http://www.vho.org/GB/Books/dth/
http://www.germarrudolf.com/
http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm
.

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