Science > Physics > Taylor & Wheeler:Spacetime Physics, Ex,L-5 Doppler Shift
| Topic: |
Science > Physics |
| User: |
"hetware" |
| Date: |
28 Jun 2007 08:09:11 AM |
| Object: |
Taylor & Wheeler:Spacetime Physics, Ex,L-5 Doppler Shift |
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?
--
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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| User: "John C. Polasek" |
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| Title: Re: Taylor & Wheeler:Spacetime Physics, Ex,L-5 Doppler Shift |
28 Jun 2007 08:40:30 AM |
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On Thu, 28 Jun 2007 09:09:11 -0400, hetware <massless@nutrino.none>
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
.
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| User: "hetware" |
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| Title: Re: Taylor & Wheeler:Spacetime Physics, Ex,L-5 Doppler Shift |
28 Jun 2007 09:51:31 AM |
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John C. Polasek wrote:
On Thu, 28 Jun 2007 09:09:11 -0400, hetware <massless@nutrino.none>
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. 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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| User: "John C. Polasek" |
|
| Title: Re: Taylor & Wheeler:Spacetime Physics, Ex,L-5 Doppler Shift |
28 Jun 2007 07:21:24 PM |
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On Thu, 28 Jun 2007 10:51:31 -0400, hetware <massless@nutrino.none>
wrote:
John C. Polasek wrote:
On Thu, 28 Jun 2007 09:09:11 -0400, hetware <massless@nutrino.none>
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?
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.
.
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| User: "hetware" |
|
| Title: Re: Taylor & Wheeler:Spacetime Physics, Ex,L-5 Doppler Shift |
29 Jun 2007 12:47:19 AM |
|
|
John C. Polasek wrote:
On Thu, 28 Jun 2007 10:51:31 -0400, hetware <massless@nutrino.none>
wrote:
John C. Polasek wrote:
On Thu, 28 Jun 2007 09:09:11 -0400, hetware <massless@nutrino.none>
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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| User: "John C. Polasek" |
|
| Title: Re: Taylor & Wheeler:Spacetime Physics, Ex,L-5 Doppler Shift |
29 Jun 2007 10:06:19 AM |
|
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On Fri, 29 Jun 2007 01:47:19 -0400, hetware <massless@nutrino.none>
wrote:
John C. Polasek wrote:
On Thu, 28 Jun 2007 10:51:31 -0400, hetware <massless@nutrino.none>
wrote:
John C. Polasek wrote:
On Thu, 28 Jun 2007 09:09:11 -0400, hetware <massless@nutrino.none>
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.
To the contrary, the wave number is the metrical analog of frequency,
being the inverse of wavelength or meters/cycle, particularly in this
context where your "nth pulse" must be the nth cycle, not radian.
You mean it hasn't been stressed to you that all variables contained
in a single parenthesis in a physics equation must, without exception,
be dimensionally identical and thus save all the discussion? (Ignore
the apostates with their 'natural' numbers.)
You can always test the hypothesis with numbers, as here, e. g., x =
10 meters and t = 10 seconds, and consider the computational
consequences for (t - x).
I've worked it out: (10 seconds - 10 meters).
It is only in mathematics where the letters x and t can stand for
anything you please.
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.
John Polasek
.
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| User: "hetware" |
|
| Title: Re: Taylor & Wheeler:Spacetime Physics, Ex,L-5 Doppler Shift |
29 Jun 2007 06:54:05 PM |
|
|
John C. Polasek wrote:
On Fri, 29 Jun 2007 01:47:19 -0400, hetware <massless@nutrino.none>
wrote:
John C. Polasek wrote:
On Thu, 28 Jun 2007 10:51:31 -0400, hetware <massless@nutrino.none>
wrote:
John C. Polasek wrote:
On Thu, 28 Jun 2007 09:09:11 -0400, hetware <massless@nutrino.none>
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.
To the contrary, the wave number is the metrical analog of frequency,
being the inverse of wavelength or meters/cycle, particularly in this
context where your "nth pulse" must be the nth cycle, not radian.
Please submit your corrections to V1 §29-3:
http://www.feynmanlectures.info/
You mean it hasn't been stressed to you that all variables contained
in a single parenthesis in a physics equation must, without exception,
be dimensionally identical and thus save all the discussion?
Citations? I've never encountered a statement of that requirement. I have
seen statements of the requirement that an entire equation be in
dimensional agreement.
(Ignore
the apostates with their 'natural' numbers.)
You can always test the hypothesis with numbers, as here, e. g., x =
10 meters and t = 10 seconds, and consider the computational
consequences for (t - x).
I've worked it out: (10 seconds - 10 meters).
It is only in mathematics where the letters x and t can stand for
anything you please.
As I stated in the original post, that is EXACTLY what the term following
(f/c) in the expression suggests.
"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."
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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| User: "John C. Polasek" |
|
| Title: Re: Taylor & Wheeler:Spacetime Physics, Ex,L-5 Doppler Shift |
30 Jun 2007 09:40:10 AM |
|
|
On Fri, 29 Jun 2007 19:54:05 -0400, hetware <massless@nutrino.none>
wrote:
John C. Polasek wrote:
On Fri, 29 Jun 2007 01:47:19 -0400, hetware <massless@nutrino.none>
wrote:
John C. Polasek wrote:
On Thu, 28 Jun 2007 10:51:31 -0400, hetware <massless@nutrino.none>
wrote:
John C. Polasek wrote:
On Thu, 28 Jun 2007 09:09:11 -0400, hetware <massless@nutrino.none>
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.
To the contrary, the wave number is the metrical analog of frequency,
being the inverse of wavelength or meters/cycle, particularly in this
context where your "nth pulse" must be the nth cycle, not radian.
Please submit your corrections to V1 §29-3:
http://www.feynmanlectures.info/
You're arguing etymology as to what 'wave number' means. In the
context of your problem, f/c = f cy per sec/c meters per sec = f/c
cycles per meter. The parens must be in meters, as must each of its
components.
You mean it hasn't been stressed to you that all variables contained
in a single parenthesis in a physics equation must, without exception,
be dimensionally identical and thus save all the discussion?
Citations? I've never encountered a statement of that requirement. I have
seen statements of the requirement that an entire equation be in
dimensional agreement.
(Ignore
the apostates with their 'natural' numbers.)
You can always test the hypothesis with numbers, as here, e. g., x =
10 meters and t = 10 seconds, and consider the computational
consequences for (t - x).
I've worked it out: (10 seconds - 10 meters).
It is only in mathematics where the letters x and t can stand for
anything you please.
As I stated in the original post, that is EXACTLY what the term following
(f/c) in the expression suggests.
"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."
You don't need clunky dimensional analysis for this, besides it
doesn't work because in the first term f(1/T) you gave away the
distinction between radians and cycles.
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.
I think my work here is done.
John Polasek
.
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| User: "hetware" |
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| Title: Re: Taylor & Wheeler:Spacetime Physics, Ex,L-5 Doppler Shift |
30 Jun 2007 10:09:58 PM |
|
|
John C. Polasek wrote:
On Fri, 29 Jun 2007 19:54:05 -0400, hetware <massless@nutrino.none>
wrote:
John C. Polasek wrote:
On Fri, 29 Jun 2007 01:47:19 -0400, hetware <massless@nutrino.none>
wrote:
John C. Polasek wrote:
On Thu, 28 Jun 2007 10:51:31 -0400, hetware <massless@nutrino.none>
wrote:
John C. Polasek wrote:
On Thu, 28 Jun 2007 09:09:11 -0400, hetware <massless@nutrino.none>
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.
To the contrary, the wave number is the metrical analog of frequency,
being the inverse of wavelength or meters/cycle, particularly in this
context where your "nth pulse" must be the nth cycle, not radian.
Please submit your corrections to V1 §29-3:
http://www.feynmanlectures.info/
You're arguing etymology as to what 'wave number' means. In the
context of your problem, f/c = f cy per sec/c meters per sec = f/c
cycles per meter. The parens must be in meters, as must each of its
components.
My mistake was that I failed to realize that f was given in "pulses or waves
per second".
As I stated in the original post, that is EXACTLY what the term following
(f/c) in the expression suggests.
"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."
You don't need clunky dimensional analysis for this, besides it
doesn't work because in the first term f(1/T) you gave away the
distinction between radians and cycles.
How do you figure that? I made no mention of radians in that statement. My
point was that if I had been working in conventional units, the equation
would have failed dimensional analysis. I did not have a problem solving
the problem, per se. I simply didn't understand the presence of c because
I thought it was 1 meter/meter.
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.
I think my work here is done.
I just need to get back in the practice of reading word problems carefully.
I find it much easier to manipulate the transformation expressions using
c=1. And the hyperbolic identities are even more helpful than the gama
beta form. The problem could just as well have occurred if I had failed to
realize that one value was given in centimeters (had that been the case)
rather than meters. On the audio of Feynman's lecture on Space-Time
(November 28, 1961), he apologizes perfusely for the proliferation of
various "standard" means of stating quantities in the physical sciences.
He then proceeds to introduced c=1 and advises "If we are ever 'frightened'
that after we have this system with c=1 we shall never be able to get our
equations right again, the answer is quite the opposite. It is much easier
to remember them without the c's in them, and it is always easy to put the
c's back, but looking after the dimensions." For now, I have to side with
that observation. I find it far more difficult to carry the c^2 around in
all my calculations than it is to work with the unitless form. I just need
to be alert to when I need to switch back and forth.
I don't,however, agree with him regarding the value of Minkowski diagrams.
I find them quite valuable when trying to reason about geometric aspects of
problems.
--
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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