Dimensions of G
Pioneer1
http://www.arxiv.org/PS_cache/arxiv/pdf/0706/0706.3359v1.pdf
Ed Witten writes on page 1 (last paragraph)
since the gravitational constant G has dimensions of length.
Can anyone explain why he is saying that G has dimensions of length?
Here's Wikipedia on dimensions of G:
http://en.wikipedia.org/wiki/Gravitational_constant#The_dimensions_of_G
Thanks.
A N Niel
Pioneer1 <1pio...@gmail.com> wrote:
See that "m" in there (meters)? That's length.
G has other dimensions as well, of course...
Rod
"Pioneer1" <1pio...@gmail.com> wrote in message
news:1183030193.7...@u2g2000hsc.googlegroups.com...
maybe he is using Natural Units, so chossn that c=1 and hbar=1
But then I thought G went as L^2 (in 4d)
hagman
length^3/(mass * time^2) = length * (length/time)^2 / mass
Using natural units for speed (speed of light) and mass
(Planck mass), we are left with a length
Peter Webb
"hagman" <goo...@von-eitzen.de> wrote in message
news:1183043865....@g4g2000hsf.googlegroups.com...
distance = time x speed
Using natural units for speed (speed of light) we are left distance as being
a time.
Pioneer1
>
> length^3/(mass * time^2) = length * (length/time)^2 / mass
> Using natural units for speed (speed of light) and mass
> (Planck mass), we are left with a length
I am confused about this. For instance, speed has dimensions L/T:
speed = L/T
If I choose speed of snail as my unit
speed of snail = 1
this does not make speed dimensionless:
speed = dimensionless in speed of snail units = false
It seems to me that regardless of units used the dimensions of a
quantity should stay the same. To express G in units of speed of light
does not remove its dimensions. What am I missing here?
Thanks
hagman
> On Jun 28, 11:17 am, hagman <goo...@von-eitzen.de> wrote:
>
>
>
> > length^3/(mass * time^2) = length * (length/time)^2 / mass
> > Using natural units for speed (speed of light) and mass
> > (Planck mass), we are left with a length
>
> I am confused about this. For instance, speed has dimensions L/T:
>
> speed = L/T
>
> If I choose speed of snail as my unit
>
> speed of snail = 1
>
> this does not make speed dimensionless:
>
Yes, and by this it identifies length and time.
Note that you set speed of snail =1 and not =1 snailunit,
hence speed is dimensionless
> speed = dimensionless in speed of snail units = false
>
> It seems to me that regardless of units used the dimensions of a
> quantity should stay the same. To express G in units of speed of light
> does not remove its dimensions. What am I missing here?
Measure in c and set c=1.
Agreed, I don't like doing that either ;)
hagman
hagman
Pioneer1
>
> - Show quoted text -
I am sorry I must be totally retarded but I don't understand this.
Can I do this:
speed * L/T
"speed" is just a label. Instead of speed I can write "s" or "z" or
whatever. Without L/T "speed" means nothing.
So if 1 speed of snail = 1L/1T then
speed = speed of snail = 1L/1T
and speed still has the dimenstions of L/T
Otherwise the dimensions of the equation will not make sense.
I would appreciate if someone could point to my misunderstanding and/
or simplify this to most elementary level.
Thanks.
hagman
One might have the idea that height is totally different from planar
distance
and thus by necessity use different units for height measurement (e.g.
use one "foot length" horizontally and one "person height"
vertically).
In that case, volume would be of dimension L^2*H and measured
in "square foot persons", and e.g. density would have
dimension M/L^2/H.
Once you tilt your head and see that height and length are "the same",
you could identify H=L.
Thus volume would be of dimension L^3 (or H^3 if you like).
What unit should one use?
Should you just identify 1 foot length = 1 person height?
There is a better method:
If a metal stick on the floor is 3 foot long, you can place
it upright which make it .5 persons high.
Thus you might decide to eliminate the "person high" unit
by replacing it with 6 ft.
All we needed was a natural H/L relation between a height and a
length,
here obtained by rotating objects from horizontal to vertical.
If you consider the speed of a snail the most natural L/T relation,
you can use this to eliminate T by L or vice versa,
i.e. (if a snail makes .5 m/h) you might say that there are about
48 meters between consecutive sunrises (or vice versa that you are
about 3 hours and 30 minutes tall).
Since L=T, speed is dimensionless, a snail has speed 1, sound has
speed approx. 600,000 etc.
Similarly, acceleration has dimension 1/T (or 1/L) and the typical
g on earth would be approx. 17,600 s^{-1}.
Coincidentally, this is the same as the frequency of a tone
near the end of the human audible spectrum :)
However, it is usual to use the speed of light as the natural speed
rather that the speed of snail.
Pioneer1
> On 30 Jun., 04:12, Pioneer1 <1pione...@gmail.com> wrote:
> If you consider the speed of a snail the most natural L/T relation,
> you can use this to eliminate T by L or vice versa,
I am totally confused. I spent about an hour with this and I still
don't understand it! I realize that there is something
constitutionally wrong with me. But let's go back to speed = L/T. How
do you make the dimensions L and T disappear?
Dimensions of speed are L and T. In fact if you could make L and/or T
disappear you would no longer have speed because L/T is the definition
of speed.
Writing L =1 and/or T=1 does not change dimensions of speed. Speed
still has dimension L and T, only the units are changing.
Can you explain this without introducing any new concepts?? Thanks.