question about planck volume

[gre]

What is the shape of the Planck volume? According to: http://en.wikipedia.org/wiki/Planck_units ... The Planck volume is just Planck_Length^3 . Does this imply the Planck volume is cubical?

[swansont]

AFAIK a shape isn't specified. It's like asking what the shape of a liter is.

[gre]

Tell me if this makes any sense.

 

The Planck angular frequency is:

planck_angular_frequency = 1.85487e43 (1/s)

 

The non-angular Planck frequency would then be:

 

planck_frequency = (1.85487e43 (1/s) / (2*pi)) = 2.952104e42 Hz

 

Which makes sense with (E=h*f): planck_frequency * h = planck_energy

 

 

Wouldn't the Planck wavelength then be:

planck_wavelength = (c / (planck_frequency)) = 1.01552129e-34 m (?)

 

 

Then If you divide this "wavelength" by (2*pi) you get the Planck Length:

planck_wavelength / (2*pi) = Planck_Length

 

Wouldn't this imply that the Planck Volume is spherical, and Planck length is a radius?

[Daecon]

Maybe as a unit of volume, it's amorphous?

[Martin]

Tell me if this makes any sense.

 

The Planck angular frequency is:

planck_angular_frequency = 1.85487e43 (1/s)

 

The non-angular Planck frequency would then be:

 

planck_frequency = (1.85487e43 (1/s) / (2*pi)) = 2.952104e42 Hz

 

Which makes sense with (E=h*f): planck_frequency * h = planck_energy

 

 

Wouldn't the Planck wavelength then be:

planck_wavelength = (c / (planck_frequency)) = 1.01552129e-34 m (?)

 

...

 

You are doing great until you get to wavelengths. Then you let the fact that both h and hbar are used in physics confuse you.

 

And no, the Planck volume doesn't have any special shape. It's like Swansont says----just a (very small) quantity of volume, like a liter. You can picture it as a cube, with planck length edges.

 

I'm aware of what bothers you. The custom is to define Planck units using hbar. hbar is more used in upperdivision coursework. In my experience people prefer hbar. So they base the units on hbar.

 

But when you use hbar it is more natural to measure frequency in radians per unit time rather than cycles per unit time.

 

So if you are studying a photon, you might give its angular frequency

(denoted omega) instead of its cyclic frequency (denoted f).

 

And instead of the old formula E= hf

you might write E = hbar omega.

 

It is the same thing because hbar is reduced by a 2 pi factor and omega is increased over f by a 2 pi factor.

 

And sometimes people define a reduced wavelength and write it lambdabar

(lambda is the usual symbol for the cycle wavelength, and you put a slash thru it.)

 

the reduced wavelength is c/omega.

It is shorter than c/f, the cycle wavelength, by a factor of 2 pi.

 

All this extra notation comes about because some physicists like to use h and some like to use hbar.

======

 

If a photon has energy E = planck energy, then

its angular frequency (radians of phase per unit time) = one per planck time.

 

and its angular wavelength (reduced, ordinary divided by 2 pi.) = one planck length.

 

Maybe you don't like? I'm offering these as a way of getting used to situations where you use hbar a lot. It's really just a matter of getting used.

[gre]

The wavelength I came up might be considered a Planck scale de Broglie wavelength. (h / (planck_mass * c)) = 1.01552129e-34 m

[Martin]

The wavelength I came up might be considered a Planck scale de Broglie wavelength. (h / (planck_mass * c)) = 1.01552129e-34 m

 

I understand. And what you calculated is a cycle length.

 

The distance over which the wavetrain goes thru one full cycle of phase.

 

There is an alternative idea of a radian of phase which is 1/(2 pi) of a full cycle. That gives an equally valid length to associate with the wave.

 

The corresponding (radian-type or angular-type) wavelength is 1/(2 pi)

of the more traditional (cycle-type) wavelength.

 

Wavelength is not God-given, it is a human-made-up concept. You have your choice, associate a length with a full cycle of wave phase, or with a radian.

 

So take your cycle-type wavelength and divide it by 2 pi. You will find it is Planck length. (I'm pretty sure you have been thru this already, so I'm not telling you anything new, but i want to make sure.)

 

The basic lesson is that Planck units are normally defined using hbar, not h.

Therefore when you calculate stuff with Planck units you should try to consistently use hbar, not h.

 

In some cases this takes a little mental effort, to kind of shift gears, if you are very used to working with h in formulas. And it means keeping two alternative ideas in mind, to refer to occasionally.

The angular freq. which is 2pi times the traditional cycle freq.

 

The angular wvlngth which is the traditional cycle wvlngth divided by 2pi.

 

These are natural quantities of freq and wvlngth to use when working with hbar.

[gre]

Thanks Martin. I am definitely having a hard time "shifting mental gears".

 

Here is something else I'm wondering about (might be the same situation), maybe you can help me.

 

The Schumann frequency of the earth is about 7.5 Hz. The wavelength of a 7.5hz electromagnetic wave is about 4e7 m , which is the circumference of the Earth. Why is this? This makes wavelength seem more fundamental.

 

7.5 Hz * 4e7 m = 3e8 (m/s)

[Martin]

The Schumann resonance is a very beautiful phenom, and I like the fact that you are learning stuff like this and sharing it.

 

http://en.wikipedia.org/wiki/Schumann_resonance

 

You are right that the cycle-type wavelength is key to figuring out standing wave modes.

 

In this case it is the earth's circumference (at the ionosphere level).

 

In order to reinforce itself constructively the wave has to stretch all the way around and come in at the same phase.

 

Keep the radian-type wavelength idea alive in the back of your mind, but often the cycle-type version is more natural (here's a case of that!) so use that.

 

The basic Schumann resonance, set off by bolts of lightning, is 7.83 cycles per second.

(Multiply that by 2pi to get the angular version of the same frequency, if you want.)

 

The cycle wvlngth of that frequency is, as you say, the circumference of the planet.

The angular wavelength, if we cared to know it, would be the radius of the planet. (out to the ionosphere)

 

It is useless to argue about which ideas of frequency and wavelength are the right ones. The ones you are using are more natural in many situations, the others more natural in other situations. But arguing is silly----like trying to say which is the "real" Planck constant. Is it hbar or is it h?

 

The cycle wvlngth is also the natural one to use when talking about organ pipes and musical tones.

Edited March 15, 2009 by Martin

[gre]

The Schumann resonance is a very beautiful phenom, and I like the fact that you are learning stuff like this and sharing it.

I agree the Schumann resonance is a very interesting phenomena, and I appreciate your comments.

 

http://en.wikipedia.org/wiki/Schumann_resonance

 

You are right that the cycle-type wavelength is key to figuring out standing wave modes.

 

In this case it is the earth's circumference (at the ionosphere level).

[/Quote]

 

This seems strange to me .. maybe I am still missing something. If the 7.83 Hz Schumann resonance is related to the Earth's circumference at the ionosphere level. Wouldn't the fundamental (mode) frequency be closer to 7 Hz?

 

Since, the Schumann_wavelength (cycle-type) = c / 7.83 = 3.828e7 meters

 

And the earth's radius is 6.372797e6 m and the ionosphere is about 300,000 m..

 

Wouldn't the standing wavelength be: 2*pi*(6.37e6 + 300,000) with a frequency of:

f = (c / (6.37e6 + 300,000)) = 7.15 hz.

[ajb]

Another thought is that the Planck scales are where quantum effects of space-time must be taken into account. It seems generally agreed that these effects will manifest themselves as noncommutativity of space-time. As such, I am not entirely sure what shape means at this level.

 

A lump of "space-time" with Planck volume would presumably be a fuzzy object.

 

A related question comes from quantum mechanics where the phase space is "broken up" into Bohr cells. What is the shape of these cells?

[gre]

What's wrong with my last post?

[Martin]

What's wrong with my last post?

 

Fair question. Someone with a practical experience with waveguides could probably give a responsible explanation.

 

In my case I can either speculate irresponsibly and guess at an explanation, or I can research it, which I don't have time to do right now.

 

But its a good question.

 

Why doesn't the electric noise from a lightning flash act as if it travels exactly along circumference great circle defined by the ionosphere? Why does it behave as if it was taking a shortcut?

 

Notice that the difference in frequency is only 7.85 versus 7.15 Hz by your calculation (which I trust so will not check.) It is not so big. It suggests our general idea is right, our general understanding of the resonance is OK.

 

Maybe the noise does somehow take a shortcut, could it go along a polygonal path? A polygon inscribed in a circle has less circumference than the circle. Or could it be sneaking sideways around the earth?

 

Waves spread out and go different paths and then come back together and reinforce. Maybe we are being too restrictive with the wave if we insist on it traveling a great circle circumference, maybe the wave knows better and gets around faster.

 

So then the resonant freq would be higher, as in fact it is, you point out.

Just quick speculation. You might check with Swansont. If he has time to answer you will get more than just this guess.

Edited March 19, 2009 by Martin
Consecutive posts merged.

[swansont]

What's wrong with my last post?

 

You have to account for the geometry and correct for the non-ideal nature of the system.

 

http://en.wikipedia.org/wiki/Schumann_resonance#Basic_theory

[gre]

You have to account for the geometry and correct for the non-ideal nature of the system.

 

http://en.wikipedia.org/wiki/Schumann_resonance#Basic_theory

Thanks.

What are the variables involved other than the non-spherical shape of the ionosphere.. And how does the frequency end up lower?

[swansont]

Thanks.

What are the variables involved other than the non-spherical shape of the ionosphere.. And how does the frequency end up lower?

 

One effect is that the speed of propagation is lower. I don't know what the effective index is for ~7 Hz waves, though.

[Martin]

... And how does the frequency end up lower?

 

Now I'm confused. I thought that your simple calculation, based on ionosphere circumference and the standard speed c, led you to expect 7.15 Hz.

And I thought you found some source that said the actual frequency was (variable but approximately) 7.85 Hz.

 

So I thought you were asking how does the frequency end up higher.

[gre]

My bad. I meant "frequency end up higher" not lower.

[swansont]

Higher than 7 Hz is because the wrong formula was being used. But if you naively plug into the formula that's on the wikipedia page, you get 10 Hz.

[Martin]

gre,

do you want to move on, with your discussion of Planck units?

We got some help from Swansont and it seems like that resonant frequency issue is under control now. Or?

 

Earlier you were "exploring" in a sense by calculating some Planck quantities and comparing natural quantities.

 

Do have any more of those random calculator explorations to show us or that you want to talk about?

 

If not that's fine---you may have more purposeful things to do. I'll ask you a question which you don't have to answer (just a playing around type question in case you want)

 

What type of physical quantity is this?: c⁴/G

 

I mean, is it a length? A volume? A mass? What sort of quantity is it? Or is it some nameless type of quantity that nobody has a use for?

 

I ask because the main equation of General Relativity uses the reciprocal of that quantity as its key coefficient.

 

The lefthand side has units of curvature (reciprocal area, like 1/square meter) and the righthand side has units of pressure (or equivalently of energy density, a pascal is a unit of either one.)

 

curvature is related to energydensity (or equivalently to stress) and the proportion that mediates between them is

[math]8 \pi G/c^4[/math]

 

So there is the reciprocal of c⁴/G

what type of physical entity is it?

Does it play a role in the Planck system of units?

If so, then is it the Planck unit of what?

 

You may already know this, but I mention it in case you dont because it's kind of nice.

[gre]

gre,

do you want to move on, with your discussion of Planck units?

We got some help from Swansont and it seems like that resonant frequency issue is under control now. Or?

 

I would like to see more on this actually (sorry I'm so thick headed). Mathematically why is what I have incorrect, and can someone show the correct math calculation is applied... Or experimental proof the ionosphere is a key part of the Schumann resonance?

 

 

Earlier you were "exploring" in a sense by calculating some Planck quantities and comparing natural quantities.

 

Do have any more of those random calculator explorations to show us or that you want to talk about?

Not right now, but If I come up with something I'll be sure to post it.

 

If not that's fine---you may have more purposeful things to do. I'll ask you a question which you don't have to answer (just a playing around type question in case you want)

 

What type of physical quantity is this?: c⁴/G

 

 

I mean, is it a length? A volume? A mass? What sort of quantity is it? Or is it some nameless type of quantity that nobody has a use for?

c^4/G Worked out it's a Planck unit of force ... Could it be the "force" holding space together, or something to do with a black hole (?).

 

By itself "c^4" might just be a representation of space-time.

 

(planck_length^4) / (planck_time^4) or (4D planck space) / (4D planck time)

 

 

And the gravitational constant by itself, or: (planck_length / (planck_mass *planck_time^2) .. Could be a few things.

 

 

 

Martin, I'll have to think about the rest of your post.

 

Thanks,

Greg

[Martin]

 

c^4/G Worked out it's a Planck unit of force ...

 

 

You're OK.:cool:

 

Could it be the "force" holding space together, or something to do with a black hole (?).

 

That's not as wacky as it sounds. Probably both those ideas are something physicists have explored at one time or another.

BTW people look at the world differently.

the way i look at the world a planck unit doesn't need to BE anything in particular. There doesn't have to be an applied interpretation or use for every planck quantity.

 

And factors like 2 pi and 8 pi don't matter. I don't feel I have to interpret a planck quantity although it is nice when an interpretation or application shows up.

 

so I'm not opposed to looking for them.

=============================

 

here's a problem for you. Calculate planck density and say approximately how many times denser than water is it. What would you guess? A million times?

A billion times denser? 10¹⁰⁰ times denser?

 

In quantum cosmology, the currently most studied model is LQC (loop quantum cosmology). That approach currently has the most highly cited papers and is getting the most researchers started working in it.

In that approach the way the big bang works is there is a collapse of a universe similar in its basics to ours, leading to a moment of very high density. And quantum corrections to gravity become dominant at that density and cause gravity to reverse briefly and repel rather than attract;

so there is a rebound, that causes the expansion we see and all the stars and galaxies have to condense and form all over again.

Now the question is, what do you suppose that critical density is, that is reached when the so-called Big Bounce happens?

The density at which quantum gravity effects become dominant, in the LQC model, and cause the collapse to rebound.

 

How many kilograms per cubic meter, or what is slightly different how many grams per cc?

Edited March 21, 2009 by Martin