Calculating the Quantum Kepler Length of a Particle

[arivero]

One knows that in classical gravity a orbiting point sweeps equal areas at equal times. It can be seen that for macroscopic distances the area swept in a plank time is a lot greater than the minimum quantum of area, which is about (plank length)^2.

Now, I ask, given a particle of mass m, for which radius will a circular gravitational orbit around the particle to have the property of sweeping one plank area in exactly one plank time unit?

Below that radius, it should be posible to use "plank time beats" to divide area into regions smaller that previous. So a fundamental break of physics will happen at "quantum kepler length of the particle m" ;-)

Of course you know which other name this length receives, do you?


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[marcus]

Originally posted by arivero
...\ ;-)

Of course you know which other name this length receives, do you?

compton wavelength
nice

 

[marcus]

Originally posted by marcus
compton wavelength
nice

maybe a little discussion would be appropriate

this is a really nice pensee

(physics haiku?)

in natural units the (reduced) compton a particle with mass M is 1/M

the circular orbit velocity around such a particle is
\sqrt{\frac{M}{R}}

so the rate area is swept out is

R\sqrt{\frac{M}{R}}= \sqrt{MR}

setting that equal to one (as Alejandro requires) tells us
that R = 1/M

this is the (reduced) compton

(when dealing with G = hbar = c = 1
units one prefers the reduced wavelength which is the cyclic
one divided by 2pi, as well as the angular format for frequency)

 

[8LPF16]

How does the compton value relate to the Red/Orange line 655nm with no mass?

LPF

 

[marcus]

Alejandro something is catchy or piquant about the implied argument.
I like it.
Alejandro says, in effect, that a circular orbit cannot sweep out
area at a rate that is slower than the Planck rate. He says it must be true for any circular orbit that:

\sqrt{MR}> 1

because say the sweeping of area is slower than unity, say that it takes 3 time units to sweep out 1 area unit
then one can divide the time interval up and this leads to
being able to describe an area smaller than the area quantum.
(one can use Planck "beats" to divide the indivisible)


so he looks for a paradox

he says fix some mass M and make the orbit radius R so small that

\sqrt{MR}< 1

now the rate of sweeping of area is less than unity
what prevents this?

and Alejandro discovers that what prevents this situation is that the orbit radius is less than the Compton for that mass.
(the central gravitating body is being localized more restrictively than quantum mechnanics permits it to be localized---every particle is spread out by at least its compton)

 

[marcus]

Originally posted by arivero
One knows that in classical gravity a orbiting point sweeps equal areas at equal times. It can be seen that for macroscopic distances the area swept in a plank time is a lot greater than the minimum quantum of area, which is about (plank length)^2.

Now, I ask, given a particle of mass m, for which radius will a circular gravitational orbit around the particle to have the property of sweeping one plank area in exactly one plank time unit?

Below that radius, it should be posible to use "plank time beats" to divide area into regions smaller that previous. So a fundamental break of physics will happen at "quantum kepler length of the particle m" ;-)

Of course you know which other name this length receives, do you?

If it is correct, Loop Quantum Gravity implies that the Planck quantities (the "natural units" as Planck called them) are real features of the environment
that they are built into spacetime just as the speed of light (which is the Planck unit of speed) is built in.

the theory pinpoints the place at which new physical effects can be expected, namely at Planck energy (2 billion joules)
and it derives quantized spectrums for area and volume where
the minimum measurable amounts of area and volume are order-one multiples of Planck area and Planck volume.

among the new physical effects expected at Planck scale are
modifications of Lorentz symmetry

It now seems that these and other Planck scale effects may be possible to probe in the next few years leading to tests of LQG.
If it checks out this will among other things infuse a greater sense of solidity into Planck units. There will be reason to think that they really are nature's units.

Alejandro's puzzle (why can't area be swept out at slower than Planck unit rate?) is an example of a "Planck Parable", as I would call it.
Or a "Planck units Haiku".

I will try to think of another such story or puzzle, to respond in kind.

If Loop Gravity develops successfully and tests out we will probably see more Planck haiku growing up as a kind of physics folklore.

 

[marcus]

The Wizard's Stove (a Loop Gravity parable)

to understand this you need to go to the NIST website and look
up Planck temperature
or take my word for it that it is 1.4168e32 kelvin

that is the temp T such that kT is Planck energy
and if Planck energy is built into spacetime as an intrinsic
threshold of some new effects
then the temp is also (in the same way) a built in intrinsic feature of nature

On that scale room temperature is about 2e-30
and a nice oven temperature is about 3e-30
(in fact if you work it out 3e-30 of the natural unit temperature
is 425 kelvin, which is 305 Fahrenheit qualifying as a "slow to moderate" oven.

Well in this short short story there is a Wizard who wants his stove to always be at 3e-30. Hope notation is clear, I mean 3x10^-30

have to go for now, continue later

 

[marcus]

the Wizard's Stove continued

Once there was a Wizard who lived in a cave that he'd hollowed out in the core of a comet and the wizard was always cold
He wanted a potbelly stove that always stayed at a steady temperature of 3E-30 natural. This would take the chill out of the cave and be convenient for baking.

What he got to stoke the stove with is interesting. He got a supply of the usual kind of black holes each of which was always at the steady temperature of
3E-30 natural. (that is 425 kelvin or 152 celsius or fahrenheit 305 depending on which conventional human scale you like)

the question is, what was the mass of each of the black holes.
and how big were they?

 

[arivero]

Yes, it is the typical Haiku one finds in elementary quantum mechanics book. Perhaps sometime we will see it in elementary LQG books :-) By the way, I had never seen it, but I do skip a lot of literature on this.

Originally posted by marcus

(physics haiku?)
(when dealing with G = hbar = c = 1
units one prefers the reduced wavelength which is the cyclic
one divided by 2pi, as well as the angular format for frequency)

It becomes even more rythmical if you allow for G, as then the one coming from plank length simplifies against the one coming from Newton universal gravitation law. At the end, Compton length contains, of course, m, h, and c, but no G around. As for author trickery, I confess that the paradox has been choosen a posteriori :-)

I am -paying time- at a cybercafe now, so I hope that by monday the wizard will have all his charcoal supply issues solved!

 

[marcus]

Originally posted by arivero
... I confess that the paradox has been choosen a posteriori :-)

I am -paying time- at a cybercafe now, so I hope that by monday the wizard will have all his charcoal supply issues solved!

A significant confession!

the wizard reflected to himself that the temperature of the usual kind of black hole (that they sell for stoves) is

\frac{1}{8\pi M}

where M is the mass of the hole

 

[marcus]

at this time a tribe of gypsies was passing through the solar system telling fortunes, selling black holes and love potions, and stealing as gypsies always do.

So the wizard solved the following equation for the mass M, so that he could ask the gypsies for holes of that mass:


\frac{1}{8\pi M} = 3*10^{-30}

 

[marcus]

\frac{1}{8\pi M} = 3*10^{-30}

8\pi M = \frac{1}{3}*10^{30}

M = \frac{1}{24\pi}*10^{30}

 

[marcus]

this means that each piece of "stove charcoal"
would be the size of a mote of dust and
would have a mass about a twenty-thousandth of that of the earth

and if it would not be too tedious I will work that out
from the previously derived value of M

 

[marcus]

the mass calculated was
M = 1.3E28 Planck

and the Earth's mass is
2.75E32 Planck

that's why I said roughly a twenty-thousandth----1/20,000 Earth mass.

------if you like everything metric------
the Planck mass unit is 22 micrograms as can be seen at NIST site,
so that E9 Planck is 22 kilograms
I get that in kilograms M = 29E19 kg = 2.9E20 kg.
and that is, in fact, roughly a 20,000th of the mass of earth
which is like about 6E24 kg.
----------------------------------------

the radius of an ordinary kind of black hole is 2M
so the ones for the Wizard's stove have a radius of
2.6E28
this is the size of a tiny mote of dust or pollengrain
because E28 is about one sixth of a micron
so we are talking about half a micron radius or micron diameter

 

[marcus]

black holes this size are always nice and warm
in fact fahrenheit 305, OK for baking cornbread
and homefry potatoes, macaroni-and-cheese,
such as a wizard might want for supper.

 

[arivero]

Fine parable. I like this kind of effort to grasp the scale of the units and objects we use; in fact some months ago I did the example of putting the elementary particle masses in amu (atomic mass units) and then to look for the nucleus having the same weight!

As you have completely given the answer to your history, let me to complete mine too. We have got Compton Length. Now, can we get quantum mechanics from it, using similar arguments?

Yes, we can: the Bohr-Sommerfeld quantum condition can be formulated, at least for circles, via a a Newton(*)-Kepler principle: any bound particle sweeps a multiple of Compton Area in a unit of Compton Time. This principle does not need gravity; it works for any central force.

The usual way, instead of this, is to use the obtained wavelength to check for destructive interference, most arguments in this line invoque De Broglie instead of Compton(**), so perhaps our whole thread could need a "fine tuning".

[EDITED:] In fact, note that the argument here above does not depend of c. Very much as the argumeng starting the thread does not depend of G.

(*)http://members.tripod.com/~gravitee/booki2.htm
(**) Indeed this was the point raised, subtly, by LPF earlier in the thread.

 

[marcus]

Originally posted by arivero
Fine parable. I like this kind of effort to grasp the scale...

Glad you approve. "kepler length" was witty and subtle (but could be stated in very simple terms) and I wish we could have more idea-puzzles like that.
the "wizard stove" idea actually did not come up to that level but was the best I could think of at the moment
why are subtle presentations of the ideas so difficult to think of?
(because the mind digs itself into ruts it can't get out of)
think of more and maybe we will start a thread.

 

[marcus]

what the Moon thinks she weighs

every lady has some idea of her weight--she may wonder if it is too much or too little, or she may be content with it, but she has some idea

the moon thinks of her weight as a force
measured by scales at her surface*

Now Alejandro you were just talking about circular orbit speed.
The moon knows the circular orbit speed right down at her surface
and she thinks of it in Planck terms (that is as a fraction of the speed of light) because that is natural
she thinks of every physical quantity that way, in those units.

what is the relation between her weight and the fourth power of that speed?

----------------------

* it is hard to imagine the moon standing on bathroom scales placed on her own surface, but she thinks of the force which is her weight as
a million times what one millionth of her would weigh on scales at her surface. or perhaps a billion times a billionth, I'm not sure which.

--------------------
EDIT: hello A. using edit button as suggested to clarify problem.
the key thing is "she thinks of every physical quantity in Planck terms" so the question is, what is the relation between two numbers:
the fourth power of the surface orbit speed and her weight (both expressed in the Planck system of units). no big deal, kind of a silly problem actually, I should solve it probably just as a check that I am stating it right. what I like is the moon having an idea of her own weight.
---------------------
MORE EDIT: well,you didnt take me up on this one, so I will say the answer.
They are the same number.
in Planck units the circ orbit velocity at surface is
\sqrt{\frac{GM}{Rc^2}}
you see I divided the dimensionful speed by c to get the speed as a fraction of c
And so the fourth power of the speed is
(\frac{GM}{Rc^2})^2
this is the same as her weight at her own surface
(\frac{GM^2}{R^2})
divided by the unit of force belonging to Planck units
\frac{c^4}{G}

 

[arivero]

I believe you have not stated the question rightly; even in plank units, the first equation (orbital velocity) gives the velocity as the square root of $M_earth/R_orbit$ while the second one gives the force as the square of $M_moon/R_moon$... Ok Ok I see.. I get you are intending to use the same Mass and same Radius in both equations, but -to me- it is not rightly stated, probably due to stress (a hard monday, as I see from other threads). You are referring to the "orbital" velocity of a stone thrown across the surface of the moon. And, actually, you only ask for G=1, neither c and h are needed.

In any case, it is fully classical gravity so we are hitting the off-topic wall :-(

[edited 10/Feb] Well, I see, you could try to connect it with the other quantization puzzles we have set up, but it seems too far fetched :-(

 

[arivero]

addenda

I have found this link

http://nedwww.ipac.caltech.edu/level5/Glossary/Essay_plancklt.html

which gives a interesting operational definition of Planck time.

There mass and time of Planck are defined as the quantitues simultanesly compatible with uncertainty principle and with gravitational collapse equation. IE a mass of plank spreaded in a plank volume takes a time of plank to collapse gravitationally into a point.

He says it is adapted from P. Coles' 1999 dictionary.

 

[marcus]

Let's try for more Loop Gravity fables!

Alejandro, thanks for the link
if that collapse-time business is from Cole's dictionary then
I prefer your definition of the "kepler length" of something.
It would make a better dictionary

Or a better "modern physics bestiary"

Let's try for more LQG fables.

What I like, one thing I like, about Loop Gravity is that
LQG and the Planck units are
mutually validating

 

[marcus]

Originally posted by marcus

LQG and the Planck units are
mutually validating

what I mean is in LQG one finds that
space (area and volume) is quantized and it turns out
that the quantization is precisely at Planck scale

so that confirms that the Planck units are a good
idea of units built into nature.

On the other hand Planck units lend some
confirmation to Loop gravity because they keep coming up
in the equations in and around the theory.

 

[arivero]

I disagree. As a mathematical aparatus, LQG does not worry about which fundamental length to choose; the theory does not depend of a concrete value. It does not turn up that quantization uses this scale, it is simply that physical intuition has suggested people to use Planck length. Examples as the one I provided in this thread are not usual in LQG treatises. Of course, I could be missing some reference... no time to read everything :-)

Personally, I am against the natural system of units, because people get used to neglect the h and the c from formulae even when they are describing approximations. For instance the fermionic lagrangian has a h multiplying it, thus it is obvious why there is not classical limit for fermions... but if you forget to put it, some students will have one or two bad days until they do the dimensional adjustment.

 

[arivero]

compton length and Zitterbewegung

Perhaps Zitterbewegung is the most general effect that depens on compton wavelength. There was some comment about it in the past, and I even started a thread recently at the Quantum zone of the forum,

https://www.physicsforums.com/showthread.php?s=&threadid=13365

 

[marcus]

A.R. this is slightly off-topic perhaps, having to do with the
article you mentioned with the quote from Lucretius
and the "relational" pivotal idea

Rovelli recaps those ideas in his book pages 150-155
which are section 5.6 "Relational interpretation of quantum theory"
5.6.1 "The observer observed"
5.6.2 "Facts are interactions"
he illustrates this section with a drawing of Archimedes
or Galileo (who can tell the difference) that it looks like he himself
drew it, or someone in his family who is not an artist. The man is fearless

anyway I am glad he did not let drop the ideas of that 1998 article
you cited, but updated and included them in the book

 

[arivero]

Originally posted by arivero
By the way, I had never seen it, but I do skip a lot of literature on this.

Doing a bit of literature search, a first surprising point is that the recent paper of Markopoulou and Smolin does not use this trick but simply prefers to postulate the existence of a stocastic phenomena.
http://xxx.unizar.es/abs/gr-qc/0311059

The "trialogue", at formula 4.2 from Veneziano, uses gravitational force between strings to underline that G and c could be more fundamental than h.
http://xxx.unizar.es/abs/physics/0110060

Also Novikov/Zeldovich are quoted pointing out that plank length and c should be enough in quantum gravity, because G and h appear always (or at least in the Einstein Hilbert action), as a product \sqrt{Gh}.

 

[marcus]

Originally posted by arivero
Doing a bit of literature search, a first surprising point is that the recent paper of Markopoulou and Smolin does not use this trick but simply prefers to postulate the existence of a stocastic phenomena.
http://xxx.unizar.es/abs/gr-qc/0311059
...
because G and h appear always (or at least in the Einstein Hilbert action), as a product \sqrt{Gh}.

Your university at Zaragoza has its own arXiv mirror site!

You refer to
"Quantum Theory from Quantum Gravity"
by Fotini Markopoulou and Lee Smolin

I am not sure what to think about this article. How can the very quantum theory itself arise from spin networks?
is the network or graph of adjacency relations such a mathematically rich structure that it can give rise to quantum theory? I did not know what to make of the abstract:

"We provide a mechanism by which, from a background independent model with no quantum mechanics, quantum theory arises
in the same limit in which spatial properties appear.

Starting with an arbitrary abstract graph as the microscopic model of spacetime..."

any clarification about what they are trying to do?

 

[arivero]

There are about 20 mirrors of arxiv around the world (so, will it survive meteorite clash?). Zaragoza happens to have the spanish one. It is remotelly supported from the original, so no maintenance needed.

It seems that M & S point out that if a stochastic trembling in coordinaates is postulated, then plank constant and schoedinger equation follow. Not surprising to us in this thread. But they do not take risk about the origin of the trembling.

From a fundamental point of view, surely they are just remarking that unification of quantum theory and general relativity is supposed to be a new theory, so both regimes should emerge in the appropiate limits. No news here neither.

 

[Orion1]

Kepler's Length...

Problem:

given a particle of mass m, for which radius will a circular gravitational orbit around the particle to have the property of sweeping one plank area in exactly one plank time unit?




Kepler's Second Law:
\frac{dA}{dt} = \frac{L}{2m} = K_k

Planck Area: (circle)
dA = \pi r_p^2 = \frac{\pi \hbar G}{c^3}

Planck Time: (period)
dt = t_p = 2 \pi \sqrt{ \frac{ \hbar G}{c^5}

Angular Momentum:
L = mrv

v = \sqrt{ \frac{Gm}{r_1}}

L = mr_1 \sqrt{ \frac{Gm}{r_1}}

integral:
\frac{\pi r_p^2}{t_p} = \frac{r_1}{2} \sqrt{ \frac{Gm}{r_1}}

r_1 = \frac{4}{Gm} \left( \frac{ \pi r_p^2}{t_p} \right)^2

Solution #1:
Compton Wavelength:
r_1 = \frac{\hbar}{mc} = \overline{ \lambda}

Planck Mass:
m = m_p = \sqrt{ \frac{\hbar c}{G}}

Solution #2:
Planck Radius:
r_1 = \sqrt{ \frac{ \hbar G}{c^3}} = \overline{ \lambda}

K_k = 2.422*10^{-27} m^2s^{-1}
L = \hbar !

[/color]

 

[marcus]

there are actually two masses in the picture
and arivero referred to the central mass
(in which case the mass of the satellite is assumed
negligible, making the angular momentum of the orbit
hard to determine)
but this analysis is very much in a good spirit and
the right direction!
Orion please try it where you have a pair of particles
of equal mass circling each other
maybe they are each m/2 so the combined mass is m
or they could each be m.
I am just waking up so not sure if this is a sensible reply
but it might be.

 

[arivero]

Originally posted by Orion1
Solution #1:
Compton Wavelength:
r_1 = \frac{\hbar}{mc} = \overline{ \lambda}

Planck Mass:
m = m_p = \sqrt{ \frac{\hbar c}{G}}

Solution #2:
Planck Radius:
r_1 = \sqrt{ \frac{ \hbar G}{c^3}} = \overline{ \lambda}

K_k = 2.422*10^{-27} m^2s^{-1}
L = \hbar !

The "standard", or at least majoritary, definition for Planck length is "the compton length of a mass of plank". I supposse that you are pointing out that this particular case has a total gravitational action of exactly h. Is it?

In any case I think it is more important to stress that the Compton Length answer happens for any mass.

Also, it is usually told that m_p is the case where Swartzchild radius and Compton radius coincide. Not sure about the meaning of this, here.

 

[Orion1]

Binary Systems...



Compton Radius/Wavelength:
r_1 = \frac{\hbar}{mc} = \overline{ \lambda}

The Compton Radius is only a solution for a Keplerian-Planck System, for which m<<M.

As m approaches M in magnitude, the system becomes binaric, and all keplerian laws fail and no longer apply.

Kepler's Laws function only as derivatives to the Binary Theorem, therefore are not actually 'laws'.

Solutions to binary systems requires a modification to Newtons Gravitational Law using a center of mass integration.

dt_1 = dt_2

\frac{dA}{dt} = K_k = 2.422*10^{-27} m^2s^{-1}

Orion1 Binary System Theorem:
\frac{dA}{dt} = \frac{Gm_2^3t_1}{4 \pi (m_1 + m_2)^2 r_1} = K_k

r_1 = \frac{Gm_2^3t_1}{4 \pi K_k (m_1 + m_2)^2}

Limit t₁ = t_p (period)
r_1 = \frac{Gm_2^3}{2 K_k (m_1 + m_2)^2} \sqrt{ \frac{\hbar G}{c^5}}

Limit m₁ = m₂ = m_p
r₁ = 2.538*10^-35 m
r_p = 1.616*10^-35 m



Also, it is usually told that m_p is the case where Swartzchild radius and Compton radius coincide.
[/color]

Compton Radius:
r_c = \frac{\hbar}{mc} = \overline{ \lambda}

Schwarzschild Radius:
r_s = \frac{2Gm}{c^2}

r_c = r_s

\frac{\hbar}{mc} = \frac{2Gm}{c^2}

Compton-Schwarzschild Mass:
m_{cs} = \sqrt{ \frac{ \hbar c}{2G}

Close, but negative.
[/color]

 

[marcus]

Hi Orion
you said:"Also, it is usually told that [planck mass] is the case where Swartzchild radius and Compton radius coincide."

I have never heard that, what I have read has been that Planck mass is the case where half the Schwarzschild radius and the Compton wavelength coincide.

if you have a link to something online with the alternative definition you could post it, then anyone who is curious about the different defintion could check it out

but it is just a factor of two that we are worrying about

half the Schw. is GM/c^2

and that has to equal the Compton, which in reduced or angular format is hbar c/Mc^2

So one sets the two equal and solves

GM/c^2 = hbar c/Mc^2

I will try to put this in LaTex, to match your clear writing.

\frac{GM}{c^2} = \frac{\hbar c}{Mc^2}

M^2 = \frac{\hbar c}{G}

 

[arivero]

factor2

Sorry marcus it was me who raised the sloopy definition; there is a factor 2 (even 4) around all the thread, but I am not worried at this level; we are playing sort of dimensional analysis.

Orion, note that the initial setup does not ask for a whole revolution. Area law forks for a sector of the circle too. Answers are the same, although to look for a complete revolution is an interesting particular case, hmm.

 

[Orion1]

Orion1 Binary Theorem...


the initial setup does not ask for a whole revolution. Area law forks for a sector of the circle too.
[/color]


dt_1 = dt_2

\frac{dA}{dt} = K_k = 2.422*10^{-27} m^2s^{-1}

Orion1 Binary System Theorem:
r_1 = \frac{4(m_1 + m_2)^2}{Gm_2^3} \left( \frac{dA}{dt} \right)^2

r_1 = \frac{}{Gm_2^3} \left( 2(m_1 + m_2) \left( \frac{dA}{dt} \right) \right)^2

r_1 = \frac{ (2K_k(m_1 + m_2))^2}{Gm_2^3}

Limit m₁ = m₂ = m_p
r₁ = 6.464*10^-35 m
r_p = 1.616*10^-35 m
r₁ = 4r_p


Below that radius, it should be posible to use "plank time beats" to divide area into regions smaller that previous. So a fundamental break of physics will happen at "quantum kepler length of the particle m"
[/color]

\frac{dA}{dt} = \frac{}{2} \sqrt{ \frac{\hbar G}{c}} = K_k = 2.422*10^{-27} m^2s^{-1}

Planck Mass Fusion:
Limit m₁ = m₂ = m_p
Limit r₁ = r_p
\frac{dA}{dt} = \frac{ \sqrt{Gm_2^3r_p}}{2(m_1 + m_2)} = \frac{}{4} \sqrt{ \frac{\hbar G}{c}} = K_1 = 1.211*10^{-27} m^2s^{-1}

Planck-Compton Radius: r₁ = r_p
r_p = \frac{\hbar}{m_pc} = \overline{ \lambda_p}

Gravitation 'quantum shutdown' below the Planck Radius: r₁ <= r_p

r₁ <= r_p
\frac{ \sqrt{m_2^3r_1}}{(m_1 + m_2)} = \frac{}{2} \sqrt{ \frac{\hbar }{c}}

r₁ <= r_p
\frac{m_2^3}{(m_1 + m_2)^2} = \frac{ \hbar}{4cr_1}

r₁ <= r_p
r_1 = \frac{ \hbar (m_1 + m_2)^2}{4cm_2^3}

[/color]

 

[arivero]

setting the record

Just to set the record straight, it is true that we must always keep the scent of two masses m_1,m_2. In fact in the first calculations from Orion the m in the angular moment is not the same m that under the square roots, but that is only a abuse of notation and the result is the right one.

Still, one does not need to go deep to Plank mass and radius too soon. For the area swept by the particle 1 around the center of mass, it suffices to substitute
m_2 \to {m_2^3 \over (m_1+m_2)^2}
and reciprocally for the particle 2. The analysis in terms of compton length is still possible.

I am unsure of which area to quantize in this case, if one or another or the sum, nor to speak of which reference frame to use. So lacking of a fundamental theory, it seems we aren't going to get some additional benefit from this additional freedom.

Perhaps a minor difference with the Keplerian case is that we have two masses to add for the total angular momentum. So the additional condition L=\hbar does not imply directly m=m_P now.

 

[Orion1]

Physical Break...



\frac{dA}{dt} = \frac{}{2} \sqrt{ \frac{\hbar G}{c}} = K_k = 2.422*10^{-27} m^2s^{-1}

Kepler's Second Law:
\frac{dA}{dt} = \frac{L}{2m} = K_k

m_2 \to {4m_2^3 \over (m_1+m_2)^2}

\frac{dA}{dt} = \frac{L(m_1 + m_2)^2}{8 m_2^3} = \frac{}{2} \sqrt{ \frac{\hbar G}{c}} = K_k

L = \frac{4m_2^3}{(m_1 + m_2)^2} \sqrt{ \frac{ \hbar G}{c}}

L = \hbar

\frac{4m_2^3}{(m_1 + m_2)^2} = \sqrt{ \frac{ \hbar c}{G}} = m_p

Kepler's Second Law fails at:
\frac{4m_2^3}{(m_1 + m_2)^2} &lt; m_p

L &lt; \hbar - impossible!

 

[marcus]

Originally posted by arivero
I have found this link

http://nedwww.ipac.caltech.edu/level5/Glossary/Essay_plancklt.html

which gives a interesting operational definition of Planck time.

There mass and time of Planck are defined as the quantitues simultanesly compatible with uncertainty principle and with gravitational collapse equation. IE a mass of plank spreaded in a plank volume takes a time of plank to collapse gravitationally into a point.

He says it is adapted from P. Coles' 1999 dictionary.

this is Barry Madore's "Level 5 Knowledgebase" for cosmology, a useful site, here are some more links to it
http://nedwww.ipac.caltech.edu/level5/cos_par.html
http://nedwww.ipac.caltech.edu/level5/toc.html
http://nedwww.ipac.caltech.edu/level5/Glossary/frames.html

 

[marcus]

Originally posted by arivero
I have found this link

http://nedwww.ipac.caltech.edu/level5/Glossary/Essay_plancklt.html

which gives a interesting operational definition of Planck time.

...

"operational definition of Planck" quantities is what this thread is really about

you have given one or more and pointed to several
I recall that Baez gives a meaningful way to look at Planck length
in a short essay at his site
(it discusses what it means for a BH Schw. radius to be comparable to its Compton wavelength, if I remember right)

people sometimes ask "what IS Planck length?" wishing for some natural object to look at or some story to be told which has in it this length, or this mass.

(these quantities are also a system of units that makes as many as possible of the most basic constants equal to one and so it is the system of units that makes nature's favorite equations appear as simple as possible---a system of units that nature likes or that is intrinsic to nature---so one can say that, and forget about answering the question. why should telling a story be necessary when they just simply are the units inherent in the universe. and yet people still ask this naive and natural question...)

"what IS Planck length, or mass, actually?"


you have offered a novel answer involving the rate that a keplerian orbit sweeps out area

maybe in this thread we can come up with some more operational definitions or
"conceptual" or "visual" definitions of Planck units

 

[marcus]

Alejandro, it occurs to me that if one is to give a
conceptual idea of "what really are the Planck unities?" then
it should work in every dimension

asking that it work in other space dimensions besides 3 is a way of discovering what is a satisfactory approach

one might say that just as in spacedimension 3 one has
GM^2 = unit force x unit area = unit force x area = hbar c

by analogy in spacedimension 2 one should have
GM^2 = unit force x unit distance = unit energy = M c^2

(it is the analogy to the inverse-square law)

and so one gets (2D space) analogs of the Planck units

GM^2 = M c^2 which solves to

M = \frac{c^2}{G}

unit energy is
E = \frac{c^4}{G}

unit freq is
\omega = \frac{c^4}{G\hbar}

That makes unit time
T = \frac{G\hbar}{c^4}

and unit distance
L = \frac{G\hbar}{c^3}

--------------
do you happen to know if these are considered the "right" Planck units to use in 2D space?
(all my friends live in 3D space and I could not find anybody to ask or any websites by people living in 2D space)

 

[arivero]

Originally posted by marcus do you happen to know if these are considered the "right" Planck units to use in 2D space?
(all my friends live in 3D space and I could not find anybody to ask or any websites by people living in 2D space)

Well, I know people living a very flat life, but then they are not really interested on Nature...

You are right that the procedure is to look for a law equivalent to the inverse square distande of 3D, then extracting from it the dimension of G... and all the other coupling constants, consider flat "electromagnetism" and all that!

A more straighforward method is to look directly in the Lagrangian density, or in the action, instead of in the force law. This is the usual method in quantum field theory, when they need to know also the dimension of each field.

I'd add that sometime such analysis inspires ways to justify the dimensionality of space-time; but perhaps the rigorous justification will really need of a Long March through Galois Theory.

 

[marcus]

Originally posted by arivero
a Long March through Galois Theory.

maybe you just received a postcard from the future

"the mountains of Algebraic Geometry are very beautiful at
this season
while there is still snow on the heights of Mount Galois
the three dimensions of space are beginning to
appear on the lower slopes
wish you were here"

 

[arivero]

Old Grothendieck (born 28/May/1928) still keeps hidden, I supposse, at some mountain range, albeit I heard he had crossed the French border a few years ago. A pity he was never interested on physics.

 

[marcus]

Originally posted by arivero
Old Grothendieck (born 28/May/1928) still keeps hidden, I supposse, at some mountain range, albeit I heard he had crossed the French border a few years ago. A pity he was never interested on physics.

Grothendieck is the bald bartender in the last scene in the movie
"Zazie dans le Metro" which I believe is very good.
it is appropriate for great mathematicians to be secretive

 

[arivero]

Anyway, let's go back to the point you have raised, marcus. Yes, it is very interesting that if we model gravity with a force G {m m \over r^q}, then only for q=2 the coupling constant will cancel when imposing our "Kepler rule".

In general, asking A(t_P) to be a multiple n of A_P we have
2n=G^{\frac12-\frac1q} m^\frac12 r^{\frac32-\frac{q}2} <br /> c^{\frac3q-1} \hbar^{-\frac1q}

For q=3, R and c dissappear from the equation. For q=0, R and m could be canceled out. But only for q=2 we get to cancel Newton constant.

Thus, if you link q to the dimensionality of space, you can say that you have a hint of why should ST have a dimensionality 3+1, or why it should to compactify to 3+1 dimensions.

 

[arivero]

related papers

Past yesterday I found a old 1949 paper from an outcast, MFM Osborne:
Phys. Rev. 75, 1579-1584 (1949) Quantum-Theory Restrictions on the General Theory of Relativity
http://link.aps.org/abstract/PR/v75/p1579 . It has passed mainly unnoticed in the literature, except for a footnote in Misner 1957 review.

Along this thread we (or at least, myself) have followed the "emerging way", expecting to get quantum mechanics out of quantum geometry. This is also the way of Asthekar (gr-qc/0207106, gr-qc/0211012) and Smolin(gr-qc/0311059, hep-th/0201031). On the contrary, Osborne takes an operative way of quantum measurement indeterminacy, he applyes it to general relativity and he concludes the existence of a Planck length and a Planck mass. And yes, also the Compton radius effect as in the beginning of the thread.

To Osborne, a Planck mass is the minimum mass whose curvature you can measure, given the traditional quantum uncertainty effects.

 

[arivero]

I am a bit slow to follow Orion1' calculations, so I have redone them, basically confirming the results. Now, it is interesting to look also to the intermediate steps, so let me play it again.

We have two bodies 1,2 circling around the center of mass, thus with a common angular velocity \omega such that \omega^2 R_i=G m_j/R^2. Here R is the sum of both radius. The equation is consistent with the center of mass condition R_1m_1=R_2m_2.

The sum of cases 1 and 2 let us to solve for \omega,
\omega=\sqrt{G{M\over R^3}}

Now we impose that the area A_i(t_P)must be a multiple n_i of Plank Area. This translates to
n_i=\frac12 \sqrt{cM\over\hbar} {R_i^2\over R^{3/2}}

Or, using the C.M. condition to substitute R,
R_i={4\hbar\over c} {M^2\over m_j^3} n_i^2

Note now that using again this condition over the already solved radiouses, we get a condition on the multiples of area, namely (m_1/m_2)^2=n_2/n_1. Or, say, m_1^2n_1=m_2^2n_2

Now let's go for the total angular momentum L=m_1\omega R_1^2+m_2\omega R_2^2. Substituting and after a little algebra we get
L={2 \hbar \over m_P} {M^{3/2}\over (m_1m_2)^{3/2} }<br /> {m_1^7 n_1^4+m_2^7n_2^4 \over(m_1^3n_1^2+m_2^3n_2^2)^{3/2}}

Which, using the relationship between n and m, simplifyes to
L={2\hbar\over m_P}{m_1+m_2\over m_1m_2} m_1^2 n_1

Orion' case L=\hbar, m_1=m_2\equiv m gives us, accordingly, m=m_P/4n

Also, if we take m1 a lot greater than m2, we recover the initial Compton formula for R2 and also we get a total angular momentum
L_{m_1&gt;&gt;m_2}\approx 2n_1\hbar {m_1^2\over m_P m_2}= 2n_2\hbar {m_2\over m_P}
which shows that plank mass keeps its role as a bound.

Last, a interesting mistake happens if we try to impose simultaneusly low quantum numbers (n1 and n2 small) and big mass differences (m1 a lot greater than m2). Then we are driven to write
L^{WRONG}_{m_1&gt;&gt;m_2}\approx 2 n_1 \hbar { m_1\over m_P} ({m_1\over m_2})^{\frac32}
that is not completely out of physical ranges, if for instance we put m1= 175 GeV then m2 is around some tenths or hundreths of eV, the maximum and minimum values of mass known in the standard model after the neutrino experiences.

The first post in the thread shows that Planck area and time can be used to refer to "low energy" scales. This calculation, althought wrong, shows that Planck mass could also be employed to refer to values in usual energy ranges.

(Personally I feel strange, because on my usual rules of thumbing I did not believe on space quantisation nor in neutrino masses, but the mathematics has invited us into these matters)

 

[marcus]

Originally posted by marcus
"operational definition of Planck" quantities is what this thread is really about

you have given one or more and pointed to several
I recall that Baez gives a meaningful way to look at Planck length
in a short essay...

... why should telling a story be necessary when they just simply are the units inherent in the universe. and yet people still ask this naive and natural question:

"what IS Planck length, or mass, actually?"


you have offered a novel answer involving the rate that a keplerian orbit sweeps out area

maybe in this thread we can come up with some more operational definitions...

Alejandro has found online the historically first description of Planck units in a reproduction of a Spring 1899 document of the
"Royal Prussian Academy of Sciences"

http://www.bbaw.de/bibliothek/digital/struktur/10-sitz/1899-1/jpg-0600/00000494.htm

I have seen this in hardcopy some years ago but did not know that it is online. Thanks Alejandro. It is wonderful to have this online!

There is something here I did not realize. The 1899 units differ by a factor of something like
\sqrt{2\pi}
(about 2.5) from the units we use now

the original 1899 Planck length, I see in this "fax"
is 4.13 x 10^-35 meters
so one must divide by about 2.5
to get the
1.6 x 10^-35 meters
we use nowadays

anyway that is what it looks like so I will check the others

 

[marcus]

so I see I have been mistaken about a factor of
sqrt 2 pi.
for some years I have thought that the Planck units
we use now are the same as those presented to the
Prussian Academy in 1899.
but this is not right.

the time unit in the original paper is also (just as
you would expect) too big by about a factor of 2.5

Planck gave its value as 1.38 x 10^-43 second
and we now say 0.539 x x 10^-43 second

his "Einheit der Temperatur" is bigger (by about factor 2.5)
than what we call Planck temperature

his "Einheit der Masse" is correspondingly bigger than
what we call Planck mass.

 

[arivero]

Originally posted by marcus
so I see I have been mistaken about a factor of
sqrt 2 pi.

Marcus, do not put attention to it.

The issue of h versus \hbar plagues all the scientific literature, and for sure this thread. Before Planck, it was already infamous with Heaviside units. It ultimatelly comes from Fourier transform, which you can normalise in some different ways, including or not a \sqrt{2\pi} factor.