# Planck length

1. Jul 20, 2004

### mee

Is the planck length a commonly accepted theory or is it controversial in any way?

2. Jul 20, 2004

### marcus

It is a length, mee. It is not a theory at all---controversial or otherwise!

Over the past 100 years, various people have expressed various intuitive hunches about the role that planck units can be expected to play in a deeper understanding of nature. Particularly in understanding gravity and spacetime at a microscopic level. It now looks as if the closer people get to a quantum theory of gravity the more likely it seems to them that Planck units will be the appropriate units to use in describing nature at that level.

Planck himself listed the Planck units in 1899
he defined them and calculated values for
Planck length
Planck time
Planck mass
Planck temperature
(he didnt call them "Planck" units, too modest, just "natural" units)

But remember that any choice of units has an arbitrary element of choice! It looks to all of us as if the speed of light c is the best unit for describing speeds in special relativity. It has a lot going for it, almost as if nature wants us to express speeds in those terms (as fractions of the speed of light) because it makes everything so easy in special relativity to do that.

But we could be pigheaded and decide to use some other unit of speed, like 2c.
then all the numbers in all the formulas would have to be half as big because the unit of measure was twice as big.
It might seem ridiculous but units are arbitrary and we could do that.

the same with Planck units, some people have proposed changine them by a factor of 8pi, or the square root of 8pi,
they think that would be more convenient
and other people say change them by 2pi and the square root of 2pi.
But at some basic level this jiggling doesnt matter. The planck units are the planck units, ignoring minor jiggling factors, and they are getting more and more used as the years go by.

You can hear mumbojumbo like "Planck length is the smallest meaningful length." and "Planck time is the briefest measureable interval of time." Just roll down the car window and you will here people clamoring like that.
But people do not have a good enough theory yet of what space and time are made of, or how they are structured and how they behave at the Planck scale, so how can they be sure? All we can say it is the most interesting scale in theoretical physics. All we can say is that new physics can be confidently expected to appear at that scale. And that this scale keeps coming up in the results people are getting (in like Loop Quantum Gravity and Loop Quantum Cosmology)----whatever the new result is about: the area and volume spectrums, the big bang non-singularity, black hole formulas, the Unruh temperature, the Planck units continue cropping up as the obvious units to write the result in.

this is not controversial, it is just what is happening---the units appear appropriate to new physics and are increasingly in use.

the mumbojumbo is sort of in the realm of popular science and is conjectural (how do we really know the smallest length we can ever measure?)----so I would not even call it controversial, it is just some
jazzy attention-grabbing talk.

Let's talk about the planck units! I mean the real stuff----what they are and what they are useful for----not their "philosophical significance".
Would you like an introduction to planck units: the length, the time, the energy, the temperature, the mass, the force, etc.?

3. Jul 20, 2004

### mee

What I was referring to is that in some popular science, the planck length is referred to as the smallest measureable distance: that distance below this amount would have no meaning in the classical sense. I was asking as to its relativity to geometry. If the planck length is the smallest measureable distance, and that distance smaller than this would have no meaning, then any line between two points perhaps has a finite number of points between the end points, measured in planck length units, each "point" becoming a unit one planck length in size.

4. Jul 20, 2004

### marcus

this is praiseworthy speculation. but I dont yet believe the "smallest measureable/meaningful" stuff so i cant help you speculate.

it seems very vague to me.. People dont have a good model of what space is like down there so small. how can we say?

If you want. we can discuss the planck length itself.

do you know the formula defining it? have you ever calculated the planck length for yourself?

have you got a scientific calculator---exponents and stuff.

you probably do.

do you know the values of constants like c and G in standard metric units?

I assume so.

maybe then you have calculated what the planck length is, to see what answer you get?

-----------
here is a preliminary exercise, if the natural units interest you.

find out what c4/G is.
if you keep track of the units it will come out in newtons (it is a force)
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but if you dont want to bother with planck units at that level of detail, its fine

5. Jul 21, 2004

### arivero

mee, besides Marcus' proposals, let me to add that most quantum gravitation theories (including, if you wish, strings) work at the level of surfaces and volumes, so the quantisation of the measurement of a lenght is beyond reach. This is not so strange if you think that curvature, the main quantity of gravitation theory, is defined for surfaces.

6. Jul 21, 2004

### mee

:)

Sorry, my scientific calculator is kaput and I wouldn't know where to start, but if you want to discuss anything, I would be happy to try to understand you. :)

7. Jul 21, 2004

### mee

curvature

Would not a line be curved if it was the shortest distance between two points in a curved space? Why would length have no meaning if a surface must be represented as at least a 2d object?

8. Jul 21, 2004

### marcus

there was a post from Baez on SPR that gave other reasoning, nevertheless in line with what you say here. I forget the exact detail but he gave different indications that area might be somehow more basic than length---the unit of length would then be just the square root of the area unit.

Indeed as it happens the formula for the planck unit of area is slightly simpler than for length, and so the area is "closer" to the fundamental constants c, hbar, and G. Sorry this is so vague but just wanted to say that your intuition is not alone about this.

mee, the formula or definition for the Planck unit of area,
or as Planck would have said, the "natural" unit of area, is

$$\frac{\hbar G}{c^3}$$

it is the only way I know of that one can combine these basic constants
of nature--- speed of light c, Planck constant hbar, and gravity constant G---to give an area,

anywhere in the universe where there is at least light and gravity then these constants are embodied and there is only one area that they tell us---so it is a pretty universal basic area

there does not even have to be a proton or electron around for miles----there will be light and gravity and that is enough---the constants are basic proportions built into nature and they are pervasive

-------------
one can ask if this area is the "smallest meaningful area"
and even tho I cannot answer because we dont yet have a sure quantum theory of gravity
still I can say that in Loop quantum gravity, which is a candidate theory under development, the spectrum of the area operator IS algebraic multiples of this Planck area

these are the possible outcomes of measuring area in THAT candidate theory----various multiples of
$$\frac{\hbar G}{c^3}$$
putting various fractions and stuff out in front of it

but some other approaches to quantum gravity do not have such an area operator with such a discrete spectrum of possible area-values (yet?)

so one has suggestive indications like that but must be careful and reserve judgement

In Gen Rel the unit of curvature is an inverse area. I think this is related to Arivero's point. All kinds of structures can have curvature but when one measures curvature in General
Relativity the basic amount is in terms of an inverse area----this too is suggestive that area could be a very basic geometric quantity at microscopic level.

9. Jul 21, 2004

### mee

marcus

Thanks for taking so much time to explain things worth thinking about. :)

10. Jul 21, 2004

### sol2

It's not that easy to dismiss philosophical discussion in light of planck measures.

There are certain anomalies that need answering, and if you have so much energy, and you used it to define particle reductionism, where does this extra energy go? I think this has to be answered orthe mental frame work about this leakeage through dimensional referencing how would we describe these lengths?

If it has indeed become fuzzy, have we then through the heisenberg uncertainty principal come to recognize another structure and realism from wave considerations, as we had done with photon recognitions.

So you develope a language and you move forward with these philosophical positions from the edge?

We would then have to remove ourselves from the discussions of LQG and strings respectively?

http://www.sciencenews.org/articles/20000219/a2113_3644.JPG

And why I would point back to emergent realities for consideration.

If the energy=graviton, then we should be able to calculate what energy is used to describe a event, and the release of the energy from those events. We would not just rely on the path integrals because these woud have been limited in what they could tell us about the nature of that energy?

We have transferred this thinking to a oscillatory nature(the energy) with a basis borne out of particle identification?

These are questions as well I need to understand better.

11. Jul 21, 2004

### marcus

Indulge me, as the bald guy in "Contact" said.

there must be someone round here with a sci/eng calculator
that's willing to calculate a certain force in metric units
(it will come out in newtons
or equivalently in kg-m-s-2)

or can get it out of any college physics text

So what about the Planck force unit? How big is it?
It is easy to calculate and nice things happen when you use it, as with
the other natural units

12. Jul 21, 2004

### sol2

and in the posture of Auguste Rodin, the Thinker......

http://store4.yimg.com/I/bronze-statue_1796_19035070

was intended to represent Dante himself at the top of the door reflecting on the scene below.

.......I have one hand raised and ask, "what the heck was Dante thinking?"

A lot of energy at this level and the relationship of gravitational consideration quite significant as well?

Cosmologically strings had to go here. The energy they need is out there.

Last edited: Jul 21, 2004
13. Jul 29, 2008

### DonJStevens

Hi marcus,
A simplistic equation can be used to determine a length I have labeled (L1). This L1 length is a circumference that has the value 2pi (Planck length) (3/2)^1/2. When the L1 value is known, the Planck length is easily determined. The L1 length is found from the ratio equation below.

L1/L2 = L4/L1

The L2 value is 1/2 of the electron Compton wavelength so its value is (h/2 mc).
The L4 value is 2pi times the photon sphere radius for the electron mass so its value is (2pi) (3Gm/c^2).

L1 = [(L4) (L2)]^1/2 = [(L4) (h/2 mc)]^1/2 = (3pi hG/c^3)^1/2

L1 (1/2pi) (2/3)^1/2 = Planck length

The Planck length has a relationship to the electron mass and the electron Compton wavelength

Last edited: Jul 29, 2008
14. Aug 3, 2008

### DonJStevens

Hi marcus,

You asked "would someone please calculate c^4/G". When using the G value stored in my TI36X calculator, (6.67259x10^-11) I find:

c^4/G = 1.210565719x10^44 newtons

When analysing electrons, I believe the critical radius is the photon sphere radius.

photon sphere radius = R1 = 3Gm/c^2 meters

(R1) c^2/ 3G = m = 9.1093819x10^-31 kg

(R1) c^4/ 3G = mc^2 = 8.187104158x10^-14 joule = mass energy

joules/1.0622x10^-19 = .5109914x10^6 electron volts = mass energy

If nature is simplistic as many theorists (and I also) expect, then the time dilation factor at the electron photon sphere circumference is equal to the value below. The L1 and L2 values were defined in the earlier post # 13.

time factor = [(3/2)^1/2 (Planck time/2pi seconds)]^1/2 = L1/L2

This time factor is correct only if the gravitatonal constant has the value (very close to) 6.6717456x10^-11.

With this G value, the photon sphere radius labeled R1 will have the length:

3Gm/c^2 = 2.02865519x10^-57 meter = R1

c^4/3G = 4.035729776x10^43 newtons

(R1) c^4/3G = mc^2 = force x distance = mass energy = joules

15. Aug 13, 2008

### DonJStevens

Hi marcus,

I entered a nunber incorrectly in post # 14. The number shown as joules/1.0622x10^-19 should be joules/1.6022x10^-19. I apologise for any confusion this may have caused.

16. Aug 16, 2008

### marcus

Great. Four years later somebody responded and calculated the Planck unit of force.

Thanks DJS.

Just to get a handle on the rough order of magnitude the nature force unit is about 10^40 tons, because 10^4 newtons is about a ton-force.

The way I remember Planck units only requires me to remember two things, that the force unit is c^4/G
and that dimensionally the product of hbar and c is energy x length or
force x area.

In other words hbar x c = Planck unit force x Planck unit area.

So if I divide hbar x c by the force c^4/G, it will get me the unit of area.

(and then taking square root will get me the unit length.)

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

DJS, how do you remember the defintion of the Planck length unit? Everybody probably has their own method that works for them.

17. Aug 16, 2008

### arivero

Hi Don,

Do you have a source for this name, "photon sphere radius", or is it your nomenclature? We worked out this same idea time ago in physicsforums, but there was no a concrete name for this quantity. We spoke then of "steeping a Planck area during a Planck time".

Of course, it does not depend of the particle being an electron or any other.

18. Aug 16, 2008

### marcus

Hi Arivero,
I am guessing what Don means is the standard nomenclature photon sphere radius, of a Schw. black hole with mass M.

In case anyone hasn't met this yet, it is 50 percent bigger than the event horizon radius. At the photon sphere, light can go in a circular orbit around the black hole.

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

I have not been following the thread so unfortunately don't know how this fits in with "stepping planck area in planck time".
But I believe I remember our discussing the latter quantity some time ago, the stepping thing---a kind of planckian-kepler idea that you introduced.

Last edited: Aug 16, 2008
19. Aug 17, 2008

### DonJStevens

Hi marcus,

You asked, How do you remember--Planck length unit?

The equation that I used first (some years ago) is the one I remember, though it is misleading. I used an angular momentum equation.

mc (2Gm/c^2) = h/4pi

m^2 = (c/2G) (h/4pi)

m = 1/2 (hc/2pi G)^1/2 = 1/2 (Planck mass)

With this mass value, the Schwarzschild radius (2Gm/c^2) is equal to the Planck length.

(2G/c^2) (1/2) (hc/2pi G)^1/2 = (hG/2pi c^3)^1/2 = Planck length

I can offer some reasons why this could be misleading if you like. Thank you for noting the SciAm article-June-08. I will look at this soon.

20. Aug 17, 2008

### DonJStevens

Hi arivero & marcus,

The photon sphere radius is the region where limit space curvature is expected for any gravitationally collapsed mass; where a photon can be confined in a closed loop (or double loop).

Alexander Burinskii has written papers that describe the electron as a gravitationally confined ring singularity without an event horizon. See paper titled "Kerr Geometry as Space-Time Structure of the Dirac Electron" (2007).

The photon sphere radius for the electron mass is critical if self-gravitational confinement is required for electron stability.