## Thursday, July 05, 2007

### The Planck Scale

The Planck scales - a length and a mass* - indicate the limits in which we expect quantum gravitational effects to become important

Gravity coupled to matter requires a coupling constant G that has units of length over mass. One finds the Planck scale if one lets quantum mechanics come into the game. For this, let us consider a quantum particle of a (so far unknown) mass mp with a Compton wavelength lp, the relation between both given by the Planck constant

This is the quantum input. Now consider that particle to be as localized as it is possible taking into account its quantum properties. That is, the mass mp is localized within a space-time region with extensions given by the particle's own Compton wavelength. The higher the mass of that particle, the smaller the wavelength. However, we know that General Relativity says if we push a fixed amount of mass together in a smaller and smaller region, it will eventually form a black hole. More general, one can ask when the perturbation of the metric that this particle causes will be of order one:

which then can be solved for the mass, and subsequently for the length scale we were looking for. If one puts in some numbers one finds

These Planck scales thus indicate the limit in which the quantum properties of our particle will cause a non-negligible perturbation of the space-time metric, and we really have to worry about how to reconcile the classical with the quantum regime. Compared to energies that can be reached at the collider (the LHC will have a center of mass energy of the order 10 TeV), the Planck mass is huge. This reflects the fact that the gravitational force between elementary particles is very weak compared to the the other forces that we know, and this is what makes it so hard to experimentally observe quantum gravitational effect.

Max Planck introduced these quantities in 1899, the paper (it's in German) is available online

(Credits to Stefan for finding it). You'll find the natural mass scales introduced on page 479ff. He didn't call them 'Planck' scales then, and it is also interesting why he found them useful to introduce, namely because the aliens would also use them

"It is interesting to note that with the help of the [above constants] it is possible to introduce units [...] which [...] remain meaningful for all times and also for extraterrestrial and non-human cultures, and therefore can be understood as 'natural units'."

Coincidentally, yesterday I saw a paper on the arxiv
What is Special About the Planck Mass?
By C. Sivaram
Abstract: Planck introduced his famous units of mass, length and time a hundred years ago. The many interesting facets of the Planck mass and length are explored. The Planck mass ubiquitously occurs in astrophysics, cosmology, quantum gravity, string theory, etc. Current aspects of its implications for unification of fundamental interactions, energy dependence of coupling constants, dark energy, etc. are discussed.

which gives a nice introduction into the appearances of various mass scales in physics, with some historical notes.

* With the speed of light set to be equal 1, in which case a length is the same as a time. It you find that confusing, just define a Planck time by dividing the length through the speed of light.

1. So if we start with Planck units and add strings we get, what, tampon theory? Brilliant!

2. Sure, that's why we need more women in physics.

3. "The higher the mass of that particle, the smaller the wavelength. However, we know that General Relativity says if we push a fixed amount of mass together in a smaller and smaller region, it will eventually form a black hole."

Bee, so what happens when two of these little blackholes meet

4. Bee, so what happens when two of these little blackholes meet

They join and form a larger one, the total horizon area increases. See e.g. here.

5. And what is peventing all mass and visible matter in the universe being clumped into a bright planet or Star - or even One massive blackhole.

Is it the myriad (millions or billions) of smaller blackholes?
Each pulled & pulling in a myriad different (spatial) directions

6. The expansion of the universe. They just don't all meet.

7. "Losing entropy by falling into a black hole is a violation of the second principle of thermodynamics. This law states that entropy is an always increasing function in a closed system - and the universe is a closed system, as nothing can escape it."

Bee, I read Sean trying to make this argument in Cosmic Variance too.

But he was missing the point that though the Universe may be a closed system - the blackhole itself becomes a closed system within that (the) Universe.

Thus on Earth you can have seawater attempting to reach entropy - but that does not mean the landmass has to reach the same entropy - nor does the air or in Earth's case atmosphere.

8. Black hole thermodynamics is based on the existence of Hawking radiation. The black hole looses energy, it's not a closed system.

9. Bee, the blackhole doesn't lose radiation from inside the blackhole

The radiation is caused by that which is falling into the blackhole

Though possibly when the blackhole reaches a certain critical volume it expels it all back out ... lava from a volcano forming new islands on the Ocean that is space

10. Bee, the blackhole doesn't lose radiation from inside the blackhole

The radiation is caused by that which is falling into the blackhole

This is not what the standard argumentation says, according to which the radiation is caused by the presence of the horizon. But either way, radiation towards infinity leads the black hole area to shrink with particles being created at the horizon. It's definitly not a closed system if you consider these quantum effects.

11. Bee, it is not a 'closed' system
if there is matter falling into.

What I was trying to differentiate is that which is inside the blackhole - and that which is outside - though they may try to attempt to reach entropy - may well be the last place where entropy is 'required' - that which is inside will attempt first to reach entropy within itself, and that which is outside has a whole rest of the universe to attempt to reach entropy with first

12. as far as I am concerned, the black hole entropy is ill defined anyhow, so I usually don't engage in these discussions. would you tell me what you think it measures, just to amuse me?

13. Hi Bee, no one has or CAN measure inside the blackhole.

But think of a massive sphere of gold or lead (1km in diameter) and the surrounding environment trying to reach entropy with it at the Perimeter Institute, in the Artic Circle, or in the Gobi desert.

The temperature of the massive solid sphere (and its immediate surround aka boundary or perimeter) will have a different temperature from the environment further and further away. In a closed environment planet or universe - it would possibly be the sphere would be the last place to reach entropy
Notwithstanding gold & lead would easily melt of course (in cosmic terms)
But for the purposes of the analogy use lesser temperatures.

14. Dear Quasar:

Correct, nobody really knows what the entropy is inside the black hole, since this region is disconnected from the outside. There is a lot of discussion about this issue, see e.g. the very nice and readable paper

Black hole entropy: inside or out?

Besides this, I think you are confused about the meaning of thermodynamic equilibrium. What exactly do you mean with a system 'tries to reach entropy with' some other system? Statements about the de- or increase of entropy are always referring to the state after the (total) system has reached thermodynamic equilibrium. A black body is by assumption in thermodynamic equilibrium with its environment.

Best,

B.

15. lol Bee, thermodynamic equilibrium can be local, and black bodies can be in thermodynamic equilibrium.

But black holes are not necessarily in thermodynamic equilibrium with outer space any more than our Sun or the stars are in equilibrium with space or the rest of the universe.

Imagine for a moment millions and billions of singularities in blackholes (as many as you like) scattered along the observable universe - each one containing a potential mini-bigbang.

Like nuclear pellets scattered across apace, They could contain as much hidden energy as is speculated was released in the big bang - waiting to be released into outer space or the observable universe.

See here for a more metaphysical implication

16. I have always felt suspicious about arguments based on dimensional analysis. The fundamental axiom of such analyses is that "there are no large pure numbers in physics, where "large" means "compared to the number of fingers and toes attached to a typical specimen of homo sapiens." Ironic that Planck wanted to be able to communicate with aliens --- better hope they don't have millions of fingers on each hand! :-)

17. "the gravitational force between elementary particles is very weak compared to the the other forces that we know"

ah yes, but the gravitational force is big in Jap.., er, the fifth dimension.
Love. Lisa (kidding)

Great article, Bee. Another post about the Planck time maybe ?

18. "the gravitational force between elementary particles is very weak compared to the the other forces that we know"

ah yes, but the gravitational force is big in Jap.., er, the fifth dimension.
Love. Lisa (kidding)

Great article, Bee. Another post about the Planck time maybe ?

19. ah sorry, hadn't noticed footnote 3. Always read the scientist's footnotes :-)

20. Your formula for the Compton wavelenght is in error, the speed of light is missing in the formula. See:

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

Furthermore, it is not the Compton wavelength which is of interest in the realm of quantum gravity, but the Planck length, which is many orders of magnitude (down to 10^-35) lower than the Compton wavelength. See:

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

21. Hi Armando,

concerning the Compton wavelength,

... the speed of light is missing in the formula.

well, depends on the convention you use for the speed of light - if you use c = 1, the formula is OK. On the other hand, since later on, c is stated explicitly, you have a point.

which is many orders of magnitude (down to 10^-35) lower than the Compton wavelength

Note that as used above (and as you can read in the text), m_P is not the Compton wavelength of the proton, or any other of the known elementary particles, but of a so-far unknown mass, later to be identified with the Planck mass, which brings into play gravity via the Schwarzschild radius.

Cheers, Stefan

22. Armando:

Your formula for the Compton wavelenght is in error, the speed of light is missing in the formula.

c=1, see footnote [3]. It's a consistent and widely used set of units. Sorry if you find it confusing there's a c appearing lateron, that's because the equations are from elsewhere.

Furthermore, it is not the Compton wavelength which is of interest in the realm of quantum gravity, but the Planck length, which is many orders of magnitude (down to 10^-35) lower than the Compton wavelength.

This statement is a) wrong and b) shows that you didn't read what I wrote. Since "the Compton wavelength" is not a fixed quantity is is meaningless to say the Planck length is a fixed ratio smaller than the Compton wavelength. What I have been referring to is specifically the wavelength of a particle with an energy so high its Compton wavelength equals its Schwarzschild radius. I'd recommend you actually read before you comment.