Planck invents a better yardstick

[tanpi]

The topic was taken from Chapter 5 in Prof. Susskind's book, The Black Hole War. Prof. Susskind pointed out that once Planck converted the unit such as length, time and mass from the Metric Standard to Planck Standard that allow the Universal constants which are light's speed (c), Gravitational constant (G), and Planck's Constant (h), all equal to one, the resulting Planck's length, time and mass have meanings. They are the size, half-life and mass of the smallest black hole.

From this fact, I felt so ignorant that I just learned this from his book and not any classes during my undergrad years. Please enlighten me with further info on the topic.

 

[Bob S]

You may recall during your undergraduate years that once e (electron charge), h-bar and c are set to one, and for that matter the permeability and permittivity of free space (which then set the impedance of free space to one ohm) that many physicists conveniently forget to lable the units of their equations. It then becomes very difficult for casual readers to interpret them. For example, it's nice to see an occasional u₀ (Henrys per meter) or h/mc (electron Compton wavelength) to help interpret the units of equations. So I for one will vote against any attempt to set everything to one, in spite of the convenience. You may recall esu (electrostatic units) or emu (electromagnetic units)

 

[isabelle]

the resulting Planck's length, time and mass have meanings. They are the size, half-life and mass of the smallest black hole.

There are semi-classical calculations to support this statement, give or take a factor of a factor of 1000, but since there is no complete quantum field theory of gravity and there is no measurements made of micro-blackhole decay, the truth is that no one knows how true those statements are, and even according to current calculations the statement is only true within a factor of 1000.

From this fact, I felt so ignorant that I just learned this from his book and not any classes during my undergrad years. Please enlighten me with further info on the topic.

The reason that you were not shown these calculations in your classes is that they require considerations from general relativity, advanced statistical mechanics, and quantum field theory. Even all of those theories working together only form an incomplete description of black holes, so the calculations are way more speculative then the kinds of things you are taught in class.

So I for one will vote against any attempt to set everything to one

I set Boltzmann's constant k to one always (measure temperature in energy units, entropy is a pure number). About 95% of the time I set hbar and c equal to one: they clutter up equations and aren't mathematically meaningful.

 

[tanpi]

Thanks isabelle and Bob_S. So basically, the resulting length, half-life and mass are like educated guesses of the bound, an induction? 1000x is kinda large error at the micro-black hole scale, right? so basically we have nothing useful from this induction. is my understanding right?

 

[alxm]

Thanks isabelle and Bob_S. So basically, the resulting length, half-life and mass are like educated guesses of the bound, an induction?

I haven't seen the actual calculation, but most likely he simply changed units to get his equations in a simpler form for solving them. That's routine, really, and has no physical significance in itself. Just an application of what you learn in basic calculus: Changing variables, scaling the problem. So you get the same results no matter what unit you actually work with; it's just a lot easier with some units.

In a way, Plank units, Atomic units (the kind I prefer, as a chemical physicist) etc aren't really less anthropocentric than any other unit. Just in a different way: That we prefer not to have our equations cluttered up with constants, and we like to deal with stuff on the 10^0 scale.

 

[Vanadium 50]

The constants are real thing - gravity is a very weak force: sure, it can break your leg, but it takes a whole planet to do it - and these constants have to somehow end up in your calculations. In non-Planck units, we measure quantities using conventional (e.g. SI) units and the constants appear in the equations. In Planck units, we measure quantities using Planck units and the constants don't appear in the equations (or rather, they are set to 1).

That's it. The Planck units mean no more, and no less than this.

One can look at the units and say "aha...something must happen when this scale nears 1", but one could have done the exact same thing with non-Planck units: only there it would be when one particular ration nears 1.

FWIW, I don't think this is a "better' yardstick. It's different, but there is a reason people don't always use it.

 

[Count Iblis]

Dimensional constants like Planck's constant are irrelevant conversion factors. See e.g. http://arxiv.org/abs/hep-th/0208093" on how even some professional physicists fail to understand that.