tutorial on natural units
Originally posted by Tsunami
...keep it comin'!
a distinctive thing about quantum gravity is that the Planck scale of area and volume come out as a result---not something put in by hand.
starting from very general principles like background independence and diffeomorphism invariance (Einstein called it "general covariance") rovelli and smolin were able to derive the area and volume spectra and they turned out to be a discret set of multiples of the Planck area and volume units---the natural units (at least up to order-one factors) of area and volume.
how to say this? It had been suspected on general theoretical grounds for more than half a century that a new picture of spacetime would emerge at Planck scale. But when LQG came along it pointed like a compass-needle at that scale, without having been told by its inventors to do that. (smolin, rovelli 1995)
so if someone wants to follow the ongoing development of quantum gravity they would do well to get a comfortable familiarity with the Planck units, hence this informal introduction.
down at the end of this post I will just directly define them in the immediate hard-ass algebraic way, as is usually done, without a serious attempt at motivation. So if you want to avoid the gradual step-by-step introduction, just scroll down and there will be the usual formulas.
-------gradualist treatment of Planck units------
one way to get a handle on them is to take a fresh look at one of the most basic equations in science, the 1915 Einstein equation, the central equation in our model of how gravity works: General Relativity. this equation relates the density of energy in a region ("joules per cubic meter", "footpounds per cubic foot") to the curvature in that region. It says the two are proportional!
Except for a factor of 8\pi which is no big deal, and the fact that they use special symbols for curvature and energy density, this equation is customarily laid out this way:
curvature = \frac{G}{c^4}energy density
Curvature in this context is measured in units of reciprocal area ("per square meter", "per square foot").
Now just suppose that we deviate a tiny bit from the customary layout and write the Einstein equation this way:
\frac{c^4}{G}curvature = energy density
This quantity c^4/G is the Planck unit of force, that is, the force unit which belongs to a system of units that Max Plack discovered in 1899 and presented to his contemporaries as a natural (rather than artificial or man-made) system of units.
So in 1915 Einstein discovered that nature knows about the force
c^4/G. It is the force that connects the amount of curvature in a region to the energy density there.
For dimensional reasons a force is the only type of quantity that can do that. Multiplying a curvature ("per sq. foot") by a force ("pounds") gives a pressure ("pounds per square foot") and that is dimensionally the same type of quantity as an energy density
("footpounds per cubic foot"). Or say the same thing substituting metric Newtons for pounds and meters for feet. Newtons per sq. meter is the same as joules per cubic meter. The longandshort is that if you multiply a curvature by a force you get an energy density and the only thing you CAN multiply a curvature by to get an energy density is a force. Einstein found that the force that works in this context
is the unit force c^4/G in Planck's 1899 system of units
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we can get all the Planck units from this force, and stuff you already know like the speed of light
People COULD have realized that Planck units were basic as early as 1915, but they did not and it was still a bit surprising in 1995 when the area and volume spectra were found to be multiples of Planck area and volume units. Our species is not quick to catch on, sometimes.
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To summarize
around 1900 Planck discovered that nature knows about a certain ratio of energy to frequency called hbar. (when using hbar you need to express frequency in angular format, radians per unit time, but that is a technicality involving a factor of 2pi and I won't belabor it)
in 1905 Einstein reminded everybody that nature knows about c the speed of light, in fact you could almost say nature is
obsessed with the speed of light. that was the year he expounded the universal speed limit and E = mc^2 and a bunch of other things involving c.
in 1915 Einstein showed that nature knows about a certain force
c^4/G which is the force unit belonging to Planck's system of natural units. It turns up as the central constant in General Relativity: the thing that relates the lefthand side to the righthand side in the main GR equation. If you have a book where you can look up the metric values of G and c, or if you just know them, then you can easily calculate what the Planck force unit is---just follow the formula c^4/G. You will get the answer in terms of the metric force unit, the so-called "Newton" which is about a tenth the weight of a kilogram in normal gravity.
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all the other Planck quantities come from these three that were already immanent and obvious in 1900, 1905, 1915.
in any system of units the unit power is always equal to the unit force multiplied by the unit speed (in our case c^4/G multiplied by c)
If you work out c^5/G with a calculator you get Planck power unit is
3.6E52 watts. Lots of watts.
In the Planck system, hbar is the ratio of unit energy to unit frequency. So unit power divided by hbar gives the square of unit frequency, namely c^5/hbarG
unit frequency = \omega = \sqrt{\frac{c^5}{Ghbar}}
this is a frequency expressed in angular format, so the convention is to use the symbol omega for it, instead of the letter f.
Anybody who wants can immediately find out what the Planck unit energy is at this point because it is hbar\omega
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in a consistent system of units the unit length is equal to the unit speed divided by the unit frequency
so in our case
unit length = \frac{c}{\omega} = \sqrt{\frac{Ghbar}{c^3}}
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that's about it for the definitions, now we have unit force, power, frequency, time (the reciprocal of frequency), and length----the rest derive in familiar ways from these. the mass unit, for instance, is equal to the energy unit divided by the square of the unit speed (the square of the speed of light) and so on like that.
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now the question is: what sizes are these units. Of course now you have the formulas for many of them so you could calculate them out in metric terms. But to save you the bother, the best way I know is just look them up at the NIST website.
BTW the NIST "fundamental constants" website has the Planck temperature unit too---which is the Planck energy unit divided by boltzmann's constant.
Beyond just always looking them up, there are some facts about them that are not too hard to remember. Like 2E-30 Planck temp is a reasonably good reference temperature to remember---it's about 10 Celsius or 50 Fahrenheit. And E38 Planck length is a mile.
-----direct no-nonsense definitions----
unit time = t_P = \sqrt{\frac{Ghbar}{c^5}}
unit length = l_P = \sqrt{\frac{Ghbar}{c^3}}
unit energy = E_P = \sqrt{\frac{c^5hbar}{G}}
unit temperature = T_P = \frac{\sqrt{\frac{c^5hbar}{G}}}{k}
-------direct no-nonsense explanation---------
the only way that it is possible to cook up a quantity with the dimension of time using the quantities G, hbar, c is this definition written here and simple constant multiples of it, but why bother to scale the thing by an extra number?
since c is going to be unit speed in the system, it must be unit length divided by unit time
so to get unit length simply multiply t
P by c (unit time by unit speed)
since hbar is the product of energy with time, and since it is going to be a unit quantity in the system, it must be equal to the unit energy multiplied by the unit time
so to get the unit energy simply divide hbar by the unit time
the Boltzmann k is a ratio of energy to temperature and it is a unitary ratio (like c and hbar) in the system
so to get the unit temperature, divide the unit energy by k
I'm trying to remember how to spell the original titles of the two basic references. Planck's 1899 paper is little-known but lays out the system and gives essentially the same values for the basic natural units that we use today
Planck (1899). "Ueber irreversible Strahlungsvorgaenge. Fuenfte Mitteilung." Koeniglich Preussische Akademie der Wissenschaften (Berlin). Sitzungsberichte: 440-480.
Einstein (1916) "Grundlage der allgemeinen Relativitaetstheorie"
