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Units
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SI Units
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The SI system uses meters-kilograms-seconds. It also defines the Coulomb as
a unit for measuring electric charge, which introduces redundant conversions
between mass-length-time units and the electric charge.
cgs Units
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The cgs system uses centimeters-grams-seconds, and also defines electric charge
in terms of the fundamental quantities of mass, length, and time. The unit of
charge is "esu" or electrostatic unit.
Natural Units
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Natural units are a system of units which replace (or re-scale) the usual mass,
length, and time bases with quantities which have "natural" (physical)
constants associated with them. The two constants usually chosen are the speed
of light (c) and Plank's constant (:m:`$\hbar$`); the gravitational constant
(G) is a possibility for the third constant/unit, but energy (in
electron-volts: eV) is often used instead because it gives more useful
relations and because there is no accepted theory of quantum gravity to unite
these three constants. See _`Plank Units` for more on using G as a unit.
Working with natural units simplifies physical relations and equations because
many conversion factors drop out.
Given the relations between cgs units (gm, cm, sec) and natural units (c,
:m:`$\hbar$` , eV), we can find the natural units of an arbitrary quantity
:m:`$[Q]=[gm]^{a}[cm]^{b}[sec]^{c}=[c]^{\alpha}[\hbar]^{\beta}[eV]^{\gamma}$`:
:m:`$$(\alpha,\beta,\gamma)=\left(\begin{array}{ccc} -2 & 1 & 0\\ 0 & 1 & 1\\ 1 & -1 & -1\end{array}\right)\left(\begin{array}{c} a\\ b\\ c\end{array}\right)=(-2a+b,b+c,a-b-c)$$`
or in reverse:
:m:`$$(a,b,c)=\left(\begin{array}{ccc} 0 & 1 & 1\\ 1 & 2 & 2\\ -1 & -1 & -2\end{array}\right)\left(\begin{array}{c} \alpha\\ \beta\\ \gamma\end{array}\right)=(\beta+\gamma,\alpha+2\beta+\gamma,-\alpha-\beta-2\gamma)$$`
Plank Units
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Plank units (defined by Plank soon after defining his constant :m:`$\hbar$`) are a version of _`Natural Units` using the gravitational constant G as the the
third unit (instead of the common measure of energy). When converted back into
mass-length-time units we get three quantities which define the "Plank Scale",
which may provide estimation of the domain where quantum gravity effects become
important (similar to how the speed of light and Plank's constant provide
estimation of when special relativistic and quantum mechanical effects become
important).