Chapter 1

Universe consists of particles with masses $(m \ge 0)$ moving in space-time $(x, y, z, ct)$ according to special relativity (SR) and quantum mechanics (QM) that interact via fundamental forces (gravity, E&M, weak, and strong). The units we use (SI or mks, cgs) to make measurements are not fundamental, but conventional. Normalizing units by fundamental constants, on the other hand, provide more "natural" units, or at least gives numbers independent of human scales.

Fundamental constants:

gravity: $G = 1.67 \times 10^{-11}~{\rm m}^3~{\rm kg}^{-1}~{\rm s}^{-2}$

SR (speed of light): $c = 3.00 \times 10^8~{\rm m~s}^{-1}$

QM (Planck's constant): $\hbar = h/2\pi = 1.05 \times 10^{-34}~{\rm J~s} = 6.58 \times 10^{-16}~{\rm eV~s}$
- electron volts (eV) are just another unit that can be used for energies, which happen to be convenient for particle masses: $1~{\rm eV} = 1.602 \times 10^{-18}~{\rm J}$

Constants can be combined to form fundamental units (called Planck units) for length $\ell$ , time $t$ , and mass $m$ . Other units (like energy) are combinations of these quantities. To isolate each of these from the constants:

G \rightarrow \frac{\ell^3}{m t^2}\, ,~~~~~~c \rightarrow \frac{\ell}{t}\, ,~~~~~~\hbar \rightarrow \frac{m \ell^2}{t}

it makes sense to eliminate each dimension one by one. For example, $G \propto m^{-1}$ and $\hbar \propto m$ , so multiplying them together will eliminate mass:

G \cdot \hbar \rightarrow \frac{\ell^5}{t^3}

Since $c$ also contains $\ell$ and $t$ , one of those can also be removed:

\frac{G \hbar}{c^3} \rightarrow \frac{\ell^5}{t^3} \frac{t^3}{\ell^3} \rightarrow \ell^2

The Planck length, $\ell_p$ , is defined as just the square root of this combination of constants. Similarly, the Planck mass $M_p$ or energy $E_p$ and Planck time $t_p$

\begin{array}{lclcl} \ell_p &=& \left(\frac{G \hbar}{c^3} \right)^{1/2} &=& 1.62 \times 10^{-35}~{\rm m} \\ M_p &=& \left(\frac{\hbar c}{G} \right)^{1/2} &=& 2.18 \times 10^{-8}~{\rm kg} \\ E_p &=& M_p c^2 &=& 1.22 \times 10^{28}~{\rm eV} \\ t_p &=& \left(\frac{G \hbar}{c^5} \right)^{1/2} &=& 5.39 \times 10^{-44}~{\rm s} \end{array}

Another way to parameterize the amount of energy in a system is through the thermodynamic quantity we call temperature, where $E = kT$ and $k$ is Boltzmann's constant ( $k = 8.62 \times 10^{-5}~{\rm eV~K}^{-1}$ ). So,

T_p = E_p / k = 1.42 \times 10^{32}~{\rm K}

If we use these units for variables in equations, the constants go away, or in other words,

c = k = \hbar = G = 1