Planck Units

Planck Units

The Planck scale marks where quantum mechanics and general relativity must both be important simultaneously. Built from the fundamental constants $G$, $\hbar$, $c$, and $k_B$, Planck units set the natural scale of quantum gravity.

The Planck Scales

$l_P = \sqrt{\frac{\hbar G}{c^3}} \approx 1.616 \times 10^{-35} \text{ m}$

$t_P = \sqrt{\frac{\hbar G}{c^5}} \approx 5.391 \times 10^{-44} \text{ s}$

$m_P = \sqrt{\frac{\hbar c}{G}} \approx 2.177 \times 10^{-8} \text{ kg}$

$E_P = m_P c^2 = \sqrt{\frac{\hbar c^5}{G}} \approx 1.22 \times 10^{19} \text{ GeV}$

$T_P = \frac{E_P}{k_B} \approx 1.417 \times 10^{32} \text{ K}$

Cosmological Significance

Inflation is thought to have occurred at energies near the Planck scale. The Big Bang singularity is a Planck-scale phenomenon where our classical description of spacetime breaks down.

The Planck time is the earliest moment after the Big Bang that our current physics can describe. Before $t < t_P \approx 5.4 \times 10^{-44}$ s, a theory of quantum gravity is required.

Constants

Use:

• $\hbar = 1.055 \times 10^{-34}$ J·s
• $G = 6.674 \times 10^{-11}$ m³/(kg·s²)
• $c = 2.998 \times 10^8$ m/s
• $1$ GeV $= 1.602 \times 10^{-10}$ J

Your Task

Implement three functions. All constants must be defined inside each function.

• planck_length_m() — returns $l_P = \sqrt{\hbar G / c^3}$ in metres
• planck_time_s() — returns $t_P = \sqrt{\hbar G / c^5}$ in seconds
• planck_energy_GeV() — returns $E_P = \sqrt{\hbar c^5 / G}$ in GeV (divide by $1.602 \times 10^{-10}$ J/GeV)