# Planck scales and fundamental constants

### Problem 1: frequency or energy?

Consider some physical quantity $A$. The multiplication of $A$ by any power of arbitrary fundamental constant, certainly changes it's dimensionality, but not the physical meaning. For example, the quantity $e\equiv E/c^2$ is energy, despite it has the dimensionality of mass. Why, then, we call the quantity $E/\hbar$ frequency, but not energy, despite that Planck constant $\hbar$, like speed of light $c$, is fundamental constant?

### Problem 2: energy, momentum and mass

In special relativity mass is determined by the relation $m^{2}=e^{2}-p^{2},\qquad e=E/c^{2}.$ This expression presents the simpliest possible relation between energy, momentum and mass. Why the relation between these quantities could not be linear?

### Problem 3: the Planck units

Construct the quantities with dimensionalities of length, time, mass, temperature, density from fundamental constants $c, G, \hbar$ and calculate their values (the corresponding quantities are called Planck units).

### Problem 4: Newton units

Perform the same procedure for just $c,G$. The considered quantities are called Newton units. Construct, in particular, the Newton force unit and Newton power unit. What is the physical meaning of these quantities? Why is there no Newton length scale?

### Problem 5: Planck time

$^*$ Compare the reception delay of an object, located at $1~\mbox{m}$ from flat mirror, with the Planck time. How much longer this time is?

### Problem 6: gravitational radius for Planck mass

Demonstrate, that gravitational radius of a particle with Planck mass coincides with it's Compton wavelength. The gravitational radius of General Relativity can be calculated in Newtonian mechanics as the radius of a spherically symmetric mass, for which the escape velocity at the surface is equal to the speed of light.

### Problem 7: 1GeV

Demonstrate, that in the units $c=\hbar=1$ $1\,GeV\approx 1.8\cdot 10^{-24}\, g;\quad 1\, GeV^{-1}\approx 0.7\cdot 10^{-24}\,c \approx 2\cdot 10^{-14}\,cm.$

### Problem 8: age of the Universe in Planck units

In units $\hbar =c=1$ estimate the energy scale, which corresponds to the current age of the Universe.

### Problem 9: Planck mass in different units

Express Planck mass in terms of $K$, $cm^{-1}$, $s^{-1}$.

### Problem 10: gravitational constant

Express Newton's constant $G$ in units $c=1$.

### Problem 11: fine structure constant

Show that the fine structure constant $\alpha=e^2/\hbar c$ is dimensionless only in the space of dimension $D=3$.

### Problem 12: dimensionless combinations

Construct a dimensionless combination from the constants $c$, $\hbar$, $e$, and $G$ in the space of arbitrary dimension.

### Problem 13: strong, weak, EM and gravity

$^*$ Compare the constants of strong, weak, electromagnetic and gravitational interactions.

### Problem 14: the Great Unification

$^*$ Estimate the order of magnitude of the temperature of Great Unification: the temperature when intensity of gravitation comes up to intensities of the three other interactions.

### Problem 15

problem id: TF_1

Construct planck units in a space of arbitrary dimension.