Difference between revisions of "Planck scales and fundamental constants"

Problem 1

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

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

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

Problem 4

Perform the same procedure for just $c,G$. Cosidered 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 there is no newton length scale?

Problem 5

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

Problem 6

Demonstrate, that gravitational radius of a particle with Planck mass coincides with it's Compton wavelength. Recall, that gravitational radius in general relativity is a radius of the spherically symmetric mass, for which the escape velocity at the surface is equal to speed of light.

Problem 7

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

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

Problem 9

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

Problem 10

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

Problem 11

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

Problem 12

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

Problem 13

• Compare the constants of strong, weak, electromagnetic and gravitational interactions.

Problem 14

• Estimate the order of magnitude of thetemperature of Great Unification (the temperature when intensity of gravitation comes up to intensities of three other interactions).