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?

solution

Let's analyze the methods for measuring of considered quantities to answer this quenstion. $E$ and $E/c^2$ are measured by the same procedure, for example, using a calorimeter, but frequency is measured in a fundamentally different way, for example, using a clock. Consequently, the expression $\omega = E/\hbar$ contains the relation between two different types of measurements, while expression $e = E/c^2$ does not. This metrological difference has mainly historical origin. Recall, that modern chronometers are based on the measurement of difference between energies of atomic levels.

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?

solution

This is impossible, since momentum $\vec p$ is a vector, while $e$ and $m$ are scalars in 3--dimensional space.

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).

solution

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?

solution

$l_{_{Pl}} = \sqrt {{G\hbar} \over {c^3}}=1.6 \times 10^{-35}\mbox{m} $, $t_{_{Pl}} = \sqrt{{G\hbar } \over {c^5 }}=5.38 \times 10^{-44}s$, $M_{_{Pl}} = \sqrt {{\hbar c} \over G}=2.18 \times 10^{-8} \mbox{kg}$, $T_{_{Pl}} = \sqrt {{\hbar c^5} \over {Gk_B^2}} = 1.4\cdot 10^{32}K$, $\rho _{_{Pl}} = {{c^5 } \over {G^2 \hbar }}=5.17 \times 10^{96} \mbox{kg/m}^3$

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?

solution

$10^{35}$

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.

solution

$r_g={{2GM_{_{Pl}}} \over {c^2}} = {\hbar \over M_{_{Pl}}c} = l_{_{Pl}}$

Problem 7

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

solution

Problem 8

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

solution

Problem 9

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

solution

$M_{Pl} = 1.42 \cdot 10^{32} \,K = 6.2 \cdot 10^{32}\mbox{\it cm}^{ - 1} = 1.8 \cdot 10^{43} s^{ - 1} $.

Problem 10

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

solution

$G = 7.4243\times 10^{-29}~\mbox{cm/g}$.

Problem 11

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

solution

Since the Coulomb force in $D$--dimensional space is $F_e \propto e^2 r^{-(D - 1)}$, the dimensionality of charge depends on the dimension of space, therefore $[\alpha ] = \left[ {{{e^2 } / {\hbar c}}} \right] = L^{D - 3}$.

Problem 12

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

solution

Recall, that dimensionalities of ($e, G$) depend on the dimension of space. The required dimensionless combination is $$\alpha^{(D)} = e^{D - 1} \hbar ^{2 - D} c^{D - 4} G^{{{3 - D} \over 2}}. $$ Gravitational constant $G$ doesn't enter this relation only for $D=3$ so that $\alpha^{(3)} = {{e^2 } \over {\hbar c}}$.

Problem 13

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

solution

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).

solution

$T^{GU} \approx 10^{28}~K$.