Planck scales and fundamental constants

1.5.1

Problem 1.5.1: 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?

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.

1.5.2

Problem 1.5.2: 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?

solution

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

1.5.3

Problem 1.5.3: 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).

1.5.4

Problem 1.5.4: 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?

solution

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

1.5.5

Problem 1.5.5: 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?

solution

$10^{35}$

1.5.6

Problem 1.5.6: 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.

solution

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

1.5.7

Problem 1.5.7: 1GeV

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

1.5.8

Problem 1.5.8: 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.

1.5.9

Problem 1.5.9: Planck mass in different units

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{cm}^{ - 1} = 1.8 \cdot 10^{43} s^{ - 1} $.

1.5.10

Problem 1.5.10: Gravitational constant

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

solution

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

1.5.11

Problem 1.5.11: Fine structure constant

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

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 $[\alpha ] = \left[ {{{e^2 } / {\hbar c}}} \right] = L^{D - 3}$.

1.5.12

Problem 1.5.12: Dimensionless combinations

Construct a dimensionless combination from the constants $c$, $\hbar$, $e$, and $G$ 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 $\alpha^{(3)} = {{e^2 } \over {\hbar c}}$.

1.5.13

Problem 1.5.13: Strong, weak, EM and gravity

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

no solution yet

1.5.14

Problem 1.5.14: 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.

solution

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

1.5.15

Problem 1.5.15

problem id: TF_1

Construct planck units in a space of arbitrary dimension.

solution

Dimensionality of the fundamental constants $c,\hbar,G_D$ in $D=4+n$ dimensions can be determined as \[[G_d]=L^{D-1}T^{-2}M^{-1},\quad \hbar=L^2T^{-1}M,\quad c=LT^{-1}.\] Note that the dimension of the space affects only the dimensionality of the Newton's constant $G_D$, because the universal gravitation law transforms with changes of dimensionality of the space as the following \[F=G_D\frac{M_1M_2}{R^{D-2}}.\] Use the combination \[[G_D^\alpha\hbar^\beta c^\gamma]= L^{\alpha(D-1)+2\beta+\gamma} T^{-2\alpha-\beta-\gamma} M^{-\alpha+\beta-\gamma}\] to find that \[L_{P(D)}=\left(\frac{G_D\hbar}{c^3}\right)^{\frac{1}{D-2}}\quad T_{P(D)}=\left(\frac{G_D\hbar}{c^{D+1}}\right)^{\frac{1}{D-2}}\quad M_{P(D)}=\left(\frac{c^{5-D}\hbar^{D-3}}{G_D}\right)^{\frac{1}{D-2}}.\]