Difference between revisions of "Planck scales and fundamental constants"
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Revision as of 19:35, 5 November 2012
Contents
- 1 Problem 1: frequency or energy?
- 2 Problem 2: energy, momentum and mass
- 3 Problem 3: the Planck units
- 4 Problem 4: Newton units
- 5 Problem 5: Planck time
- 6 Problem 6: gravitational radius for Planck mass
- 7 Problem 7: quantum gravity effects
- 8 Problem 8: 1GeV
- 9 Problem 9: age of the Universe in Planck units
- 10 Problem 10: Planck mass in different units
- 11 Problem 11: gravitational constant
- 12 Problem 12: fine structure constant
- 13 Problem 13: dimensionless combinations
- 14 Problem 14: strong, weak, EM and gravity
- 15 Problem 15: the Great Unification
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?
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: 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?
This is impossible, since momentum $\vec p$ is a vector, while $e$ and $m$ are scalars in 3--dimensional space.
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?
$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: 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?
$10^{35}$
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.
$r_g={{2GM_{_{Pl}}} \over {c^2}} = {\hbar \over M_{_{Pl}}c} = l_{_{Pl}}$
Problem 7: quantum gravity effects
Show that for any Standard Model particle quantum gravity effects are completely negligible at the particle level.
Quantum gravity effects can be neglected in the case when the Compton wavelength $\lambda _c = \frac{\hbar }[[:Template:Mc]]$ of the particle is much greater than the Schwarzschild radius $r_s = \frac[[:Template:2mG]][[:Template:C^2]]$. Parameter that defines the role of quantum gravity,\[\frac{\lambda _c }{r_s } \approx \frac{m_{Pl}^2}{m^2}\] For example, for an electron \[\frac{\lambda _c }{r_s} \approx 10^{45}\] so quantum gravity effects are completely negligible. The same is true for all other Standard Model particles.
Problem 8: 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 9: 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 10: Planck mass in different units
Express Planck mass in terms of $K$, $cm^{-1}$, $s^{-1}$.
$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 11: gravitational constant
Express Newton's constant $G$ in units $c=1$.
$G = 7.4243\times 10^{-29}~\mbox{cm/g}$.
Problem 12: fine structure constant
Show that the fine structure constant $\alpha=e^2/\hbar c$ is dimensionless only in the space of dimension $D=3$.
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 13: dimensionless combinations
Construct a dimensionless combination from the constants $c$, $\hbar$, $e$, and $G$ in the space of arbitrary dimension.
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 14: strong, weak, EM and gravity
$^*$ Compare the constants of strong, weak, electromagnetic and gravitational interactions.
Problem 15: 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.
$T^{GU} \approx 10^{28}~K$.
