Difference between revisions of "Planck scales and fundamental constants"
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=== Problem 3: the Planck units === | === 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). | 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). | ||
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=== Problem 5: Planck time === | === 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? | $^*$ 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? | ||
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| − | <p style="text-align: left;">$r_g= | + | <p style="text-align: left;">$r_g={2GM_{_{Pl}} \over {c^2}} = {\hbar \over M_{_{Pl}}c} = l_{_{Pl}}$</p> |
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| + | === Problem 7: quantum gravity effects === | ||
| + | Show that for any Standard Model particle quantum gravity effects are completely negligible at the particle level. | ||
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| + | <div class="NavHead">solution</div> | ||
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| + | <p style="text-align: left;">Quantum gravity effects can be neglected in the case when the Compton wavelength $\lambda _c = \frac{\hbar }{mc}$ of the particle is much greater than the Schwarzschild radius $r_s = \frac{2mG}{c^2}$. The 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.</p> | ||
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=== Problem 9: Planck mass in different units === | === Problem 9: Planck mass in different units === | ||
Express Planck mass in terms of $K$, $cm^{-1}$, $s^{-1}$. | Express Planck mass in terms of $K$, $cm^{-1}$, $s^{-1}$. | ||
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| − | <p style="text-align: left;">$M_{Pl} = 1.42 \cdot 10^{32} \,K = 6.2 \cdot 10^{32}\mbox{ | + | <p style="text-align: left;">$M_{Pl} = 1.42 \cdot 10^{32} \,K = 6.2 \cdot 10^{32}\mbox{cm}^{ - 1} = 1.8 \cdot 10^{43} s^{ - 1} $.</p> |
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=== Problem 10: gravitational constant === | === Problem 10: gravitational constant === | ||
Express Newton's constant $G$ in units $c=1$. | Express Newton's constant $G$ in units $c=1$. | ||
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<p style="text-align: left;">$T^{GU} \approx 10^{28}~K$.</p> | <p style="text-align: left;">$T^{GU} \approx 10^{28}~K$.</p> | ||
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| + | <div id="TF_1"></div> | ||
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| + | === Problem 15 === | ||
| + | <p style= "color: #999;font-size: 11px">problem id: TF_1</p> | ||
| + | Construct planck units in a space of arbitrary dimension. | ||
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| + | <div class="NavHead">solution</div> | ||
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| + | <p style="text-align: left;">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 | ||
| + | \[L_{P(D)}=\left(\frac{G_D\hbar}{c^3}\right)^{\frac{1}{D-2}}\quad T_{P(D)}=\left(\frac{G_D\hbar}{c^{D+1}}\right)^{\frac{1}{D-2}}\quad M_{P(D)}=\left(\frac{c^{5-D}\hbar^{D-3}}{G_D}\right)^{\frac{1}{D-2}}.\]</p> | ||
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Latest revision as of 11:37, 20 December 2013
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: 1GeV
- 8 Problem 8: age of the Universe in Planck units
- 9 Problem 9: Planck mass in different units
- 10 Problem 10: gravitational constant
- 11 Problem 11: fine structure constant
- 12 Problem 12: dimensionless combinations
- 13 Problem 13: strong, weak, EM and gravity
- 14 Problem 14: the Great Unification
- 15 Problem 15
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: 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 $K$, $cm^{-1}$, $s^{-1}$.
$M_{Pl} = 1.42 \cdot 10^{32} \,K = 6.2 \cdot 10^{32}\mbox{cm}^{ - 1} = 1.8 \cdot 10^{43} s^{ - 1} $.
Problem 10: gravitational constant
Express Newton's constant $G$ in units $c=1$.
$G = 7.4243\times 10^{-29}~\mbox{cm/g}$.
Problem 11: fine structure constant
Show that the fine structure constant $\alpha=e^2/\hbar c$ is dimensionless only in the space of dimension $D=3$.
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: 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 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.
$T^{GU} \approx 10^{28}~K$.
Problem 15
problem id: TF_1
Construct planck units in a space of arbitrary dimension.
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 \[L_{P(D)}=\left(\frac{G_D\hbar}{c^3}\right)^{\frac{1}{D-2}}\quad T_{P(D)}=\left(\frac{G_D\hbar}{c^{D+1}}\right)^{\frac{1}{D-2}}\quad M_{P(D)}=\left(\frac{c^{5-D}\hbar^{D-3}}{G_D}\right)^{\frac{1}{D-2}}.\]
