Planck scales and fundamental constants - Revision history

Cosmo All at 11:37, 20 December 2013

2013-12-20T11:37:15Z

Cosmo All at 11:35, 20 December 2013

2013-12-20T11:35:59Z

Cosmo All: /* Problem 7: 1GeV */

2013-12-20T11:35:06Z

‎Problem 7: 1GeV

Cosmo All at 11:34, 20 December 2013

2013-12-20T11:34:28Z

Igor at 17:44, 10 November 2012

2012-11-10T17:44:41Z

Igor: /* Problem 6: gravitational radius for Planck mass */

2012-11-10T17:39:12Z

‎Problem 6: gravitational radius for Planck mass

Igor: /* Problem 9: Planck mass in different units */

2012-11-10T17:37:21Z

‎Problem 9: Planck mass in different units

Cosmo All at 19:37, 5 November 2012

2012-11-05T19:37:38Z

Cosmo All: /* Problem 7: gravitational radius for Planck mass */

2012-11-05T19:35:46Z

‎Problem 7: gravitational radius for Planck mass

Cosmo All at 19:34, 5 November 2012

2012-11-05T19:34:50Z

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      <p style="text-align: left;">$T^{GU} \approx 10^{28}~K$.</p>
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 === Problem 9: Planck mass in different units ===
 Express Planck mass in terms of $K$, $cm^{-1}$, $s^{-1}$.

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 === 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?
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 === Problem 7: quantum gravity effects   ===
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 === Problem 7: 1GeV ===
 Demonstrate, that in  the units $c=\hbar=1$
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    <div class="NavHead">solution</div>
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-     <p style="text-align: left;">$r_g={{2GM_{_{Pl}}} \over {c^2}} = {\hbar  \over M_{_{Pl}}c} = l_{_{Pl}}$</p>
+     <p style="text-align: left;">$r_g={2GM_{_{Pl}} \over {c^2}} = {\hbar  \over M_{_{Pl}}c} = l_{_{Pl}}$</p>
    </div>
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+<!--<div id="razm_QG1"></div>
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 === Problem 7: quantum gravity effects   ===
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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 $r_s  = \frac{{2mG}}{{c^2 }}$. Parameter that defines the role of quantum gravity,\[\frac{\lambda _c }{r_s } \approx \frac{m_{Pl}^2}{m^2}\]
+     <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,
-For example, for an electron \[\frac{\lambda _c }{r_s} \approx 10^{45}\]
+\[\frac{\lambda _c }{r_s } \approx \frac{m_{Pl}^2}{m^2}\]
 so quantum gravity effects are completely negligible. The same is true for all other Standard Model particles.</p>
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    <div class="NavHead">solution</div>
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-     <p style="text-align: left;">$M_{Pl}  =  1.42 \cdot 10^{32} \,K = 6.2 \cdot 10^{32}\mbox{\it cm}^{ - 1} =  1.8 \cdot 10^{43} s^{ - 1} $.</p>
+     <p style="text-align: left;">$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 ===
 Express Newton's constant $G$ in units $c=1$.

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    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;">$r_g={{2GM_{_{Pl}}} \over {c^2}} = {\hbar  \over M_{_{Pl}}c} = l_{_{Pl}}$</p>
    </div>
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-=== Problem 7: 1GeV ===
+=== Problem 8: 1GeV ===
 Demonstrate, that in  the units $c=\hbar=1$
 \[1\,GeV\approx 1.8\cdot 10^{-24}\, g;\quad
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-=== Problem 8: age of the Universe in Planck units ===
+=== 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.
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-=== Problem 9: Planck mass in different units ===
+=== Problem 10: Planck mass in different units ===
 Express Planck mass in terms of $K$, $cm^{-1}$, $s^{-1}$.
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-=== Problem 10: gravitational constant ===
+=== Problem 11: gravitational constant ===
 Express Newton's constant $G$ in units $c=1$.
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-=== Problem 11: fine structure constant ===
+=== Problem 12: fine structure constant ===
 Show that the fine structure constant $\alpha=e^2/\hbar c$ is dimensionless only in the space of dimension $D=3$.
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-=== Problem 12: dimensionless combinations ===
+=== Problem 13: dimensionless combinations ===
 Construct a dimensionless combination from the constants $c$, $\hbar$, $e$, and $G$ in the space of arbitrary dimension.
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-=== Problem 13: strong, weak, EM and gravity ===
+=== Problem 14: strong, weak, EM and gravity ===
 $^*$ Compare the constants of strong, weak, electromagnetic and gravitational interactions.
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-=== Problem 14: the Great Unification ===
+=== 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.
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