Difference between revisions of "Energy balance in an expanding Universe"
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[[Category:Dynamics of the Expanding Universe|7]] | [[Category:Dynamics of the Expanding Universe|7]] | ||
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| − | + | <div id="equ_en1"></div><div style="border: 1px solid #AAA; padding:5px;"> | |
| − | + | === Problem 1: Newtonian interpretation === | |
| − | <div id="equ_en1"></div> | + | |
| − | === Problem 1 | + | |
Show that the first Friedman equation can be treated as the energy conservation law in Newtonian mechanics. Use this equation to classify the solutions describing different dynamics of the Universe. | Show that the first Friedman equation can be treated as the energy conservation law in Newtonian mechanics. Use this equation to classify the solutions describing different dynamics of the Universe. | ||
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<div class="NavHead">solution</div> | <div class="NavHead">solution</div> | ||
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| − | <p style="text-align: left;">See [[Newtonian_cosmology#equ48| | + | <p style="text-align: left;">See [[Newtonian_cosmology#equ48|here]].</p> |
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| − | === Problem 2 | + | === Problem 2: thermodynamic derivation === |
Obtain the conservation equation for the expanding Universe using only thermodynamical considerations. | Obtain the conservation equation for the expanding Universe using only thermodynamical considerations. | ||
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\[\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=0.\]</p> | \[\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=0.\]</p> | ||
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| − | === Problem 3 | + | === Problem 3: the lost energy of the Universe === |
A photon's wavelength is redshifted due to the Universe' expansion. Estimate the rate of change of the energy of the Universe due to this process. | A photon's wavelength is redshifted due to the Universe' expansion. Estimate the rate of change of the energy of the Universe due to this process. | ||
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| − | <div class="NavHead">solution</div> | + | <div class="NavHead">no solution</div> |
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Latest revision as of 16:40, 13 October 2012
Contents
Problem 1: Newtonian interpretation
Show that the first Friedman equation can be treated as the energy conservation law in Newtonian mechanics. Use this equation to classify the solutions describing different dynamics of the Universe.
See here.
Problem 2: thermodynamic derivation
Obtain the conservation equation for the expanding Universe using only thermodynamical considerations.
Let $E$ be the total energy inside the volume $V$, and $\rho$ is the energy density in the Universe, then \[dE = d(\rho V)= Vd\rho+\rho dV.\] In the first principle of thermodynamics $dQ = dE+pdV$, as the Universe is a closed system by definition, it is evident that $dQ=0$. Taking into account that $dV = 4\pi a^2da$, it is easy to see that \[\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=0.\]
Problem 3: the lost energy of the Universe
A photon's wavelength is redshifted due to the Universe' expansion. Estimate the rate of change of the energy of the Universe due to this process.
