Cosmo All at 20:01, 18 June 2015

2015-06-18T20:01:00Z

Cosmo All: Created page with "

'''Problem 1'''
problem id: 2612_1
Find dimension of t..."

2015-06-18T19:52:36Z

Created page with " <div id="2612_1"></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem 1''' <p style= "color: #999;font-size: 11px">problem id: 2612_1</p> Find dimension of t..."

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'''Problem 1'''
<p style= "color: #999;font-size: 11px">problem id: 2612_1</p>
Find dimension of the Newtonian gravitational constant $G$ and electric charge $e$ in the $N$-dimensional space.
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<p style="text-align: left;">By Gauss's theorem we have $[G]=L^NM^{-1}T^{-2}$; $[e^2]=L^NMT^{-2}$.</p>
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'''Problem 2'''
<p style= "color: #999;font-size: 11px">problem id: 2612_2</p>
In 1881, Johnson Stoney proposed a set of fundamental units involving $c$, $G$ and $e$. Construct Stoney's natural units of length, mass and time.
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<p style="text-align: left;"> \[L_S=\left(\frac{Ge^2}{c^4}\right)^{1/2};\]
\[M_S=\left(\frac{e^2}{G}\right)^{1/2};\]
\[T_S=\left(\frac{Ge^2}{c^6}\right)^{1/2}.\]</p>
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'''Problem 3'''
<p style= "color: #999;font-size: 11px">problem id: 2612_3</p>
Show that the fine structure constant $\alpha=e^2/(\hbar c)$ can be used to convert between Stoney and Planck units.
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<p style="text-align: left;">\[L_S=L_{Pl}\alpha^{1/2}\]
\[M_S=M_{Pl}\alpha^{1/2}\]
\[T_S=T_{Pl}\alpha^{1/2}\]</p>
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'''Problem 4'''
<p style= "color: #999;font-size: 11px">problem id: 2612_4</p>
Gibbons [G W Gibbons, The Maximum Tension Principle in General Relativity, arXiv:0210109] formulated a hypothesis, that in General Relativity there should be a maximum value to any physically attainable force (or tension) given by
\begin{equation}\label{2612_e_1}
F_{max}=\frac{c^4}{4G}.
\end{equation}
This quantity can be constructed with help of the Planck units: \[F_{max}=\frac14M_{Pl}L_{Pl}T_{Pl}^{-2}.\] The origin of the numerical coefficient $1/4$ has no deeper meaning. It simply turns out that $1/4$ is the value that leads to the correct form of the field equations of General Relativity. The above made assumption leads to existence of a maximum power defined by
\begin{equation}\label{2612_e_2}
P_{max}=\frac{c^5}{G}.
\end{equation}
''There is a hypothesis [C. Schiller, General relativity and cosmology derived from principle of maximum power or force, arXiv:0607090], that the maximum force (or power) plays the same role for general relativity as the maximum speed plays for special relativity.''

The black holes are usually considered as an extremal solution of GR equations, realizing the limiting values of the physical quantities. Show that the value $c^4/(4G)$ of the force limit is the energy of a Schwarzschild black hole divided by twice its radius. The maximum power $c^5/(4G)$ is realized when such a black hole is radiated away in the time that light takes to travel along a length corresponding to twice the radius.
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<p style="text-align: left;">\[F_{max}=\frac{Mc^2}{2R_S}=\frac{Mc^2}{4MG/c^2}=\frac{c^4}{4G}\approx3\times10^{43}N;\]
\[P_{max}=\frac{Mc^2}{2R_S/c}=F_{max}c=\frac{c^5}{4G}\approx9\times10^{51}W.\]</p>
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'''Problem 5'''
<p style= "color: #999;font-size: 11px">problem id: 2612_5</p>
''E. Berti, A Black-Hole Primer: Particles, Waves, Critical Phenomena and Superradiant instabilities (arXiv:1410.4481[gr-qc])''

A Newtonian analog of the black hole concept is a so-called "dark star". If we consider light as a corpuscle traveling at speed $c$, light cannot escape to infinity whenever \[V_{esc}>c \quad \left(V_{esc}^2=\frac{2GM}{R}\right)/\] Therefore the condition for existence of "dark stars" in Newtonian mechanics is
\[\frac{2GM}{c^2R}\ge1.\]
Can this condition be satisfied in the Newtonian mechanics?
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<p style="text-align: left;">A naive argument tells us that as we pile up more and more material of constant density $\rho_0$, the ratio $M/R$ increases:
\begin{equation}\label{2612_5_1}
\frac M R=\frac43\pi R^2\rho_0.
\end{equation}
This equation would seem to suggest that dark stars could indeed form. However, we must include the binding energy $U$,
\begin{equation}\label{2612_5_2}
U=-\int\frac{GMdM}{r}=-\int\frac G r \left(\frac43\pi r^3\rho_0\right)4\pi r^2\rho_0 dr = -\frac{16G\pi^2}{15}\rho_0^2R^5.
\end{equation}
The total mass $M_T$ of the hypothetical dark star is given by the rest mass $M$ plus the binding energy $U$
\begin{equation}\label{2612_5_3}
\frac{M_T}R=\frac43\pi R^2\rho_0 -\frac{16G\pi^2}{15}\rho_0^2R^4=\frac M R \left[1-\frac35\frac G{c^2}\frac M R \right]\le\frac5{12},
\end{equation}
where the upper limit is obtained by maximizing the function in the range (\ref{2612_5_1}). Thus, the dark star criterion (\ref{2612_5_1}) is never satisfied, even for the unrealistic case of constant-density matter. In fact, the endpoint of Newtonian gravitational collapse depends very sensitively on the equation of state, even in spherical symmetry.</p>
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'''Problem 6'''
<p style= "color: #999;font-size: 11px">problem id: 2612_6</p>
The lookback time is defined as the difference between the present day age of the Universe and its age at redshift $z$, i.e. the difference between the age of the Universe at observation $t_0$ and the age of the Universe, $t$, when the photons were emitted. Find the lookback time for the Universe filled by non-relativistic matter, radiation and a component with the state equation $p=w(z)\rho$.
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<p style="text-align: left;">From the definitions of redshift $1+z=1/a$ we have
\[\frac{dz}{dt}=-\frac{\dot a}{a^2}=-H(z)(1+z)\]
or
\[dt=-\frac{dz}{H(z)(1+z)}.\]
The lookback time is defined as
\[t_0-t=\int\limits_t^{t_0}dt=\int\limits_0^z\frac{dz'}{H(z')(1+z')}=\frac1{H_0}\int\limits_0^z\frac{dz'}{E(z')(1+z')}\]
where
\[E(z)=\sqrt{\Omega_r(1+z)^4+\Omega_m(1+z)^3+\Omega_k(1+z)^2+\Omega_w\exp\left(\int\limits_0^zdz'\frac{1+w(z')}{1+z'}\right)}.\]
From the definition of lookback time it is clear that the cosmological time or the time back to the Big Bang, is given by
\[t(z)=\int\limits_z^\infty\frac{dz'}{H(z')(1+z')}.\]</p>
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'''Problem 7'''
<p style= "color: #999;font-size: 11px">problem id: 2612_7</p>
Find the solutions corrected by LQC Friedmann equations for a matter dominated Universe.
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<p style="text-align: left;">Solving the corrected first Friedmann equation and the conservation equation for a matter dominated Universe
\[H^2=\frac\rho3\left(1-\frac\rho{\rho_c}\right),\quad \dot\rho+3H\rho=0\]
one obtains the following quantities
\[a(t)=\left(\frac34\rho_ct^2+1\right)^{1/3},\quad \rho(t)=\frac{\rho_c}{\frac34\rho_ct^2+1},\quad H(t)=\frac{\frac12\rho_c t}{\frac34\rho_ct^2+1}\]
For small values of the energy density ($\rho\ll\rho_c$) we recover the solutions of standard Friedmann equations.</p>
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'''Problem 8'''
<p style= "color: #999;font-size: 11px">problem id: 2612_8</p>
Show that in the case of the flat Friedmann metric, the third power of the scale factor $\varphi=a^3$ satisfies the equation
\[\frac{d^2t\varphi}{dt^2}=\frac32(\rho-p)\varphi,\quad 8\pi G=1.\]
Check validity of this equation for different cosmological components: non-relativistic matter, cosmological constant and a component with the state equation $p=w\rho$.
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<p style="text-align: left;">\[\frac{d^2a}{dt^2}=3\left(2H^2+\frac{\ddot a}a\right)a^3=3\left[\frac23\rho-\frac16(\rho+3p)\right]a^3=\frac32(\rho-p)\varphi.\]</p>
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'''Problem 9'''
<p style= "color: #999;font-size: 11px">problem id: </p>
It is of broad interest to understand better the nature of the early Universe especially the Big Bang. The discovery of the CMB in 1965 resolved a dichotomy then existing in theoretical cosmology between steady-state and Big Bang theories. The interpretation of the CMB as a relic of a Big Bang was compelling and the steady-state theory died. Actually at that time it was really a trichotomy being reduced to a dichotomy because a third theory is a cyclic cosmology model. Give a basic argument of the opponents to the latter model.
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<p style="text-align: left;">The main argument of the opponents to the cyclic cosmology model of the Universe was based on the so-called Tolman Entropy Conundrum (R.C. Tolman, Relativity, Thermodynamics and Cosmology. Oxford University Press (1934)): the entropy of the Universe necessarily increases, due to the second law of thermodynamics, and therefore cycles become larger and longer in the future, smaller and shorter in the past, implying that a Big Bang must have occurred at A FINITE time in the past.</p>
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'''Problem 10'''
<p style= "color: #999;font-size: 11px">problem id: 2612_10</p>
Express the present epoch value of the Ricci scalar $R$ and its first derivative in terms of $H_0$ and its derivatives (flat case).
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<p style="text-align: left;">We can rewrite the Ricci scalar $R$ as function of redshift $z$,
\[R=6H\left[(1+z)\frac{dH}{dz}-2H\right].\]
For $z=0$ we have
\begin{align}
\nonumber R_0 & = 6H\left(\left.\frac{dH}{dz}\right|_{z=0}-2H_0\right),\\
\nonumber \left.\frac{dR}{dz}\right|_{z=0} & =6\left[\left(\left.\frac{dH}{dz}\right|_{z=0}\right)^2-H_0 \left(3\left.\frac{dH}{dz}\right|_{z=0}-\left.\frac{d^2H}{dz^2}\right|_{z=0}\right)\right].
\end{align}</p>
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Cosmo All at 20:01, 18 June 2015

Cosmo All: Created page with " '''Problem 1''' problem id: 2612_1 Find dimension of t..."

Cosmo All: Created page with "

'''Problem 1'''
problem id: 2612_1
Find dimension of t..."