Difference between revisions of "New problems"

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<p style= "color: #999;font-size: 11px">problem id: 2612_4</p>
 
<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
 
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}
+
\begin{equation}\label{2612_e_1}
    F_{max}=\frac{c^4}{4G}.
+
F_{max}=\frac{c^4}{4G}.
    \end{equation}
+
\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
 
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}
+
\begin{equation}\label{2612_e_2}
    P_{max}=\frac{c^5}{G}.
+
P_{max}=\frac{c^5}{G}.
    \end{equation}
+
\end{equation}
{\it 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.}
+
''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.
 
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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''E. Berti, A Black-Hole Primer: Particles, Waves, Critical Phenomena and Superradiant  instabilities (arXiv:1410.4481[gr-qc])''
 
''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
+
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
  \[\frac{2GM}{c^2R}\ge1.\]
+
\[\frac{2GM}{c^2R}\ge1.\]
    Can this condition be satisfied in the Newtonian mechanics?
+
Can this condition be satisfied in the Newtonian mechanics?
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
 
     <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:
 
     <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}
+
\begin{equation}\label{2612_5_1}
    \frac M R=\frac43\pi R^2\rho_0.
+
\frac M R=\frac43\pi R^2\rho_0.
    \end{equation}
+
\end{equation}
 
This equation would seem to suggest that dark stars could indeed form. However, we must include the binding energy $U$,
 
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}
+
\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.
+
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}
+
\end{equation}
 
The total mass $M_T$ of the hypothetical dark star is given by the rest mass $M$ plus the binding energy $U$
 
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}
+
\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},
+
\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}
+
\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>
 
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>
 
   </div>
 
   </div>
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   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
 
     <p style="text-align: left;">From the definitions of redshift $1+z=1/a$ we have
 
     <p style="text-align: left;">From the definitions of redshift $1+z=1/a$ we have
\[\frac{dz}{dt}=-\frac{\dot a}{a^2}=-H(z)(1+z)\]
+
\[\frac{dz}{dt}=-\frac{\dot a}{a^2}=-H(z)(1+z)\]
 
or
 
or
\[dt=-\frac{dz}{H(z)(1+z)}.\]
+
\[dt=-\frac{dz}{H(z)(1+z)}.\]
 
The lookback time is defined as
 
The lookback time is defined as
\[t_0-t=\int\limits_t^{t_0}dt=\int\limits_0^z\frac{dz'}{H(z')(1+z')}=\frac1{H_0}\int\limits_0^z\frac{dz'}{E(z')(1+z')}\]
+
\[t_0-t=\int\limits_t^{t_0}dt=\int\limits_0^z\frac{dz'}{H(z')(1+z')}=\frac1{H_0}\int\limits_0^z\frac{dz'}{E(z')(1+z')}\]
 
where
 
where
 
\[E(z)=\sqrt{\Omega_r(1+z)^4+\Omega_m(1+z)^3+\Omega_k(1+z)^2+\Omega_w\exp\left(\int\limits_0^zdz'\frac{1+w(z')}{1+z'}\right)}.\]
 
\[E(z)=\sqrt{\Omega_r(1+z)^4+\Omega_m(1+z)^3+\Omega_k(1+z)^2+\Omega_w\exp\left(\int\limits_0^zdz'\frac{1+w(z')}{1+z'}\right)}.\]
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\[H^2=\frac\rho3\left(1-\frac\rho{\rho_c}\right),\quad \dot\rho+3H\rho=0\]  
 
\[H^2=\frac\rho3\left(1-\frac\rho{\rho_c}\right),\quad \dot\rho+3H\rho=0\]  
 
one obtains the following quantities
 
one obtains the following quantities
\[a(t)=\left(\frac34\rho_ct^2+1\right)^{1/3},\quad \rho(t)=\frac{\rho_c}{\frac34\rho_ct^2+1},\quad H(t)=\frac{\frac12\rho_c t}{\frac34\rho_ct^2+1}\]
+
\[a(t)=\left(\frac34\rho_ct^2+1\right)^{1/3},\quad \rho(t)=\frac{\rho_c}{\frac34\rho_ct^2+1},\quad H(t)=\frac{\frac12\rho_c t}{\frac34\rho_ct^2+1}\]
 
For small values of the energy density ($\rho\ll\rho_c$) we recover the solutions of standard Friedmann equations.</p>
 
For small values of the energy density ($\rho\ll\rho_c$) we recover the solutions of standard Friedmann equations.</p>
 
   </div>
 
   </div>
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<p style= "color: #999;font-size: 11px">problem id: 2612_8</p>
 
<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
 
Show that in the case of the flat Friedmann metric, the third power of the scale factor $\varphi=a^3$ satisfies the equation
\[\frac{d^2t\varphi}{dt^2}=\frac32(\rho-p)\varphi,\quad 8\pi G=1.\]
+
\[\frac{d^2t\varphi}{dt^2}=\frac32(\rho-p)\varphi,\quad 8\pi G=1.\]
 
Check validity of this equation for different cosmological components: non-relativistic matter, cosmological constant and a component with the state equation $p=w\rho$.
 
Check validity of this equation for different cosmological components: non-relativistic matter, cosmological constant and a component with the state equation $p=w\rho$.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">

Revision as of 21:54, 29 January 2015



New from 26-12-2014


Problem 1

problem id: 2612_1

Find dimension of the Newtonian gravitational constant $G$ and electric charge $e$ in the $N$-dimensional space.


Problem 2

problem id: 2612_2

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.


Problem 3

problem id: 2612_3

Show that the fine structure constant $\alpha=e^2/(\hbar c)$ can be used to convert between Stoney and Planck units.


Problem 4

problem id: 2612_4

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.


Problem 5

problem id: 2612_5

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?


Problem 6

problem id: 2612_6

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 $p=w(z)\rho$.


Problem 7

problem id: 2612_7

Find the solutions corrected by LQC Friedmann equations for a matter dominated Universe.


Problem 8

problem id: 2612_8

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 $p=w\rho$.


Problem 9

problem id:

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.


Problem 10

problem id: 2612_10

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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Exactly Integrable n-dimensional Universes

Exactly Integrable n-dimensional Universes

Problem 1

problem id: gnd_1

Derive Friedmann equations for the spatially n-dimensional Universe.


Problem 2

problem id: gnd_2

Obtain the energy conservation law for the case of n-dimensional Universe.


Problem 3

problem id: gnd_3

Obtain relation between the energy density and scale factor in the case of a two-component n-dimensional Universe dominated by the cosmological constant and a barotropic fluid.


Problem 4

problem id: gnd_4

Obtain equation of motion for the scale factor for the previous problem.


To integrate (\ref{14}), we recall Chebyshev's theorem:

For rational numbers $p,q,r$ ($r\neq0$) and nonzero real numbers $\alpha,\beta$, the integral $\int x^p(\alpha+\beta x^r)^q\,d x$ is elementary if and only if at least one of the quantities \begin{equation}\label{cd} \frac{p+1}r,\quad q,\quad \frac{p+1}r+q, \end{equation} is an integer.

Another way to see the validity of the Chebyshev theorem is to represent the integral of concern by a hypergeometric function such that when a quantity in (\ref{cd}) is an integer the hypergeometric function is reduced into an elementary function. Consequently, when $k=0$ or $\Lambda=0$, and $w$ is rational, the Chebyshev theorem enables us to know that, for exactly what values of $n$ and $w$, the equation (\ref{14}) may be integrated.


Problem 5

problem id: gnd_4_0

Obtain analytic solutions for the equation of motion for the scale factor of the previous problem for spatially flat ($k=0$) Universe.


Problem 6

problem id: gnd_5

Use the analytic solutions obtained in the previous problem to study cosmology with $w>-1$ and $a(0)=0$.


Problem 7

problem id: gnd_6

Use results of the previous problem to calculate the deceleration $q=-a\ddot a/\dot a^2$ parameter.


Problem 8

problem id: gnd_4_1

Obtain the exact solvability conditions for the case $\Lambda=0$ in equation (\ref{14}) (see problem #gnd_4 ).


Problem 9

problem id: gnd_4_2

Obtain the explicit solutions for the case $n=3$ and $w=-5/9$ in the previous problem.


Problem 10

problem id: gnd_10

Rewrite equation of motion for the scale factor (\ref{14}) in terms of the conformal time.


Problem 11

problem id: gnd_11

Find the rational values of the equation of state parameter $w$ which provide exact integrability of the equation (\ref{48}) with $k=0$.


Problem 12

problem id: gnd_12

Obtain explicit solutions for the case $w=\frac1n-1$ in the previous problem.


Problem 13

problem id: gnd_13

Obtain exact solutions for the equation (\ref{48}) with $\Lambda=0$.


Problem 14

problem id: gnd_14

Obtain inflationary solutions using the results of the previous problem.


UNSORTED NEW Problems

The history of what happens in any chosen sample region is the same as the history of what happens everywhere. Therefore it seems very tempting to limit ourselves with the formulation of Cosmology for the single sample region. But any region is influenced by other regions near and far. If we are to pay undivided attention to a single region, ignoring all other regions, we must in some way allow for their influence. E. Harrison in his book Cosmology, Cambridge University Press, 1981 suggests a simple model to realize this idea. The model has acquired the name of "Cosmic box" and it consists in the following.

Imaginary partitions, comoving and perfectly reflecting, are used to divide the Universe into numerous separate cells. Each cell encloses a representative sample and is sufficiently large to contain galaxies and clusters of galaxies. Each cell is larger than the largest scale of irregularity in the Universe, and the contents of all cells are in identical states. A partitioned Universe behaves exactly as a Universe without partitions. We assume that the partitions have no mass and hence their insertion cannot alter the dynamical behavior of the Universe. The contents of all cells are in similar states, and in the same state as when there were no partitions. Light rays that normally come from very distant galaxies come instead from local galaxies of long ago and travel similar distances by multiple re?ections. What normally passes out of a region is reflected back and copies what normally enters a region.

Let us assume further that the comoving walls of the cosmic box move apart at a velocity given by the Hubble law. If the box is a cube with sides of length $L$, then opposite walls move apart at relative velocity $HL$.Let us assume that the size of the box $L$ is small compared to the Hubble radius $L_{H} $ , the walls have a recession velocity that is small compared to the velocity of light. Inside a relatively small cosmic box we use ordinary everyday physics and are thus able to determine easily the consequences of expansion. We can even use Newtonian mechanics to determine the expansion if we embed a spherical cosmic box in Euclidean space.


Problem 1

problem id:

As we have shown before (see Chapter 3): $p(t)\propto a(t)^{-1} $ , so all freely moving particles, including galaxies (when not bound in clusters), slowly lose their peculiar motion and ultimately become stationary in expanding space. Try to understand what happens by considering a moving particle inside an expanding cosmic box


Problem 2

problem id:

Show that at redshift $z=1$ , when the Universe is half its present size, the kinetic energy of a freely moving nonrelativistic particle is four times its present value, and the energy of a relativistic particle is twice its present value.


Problem 3

problem id:

Let the cosmic box is filled with non-relativistic gas. Find out how the gas temperature varies in the expanding cosmic box.


Problem 4

problem id:

Show that entropy of the cosmic box is conserved during its expansion.


Problem 5

problem id:

Consider a (cosmic) box of volume V, having perfectly reflecting walls and containing radiation of mass density $\rho $. The mass of the radiation in the box is $M=\rho V$ . We now weigh the box and find that its mass, because of the enclosed radiation, has increased not by M but by an amount 2M. Why?


Problem 6

problem id:

Show that the jerk parameter is \[j(t)=q+2q^{2} -\frac{\dot{q}}{H} \]


Problem 7

problem id:

We consider FLRW spatially flat Universe with the general Friedmann equations \[\begin{array}{l} {H^{2} =\frac{1}{3} \rho +f(t),} \\ {\frac{\ddot{a}}{a} =-\frac{1}{6} \left(\rho +3p\right)+g(t)} \end{array}\] Obtain the general conservation equation.


Problem 8

problem id:

Show that for extra driving terms in the form of the cosmological constant the general conservation equation (see previous problem) transforms in the standard conservation equation.


Problem 9

problem id:

Show that case $f(t)=g(t)=\Lambda /3$ corresponds to $\Lambda (t)CDM$ model.