New from 26-12-2014
Problem
problem id: 0202_1
Considering the radial motion of a test particle in a spatially-flat expanding Universe show that in the Newtonian limit the radial force $F$ per unit mass at a distance $R$ from a point mass $m$ is given by \[F=-\frac{m}{R^2}-q(t)H^2(t)R.\]
Thus, the force consists of the usual $1/R^2$ inwards component due to the central (point) mass $m$ and a cosmological component proportional to $R$ that is directed outwards (inwards) when the expansion of the universe is accelerating (decelerating).The latter formula has evident origin. In order to describe the cosmological expansion one commonly uses two sets of coordinates: the physical (or Euler) coordinates ($R,\theta,\varphi$) and comoving (or fixed, Lagrangian) coordinates ($r,\theta,\varphi$). The angular coordinates are the same for both sets. The two sets are related by the formula $R(t)=a(t)r$. Therefore a point which is fixed w.r.t. cosmological expansion, i.e. with constant coordinates ($r,\theta,\varphi$), has additional radial acceleration \[\left.\frac{d^2R}{dt^2}\right|_{expansion}=R\frac{\ddot a}{a}=-qH^2R.\]
Problem
problem id: 0202_2
Consider a Universe which contains no matter (or radiation), but only dark energy in the form of a non-zero cosmological constant $\Lambda$. Find the Newtonian limit the radial force $F$ per unit mass at a distance $R$ from a point mass $m$.
In this case, the Hubble parameter and, hence, the DP become time-independent and are given by $H=\sqrt{\Lambda/3}$ and $q=-1$. Thus, the force (see problem \ref{0202_1}) also becomes time-independent, \[F=-\frac{m}{R^2}+\frac13\Lambda R.\] For the case of spatially finite (i.e. non-pointlike) spherically-symmetric massive objects the letter formula is replaced by \[F=-\frac{M(R)}{R^2}+\frac13\Lambda R.\] where $M(R)$ is the total mass of the object contained within the radius $R$. If the object has the radial density $\rho(R)$ then \[M(R)=\int\limits_0^R4\pi r^2\rho(r)dr.\]
Problem
problem id: 0202_3
Derive the expression for the force obtained in the previous problem directly from the Einstein equations.
The Einstein equation with cosmological constant $\Lambda$ is \[R_{\mu\nu}-\frac12g_{\mu\nu}R=8\pi GT_{\mu\nu}+g_{\mu\nu}\Lambda.\] Contracting with $g^{\mu\nu}$, we find \[R=-8\pi GT-4\Lambda,\] where $T\equiv T_\mu^\mu$. The Einstein equation can be transformed to \[R_{\mu\nu}=8\pi G\left(T_{\mu\nu}-\frac12T\right)-g_{\mu\nu}\Lambda.\] In the Newtonian limit, one can decompose the metric tensor as \[g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},\quad h_{\mu\nu}\ll1.\] Using the parametrization \[g_{00}=1+2\Phi,\] where $\Phi$ is the Newtonian gravitational potential, we obtain in leading order w.r.t. $\Phi$, \[R_{00}\approx\frac12\vec{\nabla}^2g_{00}=\vec{\nabla}^2\Phi.\] In the inertial frame of a perfect fluid, its $4$-velocity is given by $u_\mu=(1,0)$ and we have \[T_{\mu\nu}=(\rho+p)u_{\mu\nu}-pg_{\mu\nu}={\rm diag}(\rho +p),\] where $\rho$ is the energy density and $p$ is the pressure. For a Newtonian (non-relativistic) fluid, the pressure is negligible compared to the energy density, and hence $T\approx T_{00}=\rho$. Consequently, in the Newtonian limit, the Einstein equation reduces to \[\vec{\nabla}^2\Phi=4\pi G\rho-\Lambda.\] Assuming spherical symmetry, we have \[\vec{\nabla}^2\Phi=\frac1{R^2}\frac{\partial}{\partial R}\left(R^2\frac{\partial\Phi}{\partial R}\right),\] and this equation is easily solved to obtain \[\Phi=-\frac{GM}{R}-\frac\Lambda6R^2,\] where $M$ is the total mass enclosed by the volume $4/3\pi R^3$. The corresponding gravitational field strength is given by \[F=-\vec{\nabla}\Phi=-\frac{GM}{R^2}+\frac\Lambda3R.\]
Problem
problem id: 0202_4
Although the de Sitter background is not an accurate representation of our Universe, the SCM is dominated by dark-energy in a form consistent with a simple cosmological constant. Even in the simple Newtonian case, we see immediately that there is an obvious, but profound, difference between the cases $\Lambda=0$ and $\Lambda\ne0$. In the former, the force on a constituent particle of a galaxy or cluster (say) is attractive for all values of $R$ and tends gradually to zero as $R\to\infty$ (for any sensible radial density profile). In the latter case, however, the force on a constituent particle (or equivalently its radial acceleration) vanishes at the finite radius $R_F$. Find this radius.
\[R_F=[3M(R_F)/\Lambda]^{1/3}.\] Beyond which the net force becomes repulsive. This suggests that a non-zero $\Lambda$ should set a maximum size, dependent on mass, for galaxies and clusters.
Problem
problem id: 0202_5
Show that the structure parameter $R_F$ in general is time-dependent.
The time-dependence of the cosmological force term in the general case leads to the result that the important structure parameter $R_F$ is also time-dependent. For a central point mass, this is given explicitly by \[R_F(t)\approx\left[-\frac m{q(t)H^2(t)}\right]^{1/3}.\] provided the universal expansion is accelerating, so that $q(t)$ is negative. Time-dependent cosmological force term will act essentially differently (depending on its sign) on the formation and structure of massive objects (galaxies or galaxy clusters) as compared with the simple special case of a time-independent de Sitter background (see the previous problem). At low redshifts, where the dark energy component is dominant, we might expect that the values of $R_F$ will not differ significantly from those obtained assuming a de Sitter background. Clearly, if the expansion is decelerating (forever) then the force due to the central mass $m$ and the cosmological force are both directed inwards and so there is no radius at which the total force vanishes.
Problem
problem id: 0202_6
The radius $R=R_F$ (see previous problem) does not necessarily corresponds to the maximum possible size of the galaxy or cluster: many of the gravitationally-bound particles inside $R_F$ may be in unstable circular orbits. Therefore it is important to know the so-called "outer" radius, which one may interpret as the maximum size of the object, it is the one corresponding to the largest stable circular orbit $R_S$. The radius $R_S$ may be determined as the minimum of the (time-dependent) effective potential for a test particle in orbit about the central mass Show that $R_S=4^{-1/3}R_F$.
Remaining within the frames of the Newtonian approximation (a weak gravitational field and low velocities), the equation of motion for the test particle reads \[ \ddot R\approx-\frac m{R^2}-q(t)H^2(t)R+\frac{L^2}{R^3}.\] Extrema of the effective potential in which the test particle moves occur at the $R$-values for which $d^2R/dR^2=0$, namely the solutions of \[ -\frac{C}{R^2}+\frac{R}{t_H^2}+\frac{L^2}{R^3}=0.\] Consider the function \[ y=-t_H^{-2}R^4-CR+L^2.\] This polynomial (naturally of $R>0$) has a unique extremum --- a minimum at \[R^*=R_S=\left(-\frac M{4qH^2}\right)^{1/3}=4^{-1/3}R_F.\]
Problem
problem id: 0202_7
Find maximum value of the angular momentum $L$ for given values of $M$, $q$ and $H$, at which the stable periodic orbit still exists.
The existence condition for real roots of the equation (see the previous problem) reads $y(R^*)\le0$. The critical value of the parameter $\tau$ at which the minimum of the effective potential disappears, corresponds to the condition $y(R^*)=0$. Thus the upper bound on $L$ is \[L\le\frac{3^1/2}{2^{4/3}}\left(-\frac{M^4}{qH^2}\right)^{1/6}.\]
Problem
problem id: 0202_8
If we consider $R_S(t)$ as the maximum possible size of a massive object at cosmic time $t$, and assume that the object is spherically-symmetric and have constant density, then it follows that there exists a time-dependent minimum density (due to maximum size). Find this density.
\[ \rho_{min}(t)=\frac{3m}{4\pi R^3_S(t)}= -\frac{3q(t)H^2(t)}{\pi}.\] This relation is valid only for $q(t)<0$ (accelerating expansion). For $q(t)>0$, $\rho_{min}(t)=0$.
Problem
problem id: 0202_9
Show that the ratio \(\rho_{min}(t)/\rho_{crit}(t)\) depends only on the deceleration parameter.
\[ \frac{\rho_{min}(t)}{\rho_{crit}(t)}=-8q(t).\]
Problem
problem id: 0202_10
Estimate for the current moment of time the minimum fractional density $\delta_{min}\equiv[\rho_{min}(t)-\rho_m(t)]/\rho_m(t)$.
As $\rho_m(t)=\Omega_m(t)\rho_{crit}(t)$, then \[\delta_{min}(t)=-\left[1+\frac{8q(t)}{\Omega_m(t)}\right].\] For the current moment of time $(\Omega_{m0}\approx0.3$, $q_0\approx-0.55$) one finds that $\delta_{min\,0}\approx14$.
Problem 1
problem id: 2612_1
Find dimension of the Newtonian gravitational constant $G$ and electric charge $e$ in the $N$-dimensional space.
By Gauss's theorem we have $[G]=L^NM^{-1}T^{-2}$; $[e^2]=L^NMT^{-2}$.
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.
\[L_S=\left(\frac{Ge^2}{c^4}\right)^{1/2};\] \[M_S=\left(\frac{e^2}{G}\right)^{1/2};\] \[T_S=\left(\frac{Ge^2}{c^6}\right)^{1/2}.\]
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.
\[L_S=L_{Pl}\alpha^{1/2}\] \[M_S=M_{Pl}\alpha^{1/2}\] \[T_S=T_{Pl}\alpha^{1/2}\]
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.
\[F_{max}=\frac{Mc^2}{2R_S}=\frac{Mc^2}{4MG/c^2}=\frac{c^4}{4G}\approx3\times10^{43}N;\] \[P_{max}=\frac{Mc^2}{2R_S/c}=F_{max}c=\frac{c^5}{4G}\approx9\times10^{51}W.\]
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?
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.
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 $p=w(z)\rho$.
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 \[t(z)=\int\limits_z^\infty\frac{dz'}{H(z')(1+z')}.\]
Problem 7
problem id: 2612_7
Find the solutions corrected by LQC Friedmann equations for a matter dominated Universe.
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.
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 $p=w\rho$.
\[\frac{d^2a}{dt^2}=3\left(2H^2+\frac{\ddot a}a\right)a^3=3\left[\frac23\rho-\frac16(\rho+3p)\right]a^3=\frac32(\rho-p)\varphi.\]
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.
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.
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).
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}
