Difference between revisions of "Friedman-Lemaitre-Robertson-Walker (FLRW) metric"

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(Problem 3: FLRW metric)
(Problem 20: Christoffel symbols for FLRW metric)
 
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where $f(\chi)$ is a real-valued function which must satisfy the condition $f(\chi)\approx \chi $ at $\chi \rightarrow 0$ in the case of non-singular metrics.
 
where $f(\chi)$ is a real-valued function which must satisfy the condition $f(\chi)\approx \chi $ at $\chi \rightarrow 0$ in the case of non-singular metrics.
  
Let us fix the angle $\theta$ and consider the triangle $DGE$ in the plane $\theta=\pi/2$, shown on the figure:  
+
Let us fix the angle $\theta$ and consider the triangle $DGE$ in the plane $\theta=\pi/2$, shown on the figure: </p>
  
[[File:FLRW1.png|center|thumb|400px|Geometry in an isotropic and homogeneous space, the section $\theta=\p/2$.]]
+
[[File:FLRW1.png|center|thumb|400px|Geometry in an isotropic and homogeneous space, the section $\theta=\pi/2$.]]
  
Here $DH=HE=\chi$, $HA=\delta$, the distance $DE$ and angle $\gamma$ are assumed to be small. Angles $GDH$ and $GEH$ are both equal to $\gamma$ due to homogeneity and isotropy. Note also that \begin{equation}\label{21-1}
+
<p style="text-align: left;">Here $DH=HE=\chi$, $HA=\delta$, the distance $DE$ and angle $\gamma$ are assumed to be small. Angles $GDH$ and $GEH$ are both equal to $\gamma$ due to homogeneity and isotropy. Note also that \begin{equation}\label{21-1}
 
   EF \simeq EF' = f( 2\chi )\gamma = f( \chi)\beta
 
   EF \simeq EF' = f( 2\chi )\gamma = f( \chi)\beta
 
\end{equation}
 
\end{equation}
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     (d\theta^2+\sin^2\theta d\varphi^2) \right\},
 
     (d\theta^2+\sin^2\theta d\varphi^2) \right\},
 
\end{equation}
 
\end{equation}
where $k=0,\pm1$ is the sign of spatial curvature ([[#equ28n|see problem]]).
+
where $k=0,\pm1$ is the sign of [[#equ28n|spatial curvature]].
 
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   <div class="NavHead">solution</div>
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   </div>
 
   </div>
 
</div></div>
 
</div></div>
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<div id=""></div><div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 12: proper coordinates ===
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Express the FLRW metric in proper coordinates.
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  <div class="NavHead">solution</div>
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    <p style="text-align: left;">
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\begin{align}
 +
ds^2&  = dt^2  - a^2 (t)\left( {dr^2  + r^2 d\Omega ^2 } \right), \\
 +
d\Omega ^2  &= d\theta ^2  + \sin ^2 \theta d\phi ^2 ; \\
 +
R &= a(t)r, \\
 +
dR &= \dot ardt + adr, \\
 +
dr &= \frac{1}{a}\left( {dR - HRdt} \right); \\
 +
\end{align}
 +
\[ds^2  = \left[ {1 - R^2 H^2 (t)} \right]dt^2  + 2RH(t)dRdt - dR^2  - R^2 d\Omega ^2 .\]
 +
</p>
 +
  </div>
 +
</div></div>
 +
  
  
  
 
<div id="equ66n"></div><div style="border: 1px solid #AAA; padding:5px;">
 
<div id="equ66n"></div><div style="border: 1px solid #AAA; padding:5px;">
=== Problem 12: conformal time algebra ===
+
 
 +
=== Problem 13: conformal time algebra ===
 
Consider an arbitrary function of time $f(t)$ and express $\dot{f}$ and $\ddot{f}$ in terms of derivatives with respect to conformal time.
 
Consider an arbitrary function of time $f(t)$ and express $\dot{f}$ and $\ddot{f}$ in terms of derivatives with respect to conformal time.
 
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<div id="equ65"></div><div style="border: 1px solid #AAA; padding:5px;">
 
<div id="equ65"></div><div style="border: 1px solid #AAA; padding:5px;">
=== Problem 13: photon's geodesics in flat case===
+
=== Problem 14: photon's geodesics in flat case===
 
Obtain the equation of a photon's  worldline in terms of conformal time for the case of the isotropic and spatially flat Universe.
 
Obtain the equation of a photon's  worldline in terms of conformal time for the case of the isotropic and spatially flat Universe.
 
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   <div class="NavHead">solution</div>
 
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   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;">A worldline of a photon is defined by  equation $d{s^2} =0$. It is sufficient to consider only radial trajectories with the observer in the origin of coordinate frame. Using the metrics (\ref{FLRWconformal}), IT
+
     <p style="text-align: left;">A worldline of a photon is defined by  equation $d{s^2} =0$. It is sufficient to consider only radial trajectories with the observer in the origin of coordinate frame. Using the metric (\ref{FLRWconformal}), one obtains
one obtains
+
 
\[\chi =\pm\eta +const.\]</p>
 
\[\chi =\pm\eta +const.\]</p>
 
   </div>
 
   </div>
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<div id="equGeo1"></div><div style="border: 1px solid #AAA; padding:5px;">
 
<div id="equGeo1"></div><div style="border: 1px solid #AAA; padding:5px;">
=== Problem 14: photon's geodesics in general case (!) ===
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=== Problem 15: photon's geodesics in general case (!) ===
 
Derive the equations of geodesics in terms of conformal time and comoving coordinates for the case of radial motion in the FLRW metric.
 
Derive the equations of geodesics in terms of conformal time and comoving coordinates for the case of radial motion in the FLRW metric.
 
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   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
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   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;">Consider the FLRW metrics in  coordinates $(\eta,\chi,\theta,\varphi)$ (\ref{FLRWconformal}). In the case of radial motion $u^{\theta}=u^{\varphi}=0$, so one needs only the components of connection with the indices equal to $\eta$ and $\chi$. They are evaluated using the explicit formula in terms of the metrics ([[Equations_of_General_Relativity#GammaChristoffel|see eq.]]). The non-zero components are the following:
+
     <p style="text-align: left;">Consider the FLRW metrics in  coordinates $(\eta,\chi,\theta,\varphi)$ (\ref{FLRWconformal}). In the case of radial motion $u^{\theta}=u^{\varphi}=0$, so one needs only the components of connection with the indices equal to $\eta$ and $\chi$. They are evaluated using the [[Equations_of_General_Relativity#GammaChristoffel|explicit formula in terms of the metric]]. The non-zero components are the following:
 
\[{\Gamma^{\eta}}_{\eta\eta}
 
\[{\Gamma^{\eta}}_{\eta\eta}
 
={\Gamma^{\eta}}_{\chi\chi}
 
={\Gamma^{\eta}}_{\chi\chi}
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<div id="equGeo2"></div><div style="border: 1px solid #AAA; padding:5px;">
 
<div id="equGeo2"></div><div style="border: 1px solid #AAA; padding:5px;">
 
+
=== Problem 16: cosmological redshift (!) ===
=== Problem 15: cosmological redshift (!) ===
+
 
A comoving observer is the one that is at rest in the comoving coordinates. He sees the Universe as isotropic, and can also be called an isotropic observer. Show that the frequency of a photon and velocity of a free particle, as measured by a comoving observer$^*$ at time $t$, are proportional to $1/a(t)$.
 
A comoving observer is the one that is at rest in the comoving coordinates. He sees the Universe as isotropic, and can also be called an isotropic observer. Show that the frequency of a photon and velocity of a free particle, as measured by a comoving observer$^*$ at time $t$, are proportional to $1/a(t)$.
  
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=a^{2}(\tilde{u}^{\eta})^{2}.\]
 
=a^{2}(\tilde{u}^{\eta})^{2}.\]
  
The photon's frequency measured by this observer equals to ([[Equations_of_General_Relativity#equ_oto1a|see problem]])
+
The photon's frequency measured by this observer [[Equations_of_General_Relativity#equ_oto1a|equals to]]
 
\[\omega_{ph}=k_{\mu}\tilde{u}^{\mu}
 
\[\omega_{ph}=k_{\mu}\tilde{u}^{\mu}
 
=k_{\eta}\tilde{u}^{\eta}
 
=k_{\eta}\tilde{u}^{\eta}
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<div id="equ70"></div><div style="border: 1px solid #AAA; padding:5px;">
 
<div id="equ70"></div><div style="border: 1px solid #AAA; padding:5px;">
=== Problem 16: redshift and emission time ===
+
=== Problem 17: redshift and emission time ===
 
Express the detected redshift of a photon as a function of the cosmic time $t$ at the moment of its emission and vice versa: express the time $t$ and conformal time $\eta$ at the moment of its emission in terms of its registered redshift.
 
Express the detected redshift of a photon as a function of the cosmic time $t$ at the moment of its emission and vice versa: express the time $t$ and conformal time $\eta$ at the moment of its emission in terms of its registered redshift.
 
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<div id="equ69"></div><div style="border: 1px solid #AAA; padding:5px;">
 
<div id="equ69"></div><div style="border: 1px solid #AAA; padding:5px;">
=== Problem 17: scale factor and conformal time ===
+
=== Problem 18: scale factor and conformal time ===
 
Obtain the relation between the scale factor and conformal time using the properties of conformal time interval.
 
Obtain the relation between the scale factor and conformal time using the properties of conformal time interval.
 
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   <div class="NavHead">solution</div>
 
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     <p style="text-align: left;">As the comoving distance between the source and observer does not change, the conformal time interval between two light signals at the point of emission coincides with that at the point of detection ([[#equ65|see problem]]). Using the definition of conformal time $dt=a(\eta)d\eta$, one obtains
+
     <p style="text-align: left;">As the comoving distance between the source and observer does not change, the conformal time interval between two light signals at the point of emission [[#equ65|coincides with that at the point of detection]]. Using the definition of conformal time $dt=a(\eta)d\eta$, one obtains
 
\[\left. \frac{\Delta t}{a} \right|_{emit}
 
\[\left. \frac{\Delta t}{a} \right|_{emit}
 
= \left. \frac{\Delta t}{a}\right|_{obs}.\]
 
= \left. \frac{\Delta t}{a}\right|_{obs}.\]
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<div id="equ72"></div><div style="border: 1px solid #AAA; padding:5px;">
 
<div id="equ72"></div><div style="border: 1px solid #AAA; padding:5px;">
=== Problem 18: closed and open universes ===
+
=== Problem 19: closed and open universes ===
 
Is it possible for an open Universe to evolve into a closed one or vice versa?
 
Is it possible for an open Universe to evolve into a closed one or vice versa?
 
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<div id="equ27"></div><div style="border: 1px solid #AAA; padding:5px;">
 
<div id="equ27"></div><div style="border: 1px solid #AAA; padding:5px;">
=== Problem 19: Christoffel symbols for FLRW metric ===
+
=== Problem 20: Christoffel symbols for FLRW metric ===
 
Calculate all connection coefficients (Christoffel symbols) for the FLRW metric.
 
Calculate all connection coefficients (Christoffel symbols) for the FLRW metric.
 
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     <p style="text-align: left;">We use explicit expressions for ${\Gamma^{i}}_{kl}$ ([[Equations_of_General_Relativity#GammaChristoffel|see eq.]])
+
     <p style="text-align: left;">We use [[Equations_of_General_Relativity#GammaChristoffel|explicit expressions]]  for ${\Gamma^{i}}_{kl}$
\[\Gamma _{\alpha \beta }^{\mu }= \frac{1}{2}g^{\mu \nu }\left[
+
\begin{equation}\label{GammaGen}
\frac{\partial g_{\alpha \nu }}{\partial x^{\beta }}+\frac{\partial g_{\beta \nu }}{\partial x^{\alpha }}-\frac{\partial g_{\alpha \beta }}{\partial x^{\nu }} \right].\]
+
\Gamma _{\alpha \beta }^{\mu }= \frac{1}{2}g^{\mu \nu }\left[
 +
\frac{\partial g_{\alpha \nu }}{\partial x^{\beta }}+\frac{\partial g_{\beta \nu }}{\partial x^{\alpha }}-\frac{\partial g_{\alpha \beta }}{\partial x^{\nu }} \right].
 +
\end{equation}
 
Here we use Greek alphabet for spacetime indices $\alpha,\beta,\nu=0,1,2,3$ and Latin for spatial indices $i,j=1,2,3$.
 
Here we use Greek alphabet for spacetime indices $\alpha,\beta,\nu=0,1,2,3$ and Latin for spatial indices $i,j=1,2,3$.
  
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\quad g^{00}=1,
 
\quad g^{00}=1,
 
\quad g^{ij}=-\frac{1}{a^2(t)}{\gamma^{ij}}.\]
 
\quad g^{ij}=-\frac{1}{a^2(t)}{\gamma^{ij}}.\]
Using again the expression ([[Equations_of_General_Relativity#GammaChristoffel|see eq.]]), for the non-zero Christoffel symbols we obtain
+
Using again the expression (\ref{GammaGen}), for the non-zero Christoffel symbols we obtain
 
\[ \Gamma _{0j}^{i}
 
\[ \Gamma _{0j}^{i}
 
=\frac{1}{2}g^{ik}\frac{\partial g_{jk}}{\partial t}
 
=\frac{1}{2}g^{ik}\frac{\partial g_{jk}}{\partial t}
 
= \frac{\dot{a}}{a}\delta_{ij};
 
= \frac{\dot{a}}{a}\delta_{ij};
 
\quad\Gamma _{ij}^0=a\dot{a}{\gamma _{ij}}.\]
 
\quad\Gamma _{ij}^0=a\dot{a}{\gamma _{ij}}.\]
The symbols $\Gamma _{jk}^{i}$ are calculated with the help of ([[Equations_of_General_Relativity#GammaChristoffel|see eq.]]), where instead of the metric tensor $g^{\mu\nu}$ one uses the three-dimensional metric $\gamma_{ij}$. Therefore all the non-zero Christoffel symbols generated by the FLRW-metric are
+
The symbols $\Gamma _{jk}^{i}$ are calculated also with the help of (\ref{GammaGen}), where instead of the metric tensor $g^{\mu\nu}$ one uses the three-dimensional metric $\gamma_{ij}$. Therefore all the non-zero Christoffel symbols generated by the FLRW-metric are
 
\begin{align*}
 
\begin{align*}
 
&\Gamma _{11}^{0}=\frac{a\dot{a}}{1-k{{r}^{2}}};\quad
 
&\Gamma _{11}^{0}=\frac{a\dot{a}}{1-k{{r}^{2}}};\quad
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&\Gamma_{10}^1=\Gamma _{01}^1 =\frac{\dot a}{a};
 
&\Gamma_{10}^1=\Gamma _{01}^1 =\frac{\dot a}{a};
 
\quad
 
\quad
\Gamma _{11}^1 =-\frac{kr}{1-kr^2};\quad
+
\Gamma _{11}^1 =\frac{kr}{1-kr^2};\quad
 
\Gamma _{22}^1=-r(1-kr^2);\quad
 
\Gamma _{22}^1=-r(1-kr^2);\quad
 
\Gamma _{33}^1=-r(1-kr^2)\sin^2\theta;\\
 
\Gamma _{33}^1=-r(1-kr^2)\sin^2\theta;\\
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\Gamma^{3}_{31}=\Gamma^{3}_{13}=\frac 1 r ;\quad
 
\Gamma^{3}_{31}=\Gamma^{3}_{13}=\frac 1 r ;\quad
 
\Gamma^{3}_{32}=\Gamma^{3}_{23}
 
\Gamma^{3}_{32}=\Gamma^{3}_{23}
=\coth\theta.
+
=\cot\theta.
 
\end{align*}</p>
 
\end{align*}</p>
 
   </div>
 
   </div>
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<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
  
=== Problem 20: Ricci tensor and scalar ===
+
=== Problem 21: Ricci tensor and scalar ===
 
Derive the components of Ricci tensor, scalar curvature and the trace of energy-momentum tensor for the FLRW metric.
 
Derive the components of Ricci tensor, scalar curvature and the trace of energy-momentum tensor for the FLRW metric.
 
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<div id="equ28n"></div>
 
<div id="equ28n"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 21: spatial curvature ===
+
=== Problem 22: spatial curvature ===
 
Obtain the components of the Ricci tensor and scalar curvature ${}^{(3)}R$ of the spatial section $t=const$ of the FLRW metric. Show that $k=sign^{(3)}R$ if ${}^{(3)}R\neq 0$.
 
Obtain the components of the Ricci tensor and scalar curvature ${}^{(3)}R$ of the spatial section $t=const$ of the FLRW metric. Show that $k=sign^{(3)}R$ if ${}^{(3)}R\neq 0$.
 
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<div id="equ29n"></div><div style="border: 1px solid #AAA; padding:5px;">
 
<div id="equ29n"></div><div style="border: 1px solid #AAA; padding:5px;">
 
+
=== Problem 23: cosmological energy-momentum tensor ===
=== Problem 22: cosmological energy-momentum tensor ===
+
 
Derive the components and trace of the energy-momentum tensor which satisfies the cosmological principle.
 
Derive the components and trace of the energy-momentum tensor which satisfies the cosmological principle.
 
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<div class="NavFrame collapsed">
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in the comoving frame it will coincide with the energy-momentum tensor (\ref{2_equ29n}). As both quantities transform as tensors, they coincide in any other frame.
 
in the comoving frame it will coincide with the energy-momentum tensor (\ref{2_equ29n}). As both quantities transform as tensors, they coincide in any other frame.
 
<br/>
 
<br/>
The simplest way to generalize the expression (\ref{1_equ29n}) for the case of curved space is to replace the Minkowski space metric $\eta^{\mu\nu}$ by an arbitrary one $g^{\mu\nu} $. Indeed, for any given point of the spacetime there exists a locally Lorentzian  reference frame ([[Equations_of_General_Relativity#equ_oto3|see problem]]), in which the metric tensor locally coincides with the Minkowski tensor, and the energy-momentum tensor for matter takes the form (\ref{2_equ29n}). After transition to arbitrary reference frame one arrives to:
+
The simplest way to generalize the expression (\ref{1_equ29n}) for the case of curved space is to replace the Minkowski space metric $\eta^{\mu\nu}$ by an arbitrary one $g^{\mu\nu} $. Indeed, for any given point of the spacetime there [[Equations_of_General_Relativity#equ_oto3|exists a locally Lorentzian  reference frame]], in which the metric tensor locally coincides with the Minkowski tensor, and the energy-momentum tensor for matter takes the form (\ref{2_equ29n}). After transition to arbitrary reference frame one arrives to:
 
\begin{equation}
 
\begin{equation}
 
   T^{\mu\nu}= (\rho+p)u^\mu u^\nu - pg^{\mu\nu} . \label{3_equ29n}
 
   T^{\mu\nu}= (\rho+p)u^\mu u^\nu - pg^{\mu\nu} . \label{3_equ29n}

Latest revision as of 16:46, 3 February 2015

Problem 1: expanding baloon

Consider two points $A$ and $B$ on a two-dimensional sphere with radius $a(t)$ depending on time. Find the distance between the points $r_{AB}$, as measured along the surface of the sphere, and their relative velocity $v_{AB}={dr_{AB}}/{dt}$.


Problem 2: scale factor and Hubble's parameter

The comoving reference frame is defined so that matter is at rest in it, and the distance $\chi_{AB}$ between any two points $A$ and $B$ is constant. Show that in a homogeneous and isotropic Universe the proper (physical) distance $r_{AB}$ between two points is related to the comoving one as \[r_{AB}=a(t)\cdot \chi_{AB},\] where quantity $a$ is called the scale factor and it can depend on time only. Integrate the Hubble's law and find $a(t)$.


Problem 3: FLRW metric

Consider a spacetime with homogeneous and isotropic spatial section of constant time $dt=0$. Show that in the comoving coordinates its metric necessarily has the form of the Friedman-Lemaître-Robertson-Walker (FLRW)$^*$ metric: \begin{equation}\label{FLRW1} ds^2=dt^2-a^2(t) \left\{ d\chi^2+\Sigma^2(\chi) (d\theta^2+\sin^2\theta d\varphi^2)\right\}, \end{equation} where \[\Sigma^2(\chi)= \left\{\begin{array}{lcl} \sin^2\chi \\%\qquad \; \; k=+1\\ \chi^2 \\%\qquad \qquad k=0\\ \sinh^2 \chi. \\%\qquad k=-1,\\ \end{array}\right.\] The time coordinate $t$, which is the proper time for the comoving matter, is referred to as cosmic (or cosmological) time.

$^*$Depending on geographical or historical preferences, named after a subset of the four scientists: Alexander Friedman (also spelled Friedmann), Georges Lemaître, Howard Percy Robertson and Arthur Geoffrey Walker. Thus abbreviations FRW, RW or FL are also used.


Problem 4: another representation

Show that the FLRW metric (\ref{FLRW1}) can be presented in the form \begin{equation}\label{FLRW2} ds^2=dt^2-a^2(t) \left\{ \frac{dr^2}{1-kr^2}+r^2 (d\theta^2+\sin^2\theta d\varphi^2) \right\}, \end{equation} where $k=0,\pm1$ is the sign of spatial curvature.


Problem 5: sign of spatial curvature

Show that only the sign of spatial curvature has physical meaning, as renormalization of the scale factor rescales the curvature.


Problem 6: spatially flat Universe

Why is the normalization of the scale factor not fixed for a spatially flat Universe, for which $k=0$?


Problem 7: geometry of the closed Universe

Consider a closed Universe (with $k=+1$) and find the length of equator and full volume of its spatial section $dt=0$.


Problem 8: electric charge of the Universe

Present arguments in favor of the affirmation that the electric charge of a closed Universe should be exactly zero.


Problem 9: Hubble's law

Using the FLRW metric, derive the Hubble's law.


Problem 10: conformal time

Conformal time $\eta$ is defined as \[dt=a(\eta)d\eta.\] It can be interpreted as the time measured by a clock that decelerates along with the expansion of the Universe. Rewrite the FLRW metric in conformal time. Show that the logarithmic derivative of the scale factor with respect to conformal time determines its evolution in the physical time.


Problem 11: comoving-conformal coordinates

Express the FLRW metric in comoving coordinates and conformal time. Show that in the case $k=0$ it is conformally flat, i.e. it can be made flat (pseudo-Euclidean) by means of global stretching.


Problem 12: proper coordinates

Express the FLRW metric in proper coordinates.



Problem 13: conformal time algebra

Consider an arbitrary function of time $f(t)$ and express $\dot{f}$ and $\ddot{f}$ in terms of derivatives with respect to conformal time.


Problem 14: photon's geodesics in flat case

Obtain the equation of a photon's worldline in terms of conformal time for the case of the isotropic and spatially flat Universe.


Problem 15: photon's geodesics in general case (!)

Derive the equations of geodesics in terms of conformal time and comoving coordinates for the case of radial motion in the FLRW metric.


Problem 16: cosmological redshift (!)

A comoving observer is the one that is at rest in the comoving coordinates. He sees the Universe as isotropic, and can also be called an isotropic observer. Show that the frequency of a photon and velocity of a free particle, as measured by a comoving observer$^*$ at time $t$, are proportional to $1/a(t)$.

$^*$We will refer to these quantities as to the "physical" energy and momentum of a particle, to stress that they are the ones directly measured in the most natural way.


Problem 17: redshift and emission time

Express the detected redshift of a photon as a function of the cosmic time $t$ at the moment of its emission and vice versa: express the time $t$ and conformal time $\eta$ at the moment of its emission in terms of its registered redshift.


Problem 18: scale factor and conformal time

Obtain the relation between the scale factor and conformal time using the properties of conformal time interval.


Problem 19: closed and open universes

Is it possible for an open Universe to evolve into a closed one or vice versa?


Problem 20: Christoffel symbols for FLRW metric

Calculate all connection coefficients (Christoffel symbols) for the FLRW metric.


Problem 21: Ricci tensor and scalar

Derive the components of Ricci tensor, scalar curvature and the trace of energy-momentum tensor for the FLRW metric.


Problem 22: spatial curvature

Obtain the components of the Ricci tensor and scalar curvature ${}^{(3)}R$ of the spatial section $t=const$ of the FLRW metric. Show that $k=sign^{(3)}R$ if ${}^{(3)}R\neq 0$.


Problem 23: cosmological energy-momentum tensor

Derive the components and trace of the energy-momentum tensor which satisfies the cosmological principle.