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

From Universe in Problems
Jump to: navigation, search
Line 1: Line 1:
 
[[Category:Dynamics of the Expanding Universe|3]]
 
[[Category:Dynamics of the Expanding Universe|3]]
 
__NOTOC__
 
__NOTOC__
<div id="equ16"></div>
+
<div id="equ16"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 1. ===
 
=== Problem 1. ===
 
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 $v_{AB}={dr_{AB}}/{dt}$.
 
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 $v_{AB}={dr_{AB}}/{dt}$.
Line 7: Line 7:
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
[[File:2_16.jpg|center|thumb|400px|ghfhgfhgfh]]
+
[[File:2_16.jpg|center|thumb|400px|]]
 
     <p style="text-align: left;">When the radius of the sphere grows with time as $a(t)$, the angle $\theta_{AB}$ between two arbitrary points $A$ and $B$ is constant. Therefore the distance between the points changes as
 
     <p style="text-align: left;">When the radius of the sphere grows with time as $a(t)$, the angle $\theta_{AB}$ between two arbitrary points $A$ and $B$ is constant. Therefore the distance between the points changes as
 
$r_{AB}(t) = a(t)\theta _{AB}$ and relative velocity is $v_{AB} = \dot r_{AB} = \dot a(t)\theta _{AB} = \frac{\dot a}{a}r_{AB}.$
 
$r_{AB}(t) = a(t)\theta _{AB}$ and relative velocity is $v_{AB} = \dot r_{AB} = \dot a(t)\theta _{AB} = \frac{\dot a}{a}r_{AB}.$
 
On denoting $\frac{\dot a}{a} \equiv H(t),$ one recovers the Hubble's law.</p>
 
On denoting $\frac{\dot a}{a} \equiv H(t),$ one recovers the Hubble's law.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
  
  
<div id="equ17"></div>
 
  
 +
<div id="equ17"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 2. ===
 
=== Problem 2. ===
 
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
 
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
Line 36: Line 36:
 
The scale factor represents an analogue of radius of the two-dimensional sphere from the previous problem. Its normalization is arbitrary and it determines the unit of length in the comoving reference frame. If the normalization is fixed then the scale factor determines distance between objects or observers at a given moment of time. The comoving distance between them $\chi_{AB}$ is analogous to the angle $\theta_{AB}$ from the previous problem and it can be treated as a Lagrangian (comoving) coordinate of the point $B$ in the reference frame centered in point $A$.</p>
 
The scale factor represents an analogue of radius of the two-dimensional sphere from the previous problem. Its normalization is arbitrary and it determines the unit of length in the comoving reference frame. If the normalization is fixed then the scale factor determines distance between objects or observers at a given moment of time. The comoving distance between them $\chi_{AB}$ is analogous to the angle $\theta_{AB}$ from the previous problem and it can be treated as a Lagrangian (comoving) coordinate of the point $B$ in the reference frame centered in point $A$.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
  
  
<div id="equ21"></div>
+
 
 +
<div id="equ21"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 3. ===
 
=== Problem 3. ===
 
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-Lemaitre-Robertson-Walker (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-Lemaitre-Robertson-Walker (FLRW)$^*$ metric:
Line 92: Line 93:
 
(d\theta^2+\sin^2\theta d\varphi^2)\right\}.\]</p>
 
(d\theta^2+\sin^2\theta d\varphi^2)\right\}.\]</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
 +
 
  
  
<div id="equ20"></div>
+
<div id="equ20"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 4. ===
 
=== Problem 4. ===
 
Show that the FLRW metric (\ref{FLRW1}) can be presented in the form
 
Show that the FLRW metric (\ref{FLRW1}) can be presented in the form
Line 131: Line 133:
 
(d\theta^2+\sin^2\theta d\varphi^2) \right\}.\]</p>
 
(d\theta^2+\sin^2\theta d\varphi^2) \right\}.\]</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
  
  
<div id="equ57"></div>
+
 
 +
<div id="equ57"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 5. ===
 
=== Problem 5. ===
 
Show that only the sign of spatial curvature has physical meaning, as renormalization of the scale factor rescales the curvature.
 
Show that only the sign of spatial curvature has physical meaning, as renormalization of the scale factor rescales the curvature.
Line 146: Line 149:
 
where $\beta=const$. Then we can always introduce $\chi=\beta r$ and $a(t)=\alpha(t)/\beta$ and bring the metric to the canonical form (\ref{FLRW2}). Note, that this has sense only if $k\neq 0$.</p>
 
where $\beta=const$. Then we can always introduce $\chi=\beta r$ and $a(t)=\alpha(t)/\beta$ and bring the metric to the canonical form (\ref{FLRW2}). Note, that this has sense only if $k\neq 0$.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
  
  
<div id="equ56"></div>
+
 
 +
<div id="equ56"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 6. ===
 
=== Problem 6. ===
 
Why is the normalization of the scale factor not fixed for a spatially flat Universe, for which $k=0$?
 
Why is the normalization of the scale factor not fixed for a spatially flat Universe, for which $k=0$?
Line 157: Line 161:
 
     <p style="text-align: left;">It follows from the fact that in the case of a flat Universe there is no spatial scale to normalize the scale factor by.</p>
 
     <p style="text-align: left;">It follows from the fact that in the case of a flat Universe there is no spatial scale to normalize the scale factor by.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
 +
 
  
  
<div id="equ22"></div>
+
<div id="equ22"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 7. ===
 
=== Problem 7. ===
 
Consider a closed Universe (with $k=+1$) and find the length of equator and full volume of its spatial section $dt=0$.
 
Consider a closed Universe (with $k=+1$) and find the length of equator and full volume of its spatial section $dt=0$.
Line 183: Line 188:
 
\[V_{universe} = 2 \pi^2 a^3.\]</p>
 
\[V_{universe} = 2 \pi^2 a^3.\]</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
  
  
<div id="equ23"></div>
+
 
 +
<div id="equ23"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 8. ===
 
=== Problem 8. ===
 
Present arguments in favor of the affirmation that the electric charge of a closed Universe should be exactly zero.
 
Present arguments in favor of the affirmation that the electric charge of a closed Universe should be exactly zero.
Line 196: Line 202:
 
Let us fix some arbitrary electric field distribution $\vec E$ in the closed world and find the corresponding charge density $\varepsilon _e$ using the equation $ \mbox{div}\vec E = 4\pi \varepsilon _e$. It will always turn out that the total charge equals to zero, i.e. $Z = \int \varepsilon_e dV = 0$, as in the absence of infinity the lines of force always start from one charge and necessarily end on another charge of the opposite sign, which thus neutralizes the former charge.</p>
 
Let us fix some arbitrary electric field distribution $\vec E$ in the closed world and find the corresponding charge density $\varepsilon _e$ using the equation $ \mbox{div}\vec E = 4\pi \varepsilon _e$. It will always turn out that the total charge equals to zero, i.e. $Z = \int \varepsilon_e dV = 0$, as in the absence of infinity the lines of force always start from one charge and necessarily end on another charge of the opposite sign, which thus neutralizes the former charge.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
  
  
<div id="equ24"></div>
+
 
 +
<div id="equ24"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 9. ===
 
=== Problem 9. ===
 
Using the FLRW metric, derive the Hubble's law.
 
Using the FLRW metric, derive the Hubble's law.
Line 211: Line 218:
 
  = \frac{\dot a(t)}{a(t)}a(t)\chi  = H(t)D.\]</p>
 
  = \frac{\dot a(t)}{a(t)}a(t)\chi  = H(t)D.\]</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
  
  
<div id="equ64"></div>
+
 
 +
<div id="equ64"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 10. ===
 
=== Problem 10. ===
 
Conformal time $\eta$ is defined as
 
Conformal time $\eta$ is defined as
Line 233: Line 241:
 
=\frac{d\,\ln a}{d\eta}.\]</p>
 
=\frac{d\,\ln a}{d\eta}.\]</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
 +
 
  
  
<div id="equ71"></div>
+
<div id="equ71"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 11. ===
 
=== Problem 11. ===
 
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.
 
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.
Line 249: Line 258:
 
and it follows that $g_{\mu \nu } = a^2(\eta )\eta _{\mu \nu }$, where $\eta _{\mu \nu }$ is the Minkowski metric.</p>
 
and it follows that $g_{\mu \nu } = a^2(\eta )\eta _{\mu \nu }$, where $\eta _{\mu \nu }$ is the Minkowski metric.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
  
  
<div id="equ66n"></div>
+
 
 +
<div id="equ66n"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 12. ===
 
=== Problem 12. ===
 
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.
Line 268: Line 278:
 
is the Hubble' constant in conformal time.</p>
 
is the Hubble' constant in conformal time.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
  
  
<div id="equ65"></div>
+
 
 +
<div id="equ65"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 13. ===
 
=== Problem 13. ===
 
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.
Line 281: Line 292:
 
\[\chi =\pm\eta +const.\]</p>
 
\[\chi =\pm\eta +const.\]</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
 +
 
  
  
<div id="equGeo1"></div>
+
<div id="equGeo1"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 14. ===
 
=== Problem 14. ===
 
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.
Line 343: Line 355:
 
It is easy to see that for photons with $\epsilon^{2}=0$ and $u^{\chi}=\pm u^{\eta}$ the two equations coinside. The particles's momenta, both for massive and massless ones, are always conserved in coordinates $(\eta,\chi)$ (for massive particles $p_{\chi}=mc u_{\chi}$, for photons $k_{\chi}\sim u_{\chi}$). It is as it ought to be, as the FLRW metric is spatially homogeneous. It means that the photon's energy is conserved as well, but in the case of massive particles the Hubble's constant serves as a source of energy. It should be stressed that, though the obtained result is obviously physically meaningful, $u_{\eta}$ and $u_{\chi}$ are not the energy and momentum as measured by a comoving observer. Regarding this see  the next problem.</p>
 
It is easy to see that for photons with $\epsilon^{2}=0$ and $u^{\chi}=\pm u^{\eta}$ the two equations coinside. The particles's momenta, both for massive and massless ones, are always conserved in coordinates $(\eta,\chi)$ (for massive particles $p_{\chi}=mc u_{\chi}$, for photons $k_{\chi}\sim u_{\chi}$). It is as it ought to be, as the FLRW metric is spatially homogeneous. It means that the photon's energy is conserved as well, but in the case of massive particles the Hubble's constant serves as a source of energy. It should be stressed that, though the obtained result is obviously physically meaningful, $u_{\eta}$ and $u_{\chi}$ are not the energy and momentum as measured by a comoving observer. Regarding this see  the next problem.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
  
  
<div id="equGeo2"></div>
 
  
 +
<div id="equGeo2"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 15. ===
 
=== Problem 15. ===
 
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 $1/a(t)$.
Line 389: Line 401:
 
=\frac{\pi_0}{\sqrt{\pi_0^2+a^{2}}}.\]</p>
 
=\frac{\pi_0}{\sqrt{\pi_0^2+a^{2}}}.\]</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
  
  
<div id="equ70"></div>
 
  
 +
<div id="equ70"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 16. ===
 
=== Problem 16. ===
 
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.
Line 422: Line 434:
 
The integration constant is chosen so that $z\to\infty$ for $t\to0$. Thus the history of the Universe in the cosmic and conformal times is expressed in terms of redshift.</p>
 
The integration constant is chosen so that $z\to\infty$ for $t\to0$. Thus the history of the Universe in the cosmic and conformal times is expressed in terms of redshift.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
  
  
<div id="equ69"></div>
+
<div id="equ69"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 17. ===
 
=== Problem 17. ===
 
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.
Line 439: Line 451:
 
After substitution into the definition (\ref{RedshiftDefinition}), one has again $a(z) = \frac{1}{1 + z}$.</p>
 
After substitution into the definition (\ref{RedshiftDefinition}), one has again $a(z) = \frac{1}{1 + z}$.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
  
  
<div id="equ72"></div>
 
  
 +
<div id="equ72"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 18. ===
 
=== Problem 18. ===
 
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?
Line 451: Line 463:
 
     <p style="text-align: left;">No.</p>
 
     <p style="text-align: left;">No.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
  
  
<div id="equ27"></div>
+
 
 +
<div id="equ27"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 19. ===
 
=== Problem 19. ===
 
Calculate all connection coefficients (Christoffel symbols) for the FLRW metric.
 
Calculate all connection coefficients (Christoffel symbols) for the FLRW metric.
Line 515: Line 528:
 
\end{align*}</p>
 
\end{align*}</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
  
  
<div id="equ28"></div>
 
  
 +
<div id="equ28"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 20. ===
 
=== Problem 20. ===
 
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.
Line 548: Line 562:
 
\end{align*}</p>
 
\end{align*}</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
  
  
<div id="equ28n"></div>
 
  
 +
<div id="equ28n"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 21. ===
 
=== Problem 21. ===
 
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 ${}^{(3)}R\neq 0$.
Line 571: Line 586:
 
The spatial curvature can be obtained from $R$ by formal substitution $g_{\alpha\beta}\to-g_{\alpha\beta}$ and $a=const$ (thus $R^{0}_{0}$ also turns to zero). The scalar spacetime curvature then equals to $R=-6k/a^2$. If one changes the sign of the metric, then $\Gamma_{ikl}$ changes its sign too, but ${\Gamma^{i}}_{kl}=g^{ij}\Gamma_{j,kl}$ does not, as well as the curvature tensor ${R^{i}}_{klm}$ and Ricci tensor $R_{km}={R^{i}}_{kim}$, but the scalar curvature $R=g^{km}R_{km}$ changes its sign again. Then the spatial curvature equals to ${}^{(3)}R=6k/a^{2}$, and thus $k$ coincides with its sign.</p>
 
The spatial curvature can be obtained from $R$ by formal substitution $g_{\alpha\beta}\to-g_{\alpha\beta}$ and $a=const$ (thus $R^{0}_{0}$ also turns to zero). The scalar spacetime curvature then equals to $R=-6k/a^2$. If one changes the sign of the metric, then $\Gamma_{ikl}$ changes its sign too, but ${\Gamma^{i}}_{kl}=g^{ij}\Gamma_{j,kl}$ does not, as well as the curvature tensor ${R^{i}}_{klm}$ and Ricci tensor $R_{km}={R^{i}}_{kim}$, but the scalar curvature $R=g^{km}R_{km}$ changes its sign again. Then the spatial curvature equals to ${}^{(3)}R=6k/a^{2}$, and thus $k$ coincides with its sign.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
 +
 
  
  
<div id="equ29n"></div>
+
<div id="equ29n"></div><div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 22. ===
 
=== Problem 22. ===
 
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.
Line 626: Line 642:
 
\[T\equiv T_\mu ^\mu = \rho - 3p.\]</p>
 
\[T\equiv T_\mu ^\mu = \rho - 3p.\]</p>
 
   </div>
 
   </div>
</div>
+
</div></div>

Revision as of 21:43, 23 July 2012


Problem 1.

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.

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.

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-Lemaitre-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 Friedmann, Georges Lemaitre, Howard Percy Robertson and Arthur Geoffrey Walker. Thus abbreviations FRW, RW or FL are also used.


Problem 4.

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 (see problem).


Problem 5.

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


Problem 6.

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


Problem 7.

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


Problem 8.

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


Problem 9.

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


Problem 10.

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.

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.

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 13.

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


Problem 14.

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


Problem 15.

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 16.

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 17.

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


Problem 18.

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


Problem 19.

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


Problem 20.

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


Problem 21.

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 22.

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