Friedman-Lemaitre-Robertson-Walker (FLRW) metric

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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.