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

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(Problem 22.)
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\end{equation}
 
\end{equation}
 
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/>
 
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 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:
 
\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}
 
\end{equation}
 
\end{equation}
It is worth noting that in general the expression (\ref{3_equ29n}) is valid only in the case of weak gravity;  otherwise the expression for the energy-momentum tensor may contain additional terms depending on the curvature tensor.
+
It is worth noting that in general the expression (\ref{3_equ29n}) is valid only in the case of weak gravity;  otherwise the expression for the energy-momentum tensor may contain additional terms depending on the curvature tensor.<br/>
  
 
Note also that the simplest way to present explicitly the components of energy-momentum tensor (\ref{3_equ29n}) is the following
 
Note also that the simplest way to present explicitly the components of energy-momentum tensor (\ref{3_equ29n}) is the following

Revision as of 11:10, 24 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.