Friedman-Lemaitre-Robertson-Walker (FLRW) metric - Revision history

Igor: /* Problem 20: Christoffel symbols for FLRW metric */

2015-02-03T16:46:53Z

‎Problem 20: Christoffel symbols for FLRW metric

Igor: /* Problem 20: Christoffel symbols for FLRW metric */

2014-02-26T09:14:34Z

‎Problem 20: Christoffel symbols for FLRW metric

Igor: /* Problem 12: proper coordinates */

2012-11-10T17:42:30Z

‎Problem 12: proper coordinates

Cosmo All at 20:00, 5 November 2012

2012-11-05T20:00:33Z

Cosmo All: /* Problem 12: proper coordinates */

2012-11-05T19:58:52Z

‎Problem 12: proper coordinates

Cosmo All at 19:58, 5 November 2012

2012-11-05T19:58:17Z

Igor: /* Problem 19: Christoffel symbols for FLRW metric */

2012-10-11T18:20:05Z

‎Problem 19: Christoffel symbols for FLRW metric

Igor: /* Problem 19: Christoffel symbols for FLRW metric */

2012-10-11T18:00:45Z

‎Problem 19: Christoffel symbols for FLRW metric

Igor at 17:59, 11 October 2012

2012-10-11T17:59:27Z

Igor: /* Problem 3: FLRW metric */

2012-10-11T17:45:03Z

‎Problem 3: FLRW metric

@@ Line 554: / Line 554: @@
 	\Gamma^{3}_{31}=\Gamma^{3}_{13}=\frac 1 r ;\quad
 	\Gamma^{3}_{32}=\Gamma^{3}_{23}
-	=\coth\theta.
+	=\cot\theta.
 \end{align*}</p>
    </div>

@@ Line 543: / Line 543: @@
 	&\Gamma_{10}^1=\Gamma _{01}^1 =\frac{\dot a}{a};
 	\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 _{33}^1=-r(1-kr^2)\sin^2\theta;\\
@@ Line 563: / Line 563: @@
 <div id="equ28"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === 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.

@@ Line 269: / Line 269: @@
 <div id=""></div><div style="border: 1px solid #AAA; padding:5px;">
 === Problem 12: proper coordinates ===
-Express FRW metric in proper coordinates.
+Express the FLRW metric in proper coordinates.
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">$$\begin{array}{l}
+     <p style="text-align: left;">
-  ds^2  = dt^2  - a^2 (t)\left( {dr^2  + r^2 d\Omega ^2 } \right), \\
+\begin{align}
-  d\Omega ^2  = d\theta ^2  + \sin ^2 \theta d\phi ^2 ; \\
+  ds^2&  = dt^2  - a^2 (t)\left( {dr^2  + r^2 d\Omega ^2 } \right), \\
-  R = a(t)r, \\
+  d\Omega ^2  &= d\theta ^2  + \sin ^2 \theta d\phi ^2 ; \\
-  dR = \dot ardt + adr, \\
+  R &= a(t)r, \\
-  dr = \frac{1}{a}\left( {dR - HRdt} \right); \\
+  dR &= \dot ardt + adr, \\
-  \end{array}$$
+  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 .
+\[ds^2  = \left[ {1 - R^2 H^2 (t)} \right]dt^2  + 2RH(t)dRdt - dR^2  - R^2 d\Omega ^2 .\]
 \]
 </p>
    </div>
@@ Line 291: / Line 290: @@
 <div id="equ66n"></div><div style="border: 1px solid #AAA; padding:5px;">
 === 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.

@@ Line 267: / Line 267: @@
-<div id="equ71"></div><div style="border: 1px solid #AAA; padding:5px;">
+<div id=""></div><div style="border: 1px solid #AAA; padding:5px;">
 === Problem 12: proper coordinates ===
 Express FRW metric in proper coordinates.
@@ Line 291: / Line 291: @@
 <div id="equ66n"></div><div style="border: 1px solid #AAA; padding:5px;">
+=== Problem 13: conformal time algebra ===
-=== Problem 12: 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.
 <div class="NavFrame collapsed">
@@ Line 312: / Line 311: @@
 <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.
 <div class="NavFrame collapsed">
@@ Line 325: / Line 324: @@
 <div id="equGeo1"></div><div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 14: photon's geodesics in general case (!) ===
+=== 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.
 <div class="NavFrame collapsed">
@@ Line 388: / Line 387: @@
 <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 $1/a(t)$.
@@ Line 435: / Line 433: @@
 <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.
 <div class="NavFrame collapsed">
@@ Line 467: / Line 465: @@
 <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.
 <div class="NavFrame collapsed">
@@ Line 485: / Line 483: @@
 <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?
 <div class="NavFrame collapsed">
@@ Line 497: / Line 495: @@
 <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.
 <div class="NavFrame collapsed">
@@ Line 565: / Line 563: @@
 <div id="equ28"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
+=== Problem 21: Ricci tensor and scalar ===
-=== Problem 20: Ricci tensor and scalar ===
 Derive the components of Ricci tensor, scalar curvature and the trace of energy-momentum tensor for the FLRW metric.
 <div class="NavFrame collapsed">
@@ Line 599: / Line 596: @@
 <div id="equ28n"></div>
 <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 ${}^{(3)}R\neq 0$.
 <div class="NavFrame collapsed">
@@ Line 622: / Line 619: @@
 <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.
 <div class="NavFrame collapsed">

@@ Line 273: / Line 273: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">$\begin{array}{l}
+     <p style="text-align: left;">$$\begin{array}{l}
   ds^2  = dt^2  - a^2 (t)\left( {dr^2  + r^2 d\Omega ^2 } \right), \\
   d\Omega ^2  = d\theta ^2  + \sin ^2 \theta d\phi ^2 ; \\
@@ Line 279: / Line 279: @@
   dR = \dot ardt + adr, \\
   dr = \frac{1}{a}\left( {dR - HRdt} \right); \\
-  \end{array}$
+  \end{array}$$
 \[
 ds^2  = \left[ {1 - R^2 H^2 (t)} \right]dt^2  + 2RH(t)dRdt - dR^2  - R^2 d\Omega ^2 .
@@ Line 291: / Line 291: @@
 <div id="equ66n"></div><div style="border: 1px solid #AAA; padding:5px;">
 === Problem 12: 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.

← Older revision		Revision as of 19:58, 5 November 2012
Line 264:		Line 264:
	</div>		</div>
	</div></div>		</div></div>
		+
		+
		+
		+	<div id="equ71"></div><div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 12: proper coordinates ===
		+	Express FRW metric in proper coordinates.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">$\begin{array}{l}
		+	ds^2 = dt^2 - a^2 (t)\left( {dr^2 + r^2 d\Omega ^2 } \right), \\
		+	d\Omega ^2 = d\theta ^2 + \sin ^2 \theta d\phi ^2 ; \\
		+	R = a(t)r, \\
		+	dR = \dot ardt + adr, \\
		+	dr = \frac{1}{a}\left( {dR - HRdt} \right); \\
		+	\end{array}$
		+	\[
		+	ds^2 = \left[ {1 - R^2 H^2 (t)} \right]dt^2 + 2RH(t)dRdt - dR^2 - R^2 d\Omega ^2 .
		+	\]
		+	</p>
		+	</div>
		+	</div></div>
		+

@@ Line 479: / Line 479: @@
    <div style="width:100%;" class="NavContent">
      <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[
 Here we use Greek alphabet for spacetime indices $\alpha,\beta,\nu=0,1,2,3$ and Latin for spatial indices $i,j=1,2,3$.
@@ Line 507: / Line 509: @@
 	\quad g^{00}=1,
 	\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
 \[ \Gamma _{0j}^{i}
 	=\frac{1}{2}g^{ik}\frac{\partial g_{jk}}{\partial t}
 	= \frac{\dot{a}}{a}\delta_{ij};
 	\quad\Gamma _{ij}^0=a\dot{a}{\gamma _{ij}}.\]
-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*}
 	&\Gamma _{11}^{0}=\frac{a\dot{a}}{1-k{{r}^{2}}};\quad

@@ Line 111: / Line 111: @@
      (d\theta^2+\sin^2\theta d\varphi^2) \right\},
 \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]].
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
@@ Line 293: / Line 293: @@
    <div class="NavHead">solution</div>
    <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>
    </div>
@@ Line 307: / Line 306: @@
    <div class="NavHead">solution</div>
    <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:
 \[{\Gamma^{\eta}}_{\eta\eta}
 	={\Gamma^{\eta}}_{\chi\chi}
@@ Line 387: / Line 386: @@
 	=a^{2}(\tilde{u}^{\eta})^{2}.\]
-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}
 	=k_{\eta}\tilde{u}^{\eta}
@@ Line 449: / Line 448: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <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
 \[\left. \frac{\Delta t}{a} \right|_{emit}
 	= \left. \frac{\Delta t}{a}\right|_{obs}.\]
@@ Line 624: / Line 623: @@
 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 [[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}
    T^{\mu\nu}= (\rho+p)u^\mu u^\nu - pg^{\mu\nu} . \label{3_equ29n}

@@ Line 70: / Line 70: @@
 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 $\theta=\pi/2$.]]
-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
 \end{equation}