The Milne Universe - Revision history

Igor: /* Problem 2: alternative derivation */

2014-02-27T00:00:17Z

‎Problem 2: alternative derivation

Igor: /* Problem 1: the solution */

2014-02-26T23:59:34Z

‎Problem 1: the solution

Cosmo All at 22:03, 12 November 2012

2012-11-12T22:03:38Z

Igor at 12:17, 14 October 2012

2012-10-14T12:17:15Z

Cosmo All at 16:34, 23 July 2012

2012-07-23T16:34:32Z

Igor: /* Problem 3: deeper relation */

2012-07-20T15:27:26Z

‎Problem 3: deeper relation

Igor at 15:26, 20 July 2012

2012-07-20T15:26:41Z

Igor at 15:25, 20 July 2012

2012-07-20T15:25:49Z

Igor at 15:23, 20 July 2012

2012-07-20T15:23:42Z

Cosmo All at 20:21, 19 July 2012

2012-07-19T20:21:41Z

@@ Line 57: / Line 57: @@
 \[ds^{2}=dT^{2}-dR^{2}-R^{2}d\Omega^{2}
 	=d\tau^{2}
-	-\tau^{2}\big[dr^{2}+r^{2}d\Omega^{2}\big].\]
+	-\tau^{2}\big[dr^{2}+\sinh^2 r d\Omega^{2}\big].\]
 As the Milne Universe is just a re-parametrized Minkowski, not only its scalar curvature is zero, but the curvature tensor as well (as should be in the absence of any matter). Its spatial slices $T=const$ are, obviously, flat, while the slices $\tau=const$, which are ''hyperboloids'' $T^{2}-R^{2}=\tau^{2}$, have constant ''negative'' (scalar) curvature $k=-1$.
@@ Line 68: / Line 68: @@
 <div id="Milne3"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 3: deeper relation ===
 Let the density of matter in the Milne Universe (in the comoving frame) be small but finite. Find the dependence of (number) density on the distance to the horizon $R=T$ in the Minkowski spacetime (the laboratory frame with regard to the experiment of the Big Bang), if the distribution in the Milne Universe is homogeneous. What is the total number of particles (galaxies) in each of the frames of reference?

@@ Line 18: / Line 18: @@
 \begin{equation}\label{Milne-metric}
 	ds^{2}=dt^{2}
-	-t^{2}\big[dr^{2}+r^{2}\sinh^{2}r d\Omega^{2}\big].
+	-t^{2}\big[dr^{2}+\sinh^{2}r d\Omega^{2}\big].
 \end{equation}
 Using the [[Friedman-Lemaitre-Robertson-Walker (FLRW) metric#equ28n|expression for the scalar curvature]] in the FLRW metric, we get
@@ Line 30: / Line 30: @@
 <div id="Milne2"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 2: alternative derivation ===
 Let us consider the Minkowski spacetime in spherical spatial coordinates $(T,R,\theta,\phi)$. Let at some moment of initial explosion a cloud of particles emerge from the origin with all possible velocities $v<c$ in all directions, which stay constant. Their mass is considered negligible, so that they do not interact and do not affect the underlying spacetime. The larger is the velocity of a particle, the further away from the origin it is at a given moment of time, so the velocity of a particle $v$, or alternatively its "rapidity"

@@ Line 1: / Line 1: @@
-[[Category:Dynamics of the Universe in the Big Bang Model|4]]
+[[Category:Dynamics of the Universe in the Big Bang Model|5]]
 __NOTOC__
 <div id="Milne1"></div>

@@ Line 20: / Line 20: @@
 	-t^{2}\big[dr^{2}+r^{2}\sinh^{2}r d\Omega^{2}\big].
 \end{equation}
-Using the expression for the scalar curvature obtained in problem \ref{equ28n} of chapter 2, we get
+Using the [[Friedman-Lemaitre-Robertson-Walker (FLRW) metric#equ28n|expression for the scalar curvature]] in the FLRW metric, we get
 \[R =  - 6\left( \frac{\ddot a}{a}
 	+ \frac{\dot a^2}{a^2} + \frac{k}{a^2} \right)=0,\]
 as it should be in a spacetime with no matter whatsoever.</p>
    </div>
-</div>
+</div></div>
 </div>
-−
@@ Line 64: / Line 62: @@
 It should be noted however, that though the (part of) spacetime is the same, the Milne and Minkowski metrics describe quite different frames of reference, and the observables in them (redshifts for example) will be different too.</p>
    </div>
-</div>
+</div></div>
 </div>
-−

@@ Line 1: / Line 1: @@
-[[Category:Dynamics of the Universe in the Big Bang Model]]
+[[Category:Dynamics of the Universe in the Big Bang Model|4]]
 __NOTOC__
 <div id="Milne1"></div>

@@ Line 90: / Line 90: @@
 b) In the Minkowski frame the spatial metric of the slice $T=const$ is
 \[dl^{2}_{T}=dR^{2}+R^{2}d\Omega,\]
-so the volume element is $dV_{T}=4\pi R^2 dR$. We can rewrite it in terms of $r$, while keeping in mind that we are talking of the slice $T=const$, and thus from (\ref{Milne-Mink}) $R=T\tanh r$ and $dR=Tdr/\cosh^{2}r$ we get
+so the volume element is $dV_{T}=4\pi R^2 dR$. We can rewrite it in terms of $r$, while keeping in mind that we are talking of the slice $T=const$, and thus from (\ref{Milne-Mink}) $R=T\tanh r$ and $dR=Tdr/\cosh^{2}r$   we get
 \[dV_{T}=4\pi R^{2}dR
 	=4\pi T^{3}\frac{\sinh^{2}r\,dr}{\cosh^{4}r}

@@ Line 114: / Line 114: @@
 	=4\pi\;n_{0}T_{0}^{3} 	\int\limits_{0}^{T}
 	\frac{R^2\,dR /T^3}{(1-R^2 /T^2)^{2}}=\\
-	&=	\Big\lwavy \xi=R/T\;\;\Big\rwavy
+	&=	\Big\| \xi=R/T\;\;\Big\|
 	=4\pi\;n_{\tau}\tau^{3} \int\limits_{0}^{1}
 	\frac{\xi^{2}\,d\xi}{(1-\xi^2)^{2}}=\\

@@ Line 60: / Line 60: @@
 	-\tau^{2}\big[dr^{2}+r^{2}d\Omega^{2}\big].\]
-As the Milne Universe is just a re-parametrized Minkowski, not only its scalar curvature is zero, but the curvature tensor as well (as should be in the absence of any matter). Its spatial slices $T=const$ are, obviously, flat, while the slices $\tau=const$, which are \emph{hyperboloids} $T^{2}-R^{2}=\tau^{2}$, have constant ''negative'' (scalar) curvature $k=-1$.
+As the Milne Universe is just a re-parametrized Minkowski, not only its scalar curvature is zero, but the curvature tensor as well (as should be in the absence of any matter). Its spatial slices $T=const$ are, obviously, flat, while the slices $\tau=const$, which are ''hyperboloids'' $T^{2}-R^{2}=\tau^{2}$, have constant ''negative'' (scalar) curvature $k=-1$.
 It should be noted however, that though the (part of) spacetime is the same, the Milne and Minkowski metrics describe quite different frames of reference, and the observables in them (redshifts for example) will be different too.</p>
@@ Line 71: / Line 71: @@
 <div id="Milne3"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 3. ===
+=== Problem 3: deeper relation ===
 Let the density of matter in the Milne Universe (in the comoving frame) be small but finite. Find the dependence of (number) density on the distance to the horizon $R=T$ in the Minkowski spacetime (the laboratory frame with regard to the experiment of the Big Bang), if the distribution in the Milne Universe is homogeneous. What is the total number of particles (galaxies) in each of the frames of reference?
 <div class="NavFrame collapsed">
@@ Line 95: / Line 95: @@
 	=4\pi \tau^{3}\frac{\sinh^{2}r\,dr}{\cosh r}.\]
-As $r$ is the \emph{comoving} coordinate, and the number of particles is conserved, it stays the same in the layer between $r$ and $r+dr$ regardless of the chosen spatial slice, so
+As $r$ is the ''comoving'' coordinate, and the number of particles is conserved, it stays the same in the layer between $r$ and $r+dr$ regardless of the chosen spatial slice, so
 \[dN[r,dr]=n_{\tau}dV_{\tau}=n_{T}dV_{T}\]
 and
@@ Line 117: / Line 117: @@
 	=4\pi\;n_{\tau}\tau^{3} \int\limits_{0}^{1}
 	\frac{\xi^{2}\,d\xi}{(1-\xi^2)^{2}}=\\
-	&=\Big\lwavy \xi=\tanh r\Big\rwavy
+	&=\Big\| \xi=\tanh r\Big\|
 	=4\pi\;n_{\tau}\tau^{3} \int\limits_{0}^{\infty}
 	\sinh^{2}r\,dr\to\infty.

@@ Line 3: / Line 3: @@
 <div id="Milne1"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1. ===
+=== Problem 1: the solution ===
 Find the solutions of the Friedman equations for the Milne Universe: the expanding Universe with $\rho\to 0$ and $k=-1$. Why is it necessarily spatially open? What is the scalar curvature of this spacetime?
 <div class="NavFrame collapsed">
@@ Line 32: / Line 32: @@
 <div id="Milne2"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 2. ===
+=== Problem 2: alternative derivation ===
 Let us consider the Minkowski spacetime in spherical spatial coordinates $(T,R,\theta,\phi)$. Let at some moment of initial explosion a cloud of particles emerge from the origin with all possible velocities $v<c$ in all directions, which stay constant. Their mass is considered negligible, so that they do not interact and do not affect the underlying spacetime. The larger is the velocity of a particle, the further away from the origin it is at a given moment of time, so the velocity of a particle $v$, or alternatively its "rapidity"
 \[r=\text{artanh}\, v
 	\equiv\tfrac{1}{2}\ln\frac{1+v}{1-v}\]
-serve as radial coordinates in the region $R<T$. Let $\tau$ be the proper time of the particle. Show that the region $R<T$ in coordinates $(\tau,r,\theta,\phi)$ \emph{is} the Milne Universe$^*$.
+serve as radial coordinates in the region $R<T$. Let $\tau$ be the proper time of the particle. Show that the region $R<T$ in coordinates $(\tau,r,\theta,\phi)$ ''is'' the Milne Universe$^*$.
 $^*$In fact, this is the way Milne in his papers of 1935 and 1948 introduced this spacetime, trying to show that Big Bang can be described by pure kinematics and in the frame of Special Theory of Relativity only. This is in general not possible, but his renowned example is very instructive.
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
@@ Line 59: / Line 60: @@
 	-\tau^{2}\big[dr^{2}+r^{2}d\Omega^{2}\big].\]
-As the Milne Universe is just a re-parametrized Minkowski, not only its scalar curvature is zero, but the curvature tensor as well (as should be in the absence of any matter). Its spatial slices $T=const$ are, obviously, flat, while the slices $\tau=const$, which are \emph{hyperboloids} $T^{2}-R^{2}=\tau^{2}$, have constant \emph{negative} (scalar) curvature $k=-1$.
+As the Milne Universe is just a re-parametrized Minkowski, not only its scalar curvature is zero, but the curvature tensor as well (as should be in the absence of any matter). Its spatial slices $T=const$ are, obviously, flat, while the slices $\tau=const$, which are \emph{hyperboloids} $T^{2}-R^{2}=\tau^{2}$, have constant ''negative'' (scalar) curvature $k=-1$.
 It should be noted however, that though the (part of) spacetime is the same, the Milne and Minkowski metrics describe quite different frames of reference, and the observables in them (redshifts for example) will be different too.</p>

@@ Line 8: / Line 8: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;"></p>
+     <p style="text-align: left;">The Friedman equations with $\rho\to 0$ reduce to
    </div>
 </div>
@@ Line 27: / Line 42: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;"></p>
+     <p style="text-align: left;">The velocity and Lorentz-factor of a particle expressed through its rapidity are
    </div>
 </div>
@@ Line 41: / Line 75: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;"></p>
+     <p style="text-align: left;">a) In the Milne frame the spatial metric of the slice $\tau=const$ is
    </div>
 </div>
 </div>