Difference between revisions of "The Milne Universe"

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[[Category:Dynamics of the Universe in the Big Bang Model]]
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[[Category:Dynamics of the Universe in the Big Bang Model|5]]
 
__NOTOC__
 
__NOTOC__
 
<div id="Milne1"></div>
 
<div id="Milne1"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1. ===
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=== 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?
 
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?
 
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\begin{equation}\label{Milne-metric}
 
\begin{equation}\label{Milne-metric}
 
ds^{2}=dt^{2}
 
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}
 
\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}
 
\[R =  - 6\left( \frac{\ddot a}{a}
 
+ \frac{\dot a^2}{a^2} + \frac{k}{a^2} \right)=0,\]
 
+ \frac{\dot a^2}{a^2} + \frac{k}{a^2} \right)=0,\]
 
as it should be in a spacetime with no matter whatsoever.</p>
 
as it should be in a spacetime with no matter whatsoever.</p>
 
   </div>
 
   </div>
</div>
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</div></div>
</div>
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<div id="Milne2"></div>
 
<div id="Milne2"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 2. ===
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 +
=== 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"
 
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
 
\[r=\text{artanh}\, v
 
\equiv\tfrac{1}{2}\ln\frac{1+v}{1-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.
 
$^*$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.
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   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
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\[ds^{2}=dT^{2}-dR^{2}-R^{2}d\Omega^{2}
 
\[ds^{2}=dT^{2}-dR^{2}-R^{2}d\Omega^{2}
 
=d\tau^{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 \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 ''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>
 
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>
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</div></div>
</div>
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+
  
  
 
<div id="Milne3"></div>
 
<div id="Milne3"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 3. ===
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 +
=== 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?
 
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?
 
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b) In the Minkowski frame the spatial metric of the slice $T=const$ is
 
b) In the Minkowski frame the spatial metric of the slice $T=const$ is
 
\[dl^{2}_{T}=dR^{2}+R^{2}d\Omega,\]
 
\[dl^{2}_{T}=dR^{2}+R^{2}d\Omega,\]
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
 
\[dV_{T}=4\pi R^{2}dR
 
=4\pi T^{3}\frac{\sinh^{2}r\,dr}{\cosh^{4}r}
 
=4\pi T^{3}\frac{\sinh^{2}r\,dr}{\cosh^{4}r}
 
=4\pi \tau^{3}\frac{\sinh^{2}r\,dr}{\cosh r}.\]
 
=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
 
\[dN[r,dr]=n_{\tau}dV_{\tau}=n_{T}dV_{T}\]
 
\[dN[r,dr]=n_{\tau}dV_{\tau}=n_{T}dV_{T}\]
 
and
 
and
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=4\pi\;n_{0}T_{0}^{3} \int\limits_{0}^{T}
 
=4\pi\;n_{0}T_{0}^{3} \int\limits_{0}^{T}
 
\frac{R^2\,dR /T^3}{(1-R^2 /T^2)^{2}}=\\
 
\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}
 
=4\pi\;n_{\tau}\tau^{3} \int\limits_{0}^{1}
 
\frac{\xi^{2}\,d\xi}{(1-\xi^2)^{2}}=\\
 
\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}
 
=4\pi\;n_{\tau}\tau^{3} \int\limits_{0}^{\infty}
 
\sinh^{2}r\,dr\to\infty.
 
\sinh^{2}r\,dr\to\infty.

Latest revision as of 00:00, 27 February 2014


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?


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 $(\tau,r,\theta,\phi)$ 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.


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?