Difference between revisions of "The Milne Universe"

The Friedman equations with $\rho\to 0$ reduce to \begin{align*} (\dot{a})^{2}=-k,\\ \ddot{a}=0, \end{align*} and thus we have that $k=-1$, and the solution for the scale factor is $a=t$ (or $a(t)=ct$ if we preserve $c$). The metric of the Milne Universe therefore has the form \begin{equation}\label{Milne-metric} ds^{2}=dt^{2} -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 \[R = - 6\left( \frac{\ddot a}{a} + \frac{\dot a^2}{a^2} + \frac{k}{a^2} \right)=0,\] as it should be in a spacetime with no matter whatsoever.

The velocity and Lorentz-factor of a particle expressed through its rapidity are \[v=\tanh r,\qquad \gamma=\cosh r.\] Then taking into account the time dilation \[\tau=\gamma^{-1}T=T/\cosh r,\] and the obvious $R=vT$, we obtain \begin{align} T&=\tau \cosh r,\nonumber\\ \label{Milne-Mink} R&=\tau \sinh r. \end{align} Now it is not hard to see that \[dT^{2}-dR^{2}=d\tau^{2}-\tau^{2}dr^{2},\] and the metric of the original Minkowski spacetime turns out to describe the Milne Universe: \[ds^{2}=dT^{2}-dR^{2}-R^{2}d\Omega^{2} =d\tau^{2} -\tau^{2}\big[dr^{2}+r^{2}d\Omega^{2}\big].\] 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.

a) In the Milne frame the spatial metric of the slice $\tau=const$ is \[dl^{2}_{\tau} =\tau^2 dr^2 +\tau^2 \sinh^{2}r d\Omega^2,\] so the volume element is (integrating over $d\Omega$) \[dV_{\tau}=4\pi \tau^{3}\sinh^{2}r\, dr.\] If $n_{\tau}=const$ is the particles' number density, the total number of particles within a sphere of radius (radial coordinate, to be precise) $r_{\star}$ is \[N_{\tau}(r_{\star})=4\pi\; \underbrace{n_{\tau}(\tau)\tau^{3}}_{const} \int\limits_{0}^{r_{\star}}dr\,\sinh^{2}r \xrightarrow[r_{\star}\to\infty]{}\infty,\] which is natural, as we assume the number density is constant at a given time on an unbounded hyperboloid of infinite volume. 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 \[dV_{T}=4\pi R^{2}dR =4\pi T^{3}\frac{\sinh^{2}r\,dr}{\cosh^{4}r} =4\pi \tau^{3}\frac{\sinh^{2}r\,dr}{\cosh r}.\] 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 \[dN[r,dr]=n_{\tau}dV_{\tau}=n_{T}dV_{T}\] and \[n_{T}=n_{\tau}\cosh r =n_{\tau}\gamma =\frac{n_{\tau}}{\sqrt{1-R^2 /T^2}}.\] We could have obtained this result by simple observation that the number (or mass) density in the Minkowski frame changes due to the contraction of the radial distance between the particles by the Lorentz-factor $\gamma$. However, when considering the slice $T=const$, we now also have to take into account that $\tau$ is a function of $R$: \[\tau^{2}=T^{2}-R^{2},\] so due to $n_{\tau}\tau^{3}=const$, we get \begin{equation}\label{Milne-density} n_{T}=n_{0}\frac{(T_{0}/T)^{3}} {\big[1-R^2 /T^2\big]^{2}}, \end{equation} which leads to the correct divergence of the integral for the total number of particles on the horizon $R=T$: \begin{align*} N_{T}&=\int n_{T}dV_{T} =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 =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 =4\pi\;n_{\tau}\tau^{3} \int\limits_{0}^{\infty} \sinh^{2}r\,dr\to\infty. \end{align*} In terms of mass density the divergence of $\rho_{T}$ at the horizon means that even for arbitrarily small density $\rho_{0}$ in the comoving frame the density in the laboratory frame becomes arbitrarily large close enough to the horizon. Thus in the neighborhood of the horizon we cannot actually consistently neglect the mass distribution and the curvature of spacetime it induces, and even for small $\rho_0$ we cannot define the world time $T$ in all of the originally Minkowski spacetime.