The Milne Universe

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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?

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?