We continued our discussion of how the cosmological principle limits both the spacetime metric and stress-energy tensor of our universe; these allowed forms then become input to Einstein's equations.

Last time we showed that the cosmological principle implied a spacetime metric which is block diagonal between time and spatial components, and whose spatial metric determines a spatial submanifold of constant curvature. We elaborated the defining equation of such a spacetime: $\nabla_m\ R_{ijkl} = 0$, and the resulting form of $R_{ijkl},\ R_{ij}$, and $R$, all fixed by the metric, the spatial dimension, and an overall constant $K$. In particular, $R_{ij}$ is simply proportional to $g_{ij}$. We then used a pure mathematics result to simplify our search for all such metrics: Eisenstein showed in 1949 that all spaces of constant curvature that share the same spatial dimension, metric signature, and value of $K$ are equivalent; that is, they are coordinate transformations of each other. Thus our task simplifies to finding one representative 3-dimensional Euclidean space of constant curvature, for each value of $K$.

We went through Berry's derivation (Berry p. 68-69) of such spaces, which uses isotropy explicitly in a very geometric construction. He notes that, just as we defined a circle for a 2-d surface as a collection of points at fixed distance from an origin, we can define a sphere in a 3-d manifold as again, a collection of points at fixed distance from an origin. We label such spheres about a single origin by the variable $r$, related to the sphere's surface area by $A = 4\pi r^2$ (note that, because the space is curved, $r$ is generally not the radial distance from the origin). Because space is isotropic, distances on the sphere must depend on its angular coordinates in the usual way, with distance element $r^2\, d^2\Omega$ (the coefficient must be $r^2$ to get the surface area as an integral over area elements). The full metric is then $ds^2 = f(r) \, dr^2 + r^2\, d\Omega$. Berry then calculates the Gaussian curvature of a cleverly chosen subsurface of the manifold (by isotropy all subsurfaces will give the same value), obtaining a differential equation relating $f(r)$ to $K$. By solving it and imposing the condition that flat space give $K=0$, he finds the solution

$$ds^2 = \frac{dr^2}{1-Kr^2} + r^2 d\Omega\ \ .$$

We discussed the geometry of this metric -- solving for radial distances and noting that surface area scales more slowly with radial distance for positive $K$, the same for $K=0$, and faster for negative $K$, compared to Euclidean space. We also discussed the periodicity in radial distance of the positive $K$ (closed) 3-manifold, due to circumnavigation of the entire manifold.

We then wrote the above solutions for allowed spatial metrics in comoving coordinates, so that physical distances and surface areas explicitly scale with the universe's expanding scale factor $R(t)$ (and curvature as $1/R^2$). We noted the proper time component of the metric $g_{oo}$, ending up with the Friedmann-Robertson-Walker metric, whose 3 possible comoving curvatures $k=0, \pm 1$ describe all possible spacetime metrics consistent with spatial homogeneity and isotropy in the universe.

This completes our derivation of the most general stress-energy tensor and metric, in comoving coordinates, consistent with homogeneity and isotropy of the universe. Next we will apply Einstein's equations to this $T_{\mu\nu}$ and $g_{\mu\nu}$ (called a Friedmann-Robertson-Walker universe) to obtain Friedmann's equations, then examine the dynamics and features of FRW universes.

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KB

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