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In General Relativity, spacetime itself is described as a single 4-surface (a four-dimensional manifold with a Lorentzian metric)
Using the generalized definition of an n-surface (where an n-dimensional manifold is treated as a single geometric object), how many surfaces does a single four-dimensional pseudo-Riemannian manifold constitute?
The answer is 1.
The relationship between mean density, surface density, and the radius is ρ = 3σ / r
For a manifold with only 1 surface, σ is a constant.
If 3σ is a constant,
ρ ∝ 1/r
What this means is that for a constant surface density, there is a 1/r distribution of matter.
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