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  Covariant version of the cosmological principle?

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Wikipedia states:

 In modern physical cosmology, the cosmological principle is the notion
 that the spatial distribution of matter in the universe is homogeneous
 and isotropic when viewed on a large enough scale, since the forces
 are expected to act uniformly throughout the universe, and should,
 therefore, produce no observable irregularities in the large-scale
 structuring over the course of evolution of the matter field that was
 initially laid down by the Big Bang.

This seems to imply some average or something related to the stress energy tensor $T_{\mu \nu}$ is periodic/ quasi periodic/ almost periodic (we do not consider the perfect fluid stress energy tensor) but instead consider:

$$T^\mu_\nu = (\rho + p)u^\mu u_\nu - p\delta^\mu_\nu - \eta(P^\mu_\gamma u^{;\gamma}_\nu+P^\gamma_\nu u^\mu_{;\gamma}-\frac{2}{3}P^\mu_\nu u^{\gamma}_{;\gamma})$$ 

where $\rho$ is the rest-frame mass density, $p$  pressure, and $u$ the four velocity, $\eta$ is the shear viscosity coefficient and $P \equiv \delta^\mu_\nu - u^\mu u_\nu$ is the projection operator.For a rough sketch of proof see here

On some length scale in the limit of a $L$ where

$${\displaystyle L=\min c\int _{P}{\sqrt {g_{\mu \nu }dx^{\mu }dx^{\nu }}}}$$

where $g_{\mu \nu}$ is the metric, $c$ is the speed of light the signature is $-+++$ between points $x_1$ and $x_2$ where $P$ is the a possible path and $\min$ is an operator which minimises.

Or some the volume scale is considered:

$${\displaystyle \mathrm {vol} _{g}=\pm {\sqrt {|\det[g_{\mu \nu }]|}}\,dx^{0}\wedge dx^{1}\wedge dx^{2}\wedge dx^{3}}$$

I am aware of the derivation of the "The Friedmann Model of the universe" I am asking for the covariant version of the cosmological principle and the error term in finite $L$ or the integrated version of $\mathrm {vol} _{g}$ is finite?

asked Nov 3 in Astronomy by Asaint (50 points) [ no revision ]

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