A uniform dust cloud has

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Newtonian gravitation as a field theory · Newtonian gravitation as a field theory

The Geometry of Gravity, Ch The tidal field: what free fall cannot remove

Problem

A uniform dust cloud has, in its own rest frame, the stress-energy component

T^{00} = 4.5\times10^{20}\,\mathrm{J\,m^{-3}}

, which in that frame equals its proper energy density,

T^{00} = \rho

. ((a)) Find the Newtonian mass density

\rho_{\rm m} = \rho/c^2

of the cloud. ((b)) Use Poisson's equation to find the trace

\mathcal{E}_{ii} = \nabla^2\Phi

of the tidal tensor at a point inside the cloud. ((c)) State how many of the six independent components of

\mathcal{E}_{ij}

at that point are fixed by this result, and what fixes the remaining components.

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