Are most galaxies expanding from us as previously calculated?

Question

In this question: Can a photon that is emitted from a denser part of the universe to a less dense part appear redshifted? The answers there were yes, yes and yes primarily. With our space less dense than most, is it possible that our own region of galaxy or universe space could be collapsing much faster, becoming denser faster more than metric expansion as a whole, to a point where it makes the rest of the universe appear to be accelerated expanding and red shifted?

Is it possable that matter in the universe is just expanding and collapsing imperceptibly in our existence on Earth?

Answer 1

It is not hard to see that the redshift due to the mass of a galaxy is not that significant. A galaxy has around $10^{11}$ stars with a mass $\sim~10^{41}kg$. We think of the photons as leaving a region at the outer reaches of a galaxy $\sim~10^5$ light years or $10^{21}$m. We now evaluate $2GM/rc^2$, with $G~=~6.7\times 10^{-11}Nm^2/kg^2$. This gives an order of magnitude estimate that $$ \frac{2GM}{rc^2}~\simeq~10^{-9}. $$ The Schwarzschild factor $1~-~2GM/rc^2$ gives the redshift as $$ \frac{\lambda_\infty}{\lambda}~=~\frac{\lim_{r\rightarrow\infty}(1~-~2GM/rc^2)}{1~-~2GM/rc^2} $$ $$ \simeq~1~-~2GM/rc^2~\simeq~1~+~10^{-9}. $$ Here $r~\rightarrow~\infty$ means the radius from the mass to the observer very far away.

This is very small and not significant as a source of redshift. The gravitational field of a galaxy is though large enough over a region of space to deflect light rays from distant sources.