It is worth noting that the redshift factor indicates this galaxy is moving outwards 7 times faster than the speed of light. However, it is really framed dragged along by the expansion of space itself. There is however no situation where two galaxies in the same local region pass each other at some velocity faster than light.

So what is going on? Why is there this discrepancy? We can go back to Newton’s law of motion with gravity. The motion of an object in a gravity field is given by the kinetic energy of motion T = 1/2mv^2 and its potential energy V = -GMm/d, where M is all the other masses. The sum is then equal to the total energy

E = 1/2mv^2 – GMm/d.

We now consider the motion of a galaxy according to a scale factor a. So a distance d is scale factored in time by ad or d(t) = a(t)d_0. The velocity is then v = (da/dt)d_0, we now write as a’d_0. The mass M is all the matter from all other galaxies or from the vacuum etc in a region with a radius d, which has a volume V = (4?/3)d^3 = (4?/3)a^3d_0^3. The matter in this region has an average density ? and so M = ?V We now put all of this together in our total energy equation to get

E/m = 1/2a’^2 – (4?G?a^2/3).

This is then set to zero in order to model a cosmology with a flat space, and with more work this gives

(a’/a)^2 = 8?G?/3.

This is the Friedmann-Lemaitre-Robertson-Walker (FLRW) energy equation for the motion of a flat spatial universe. It is interesting that basic Newtonian mechanics captures something which is computed more completely in general relativity. The left hand term is written this way because the Hubble parameter H/c = (a’/a).

Assume there is a vacuum energy that fills space as a constant energy density ?. Further assume this dominates the dynamics of the universe. This equation can be represented as a differential equation

a’ = sqrt{8?G?/3}a,

which has the solution a(t) = a_0exp(t sqrt{8?G?/3}). Sometimes this is written with ? = 8?G?/3, which is the cosmological constant, or with the Hubble parameter (H/c)^2 = 8?G?/3. This the most general solution, but for small time t we can expand this by Taylor series as

a(t) =~ a_0(1 + tH/c}.

here the speed of light converts the time to a distance and so we have the Hubble relation

a(d) ~= a_0(1 + Hd).

If we carry the Taylor expansion to second order we have

a(d) ~= a_0(1 + Hd + (Hd)^2/2).

For d large enough this second order term becomes larger. The distance d is solved by a quadratic equation for a more exact result.

LC

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