Question

Hubble’s law can be stated in vector form. Outside the local group of galaxies, all objects are moving away from us with velocities proportional to their positions relative to us. In this form, it sounds as if our location in the Universe is specially privileged. Prove that Hubble’s law is equally true for an observer
elsewhere in the Universe. Proceed as follows. Assume we are at the origin of coordinates, one galaxy cluster is at location and has velocity relative to us, and another galaxy cluster has position vector and velocity Suppose the speeds are nonrelativistic. Consider the frame of reference of an
observer in the first of these galaxy clusters.

(a) Show that our velocity relative to her, together with the position vector of our galaxy cluster from hers, satisfies Hubble’s law.

(b) Show that the position and velocity of cluster $2$ relative to cluster $1$ satisfy Hubble’s law.

Answer 1

In our frame of reference, Hubble’s law is exemplified by ${\overrightarrow{v}}_1=H{\overrightarrow{R}}_1$ and ${\overrightarrow{v}}_2=-H{\overrightarrow{R}}_2$.
(a) From the first equation ${\overrightarrow{v}}_1=H{\overrightarrow{R}}_1$ we may form the equation $-{\overrightarrow{v}}_1=-H{\overrightarrow{R}}_1$ . This equation expresses Hubble’s law as seen by the observer in the first galaxy cluster, as she looks at us to find our velocity relative to her (away from her) is $-{\overrightarrow{v}}_1=H(-{\overrightarrow{R}}_1)$.
(b) From both equations, we may form the equation ${\overrightarrow{v}}_2-{\overrightarrow{v}}_1=H({\overrightarrow{R}}_2-{\overrightarrow{R}}_1)$ . This equation expresses Hubble’s law as seen by the observer in the first galaxy cluster, as she looks at cluster two to find the relative velocity of cluster 2 relative to cluster 1 is ${\overrightarrow{v}}_2-{\overrightarrow{v}}_1=H({\overrightarrow{R}}_2-{\overrightarrow{R}}_1)$ .