Light Propagation

 

Consider an observer located at the origin O of our coordinate system. By the Cosmological Principle this assumption is without loss of generality. This observer sees light coming from a distant galaxy P located at the point . We want to describe the path of the light ray that we observe. We impose the null radial condition on the Robertson-Walker line element

 

(34.1)                                                              

 

as follows

 

(34.2)

 

 

This leads to the differential relation

(34.3)

 

The + sign is associated with a receding light ray and the - sign is associated with an approaching light ray.  We assume that the galaxy P releases its ray of light at coordinate time  and that we observe the light ray at coordinate time . We then set up the integration as follows

 

 

 

(34.4)

 

 

The right hand side can be integrated for the three curvature cases:

 

(34.5)

 

 

 

 

(34.6)

 

 

(34.7)

 

 

We now consider two closely timed light rays coming from galaxy P: one released at time  and the next released at the next wave crest time . The observer receives these rays at respective times  and . Assuming little local motion of galaxy P we get the following equivalence

 

 

 

 

(34.8)

 

By using intermediate integration limits it can easily be shown that

 

 

(34.9)

 

 

 

We can take the scale factor function out of the integral in the last relation if we assume that the scale factor is not changing very much in the light travel time-intervals that we are considering. With that assumption the above relation implies that

 

(34.10)

 

 

 

This implies that

 

 

(34.11)

 

 

The intervals  and  are the proper time intervals between the rays as measured respectively in the observers frame and the galaxy P's frame. We have assumed a special ‘go with the flow’ coordinate system where each fundamental particle (i.e. galaxy) is at rest relative to the homogeneous time slice. Hence all galaxies have coordinates with

 

(34.12)

 

It then follows that the coordinate time for each particle must be the proper time since

 

 

(34.13)

 

 

in these co-moving coordinates.

 

Since we linked the small time differences with consecutive wave crests we can introduce the frequency, , of the light waves as follows:

 

 

(34.14)

 

 

We can relate this ratio expression to the redshift parameter z using the definition

 

 

(34.15)

Note that

 

(34.16)

 

and

 

(34.17)

 

 

Since

(34.18)

 

we get

(34.19)

 

 

Note that in an expanding universe we have the condition

 

 

Thus an expanding universe is associated with a redshift since we have .

 

Derivation of the Hubble Redshift Relation

 

Our redshift formula can be written as

(34.20)

 

We now make the assumption that cosmologically our observer in galaxy O is not too far away from galaxy P on cosmological length scales.

 

(34.21)

 

The redshift equation then becomes

(34.22)

 

 

 

Then using a Taylor expansion of the bottom term and expanding expanding to first order we find

 

 

(34.23)

 

 

Rearranging we get to first order the following relation

 

(34.24)

 

 

 

 

We now find how this redshift equation relates to the distance between the two galaxies. We know that the following is true:

 

 

(34.25)

 

 

 

Since we assume that cosmologically  is very small 

 

 

(34.26)

 

Expanding and rearranging we get the following relation

 

 

 

(34.27)

 

 

 

We have shown previously that this integral relates to a function of r as follows:

 

 

 

(34.28)

 

 

If the distance value  is cosmologically small then the three terms on the right hand side of the previous equation are all essentially . Therefore we now get the following relation which can be used in the redshift equation.

 

(34.29)

 

The redshift equation becomes

 

(34.30)

 

This is the famous Hubble redshift formula. It says that for a given cosmological epoch the red shift z of a particular galaxy is proportional to the cosmological distance to that galaxy. Click here for more information about Edwin Hubble. Below is a picture of Edwin Hubble (1889- 1953).