Blog Post 26, Worksheet 8.1, Problem 1

1. Before we dive into the Hubble Flow, let's do a thought experiment. Pretend that there is an infinitely long series of balls sitting in a row. Imagine that during a time interval \(\Delta t\) the space between each ball increases by \(\Delta x\). 

(a) Look at the shaded ball, Ball C, in the figure above. Imagine that Ball C is sitting still (so we are in the reference frame of Ball C). What is the distance to Ball D after time \(\Delta t\)? What about Ball B?

After some \(\Delta x\), the distance between Ball D and Ball C will be \(\Delta x\), and the distance between Ball B and Ball C will also be \(\Delta x\), but in the opposite direction.

(b) What are the distances from Ball C to Ball A and Ball E?

As the distance between Ball C and Ball A was originally twice the distance from Ball C to Ball B, the distance between Ball C and Ball A is now \(2\Delta x\). The same is true for Ball C and Ball E.

(c) Write a general expression for the distance to a ball N balls away from Ball C after time \(\Delta t\). Interpret your finding.

After some time \(\Delta t\), a ball originally N balls away from Ball C will be \(N \Delta x\) away from Ball C. This means physically that the space between any two balls x and y a distance N apart will also be \(N \Delta x\) away from each other.

(d) Write the velocity of a ball N balls away from Ball C during \(\Delta t\). Interpret your finding. 

The velocity of a ball N balls away from Ball C during this time interval is simply given by \(\frac{\text{distance}}{\text{time}}\): \[v = \frac{N\Delta x}{\Delta t}\] As the distance between Ball C and some other ball increases, the velocity of that ball increases. This means that in a uniformly expanding universe, objects farther away from an object appear to be moving at a greater velocity.