Blog Post 26, WS 8.1, Problem 1: Spatial Expansion

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 ∆t the space between each ball increases by ∆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 ∆t? What about Ball B?

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

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

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

This problem is very hypothetical but its a great introduction into spatial expansion of the universe.

(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 ∆t? What about Ball B? 

This pretty easy, the problem says that the distance increases by \(\Delta\)x every \(\Delta\)t, also D and B are both 1 ball away from C. Thus:

\[d_{D,C}(\Delta t)=d_{B,C}(\Delta t)=\Delta x\]

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

This requires a little bit more thought.  Since A and E are two balls away, each will move a distance 2\(\Delta\)x.  Thus:

\[d_{E,C}(\Delta t)=d_{A,C}(\Delta t)=2\Delta x\]

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

Let's look at our last two answers.  The distance travelled by a ball seems to be proportionate to the number of balls between the two endpoints.  Thus, we can write:

\[d_N(\Delta t)=N\Delta x\]
This implies that the further away an object is, the more it moves away in a time-step.

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

Velocity can be written as \(\frac{\Delta x}{\Delta t}\).  Thus we can modify our previous expression:

\[v_N(\Delta t)=N\frac{\Delta x}{\Delta t}=N\Delta v\]

Let's interpret our findings.  We can come up with 2 general rules for our forceless expanding space:

1. Objects always move away for each other

2. The farther away an object is, he faster it moves further away.  

These rules hold true in our real universe provided that no other forces act upon the objects.  For instance, distant galaxies and quasars are moving away from us faster and faster every day.  However, 

nearby galaxies such as Andromeda are close enough to be affected by the force of gravity, thus Andromeda is ready to collide with us in the Milkomeda collision.  This spatial expansion is even occurring on the space between our cells and between our very atoms, but the fundamental forces are more than strong enough to make spatial expansion not even register of such small scales.

I worked with B. Brzycki, G. Grell, and N. James on this problem.