Introduction to growth in infinite directions

We are bounded in a nutshell of Infinite space: Blog Post #26, Worksheet # 8.1, Problem #1: Introduction to growth in infinite directions

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?

(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 \(\Delta t\). Interpret your finding.

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

(a) As Edwin Hubble trained his telescope onto the greater heavens, objects thousands if not hundreds of thousands of light years away, he began to see the hints of something in play in the universe, something few had ever thought of conceiving as possible. Since ancient times, the Earth was the center of the cosmos, with everything rotating around it, an idea disproven during the Scientific Revolution. But even these great minds did not envision how humanity would later understand how there were galaxies and clusters and unimaginably large structures in the cosmos which left Hubble with a clear idea: the universe was growing, every second, of every day. The expansion Hubble saw can be described by various methods, mainly by his own Hubble Constant, but more on that in the next post.

First let us discuss how this expansion works, how objects that are spaced from each other all move at a same speed, but because of their original place, they continually look farther and farther away.

In the example of a row of balls, we have the question of how much distance separates B and D from C, in its point of view. For each case, the distance to B or D is \(\Delta x\), as is established by the question after a time \(\Delta t\).

(b) The same reasoning applies to balls A and E, where if the space from B to A has to be \(\Delta x\) and if the distance from C to B is already \(\Delta x\), then the distance from C to A has to be \(2 \Delta x\), a case identical to ball E.

(c) So, if the balls’ position away from the point of reference is the guiding force driving the distance it has after a time \(\Delta t\), a ball N balls away will always be \(N \Delta x\) away after a time \(\Delta t\). So the equation for the distance from the ball C will always be: \[D(\Delta t) = N\Delta x .\] This shows, and we’ll get more into this in just a bit, that the farther an object was at the start, the farther it will move after a time \(\Delta t\)

(d) Now that we have a general equation for the distance from the point of reference, we can use it to find the velocity of the object for any original condition. From kinematics we already know the equation for distance traveled by an object with constant velocity is: \[X_f = X_i + v(\Delta t) ,\] which can be rewritten as: \[ v = \frac{X_f – X_i }{\Delta t} ,\] \[ v = \frac{\Delta x}{\Delta t}, \] which happens to be very similar to the equation for distance we have already found. Now we simply plug in the actual distance equation for the case of a constant expansion as we have been describing: \[v = \frac{N \Delta x}{\Delta t}, \] which describes the velocity of an object we are viewing from a particular reference point. From it, we can clearly see that an object that is farther away is perceived to be moving increasingly faster, a fact which has helped astronomers many times.

As we will see in the next post, apparent and actual velocities help a great deal, permitting us to understand just how far away an object is (giving us a new rung on the distance ladder to use as necessary), as well as how long its light has been traveling, but more on this next time.