This type of expansion is more generally called homologous expansion. Ned Wright as some graphics describing various types of expansions on his cosmology site: "Distortion during Expansion of the Universe"
Stepping Through the Expansion
Paralleling the treatment in Krauss, we'll first take a few snapshots of the positions of our galaxies at three different times. The positions of the galaxies, where our observers are located, are called the co-moving frame, and are carried with the expansion. At the first time, all the points (galaxies) are located at the vertices of unit squares, so we have galaxies at (x,y)= (0,0), (1,0), (1,1), etc. which are represented by the red dots in this graphic. At each time step we expand the distance between each of the points.
|Figure 1: scale = 1.0|
At the next timestep, the distances between each of the points is expanded by a factor of 1.2. The co-moving coordinates of each galaxy remain the same, but the measured distances between the galaxies has increased.
|Figure 2: scale = 1.2|
For good measure, we'll expand the scale once more by a factor of 1.4 over the original scale. Assuming the same time between each step, this corresponds to a constant rate of expansion.
|Figure 3: scale = 1.4|
To explore this exercise yourself, print out the three graphs above on thin paper (sufficiently translucent that you can read other material trough it, or are able to hold it up to a light).
Now we overlay the three plots, lining up the dots at (0,0), representing our home galaxy, for each plot. If you've printed these out on sufficiently translucent paper, you should be able to line up the three pages (or at least two of the pages) and see through them to see the overall picture. When properly aligned, it would look something like Figure 4:
|Figure 4: Overlay at (0,0)|
Here we see an analog of the cosmological motions with how the dots overlap, corresponding to later times. The galaxy at (0,0) stays fixed, the (apparent) center of the motion. But as we look at objects away from the center, the dots marking the galaxies nearest the center are still so close, the dots corresponding to different times slightly overlap - their velocities are low. But further out, the distances between the corresponding galaxies (red-to-green-to-blue) get further apart for the same amount of time. Therefore the galaxies further away from the center appear to be moving faster. The greater separation will be in proportion to the distance of the object from the center, analogous to the Hubble law.
But suppose we want to see how things look from one of the other galaxies?
Let's adjust the plot so we observe the expansion from the galaxy at appears to start at the position (2,1) in the first graphic. We can do the same with our printed pages. Be careful to align the SAME co-moving galaxy of the observer on each page. So we overlay the corresponding point in the remaining plots and get
|Figure 5: Overlay at (2,1)|
Here we see the exact same thing as before. The galaxy at (2,1) corresponding to our observer remains motionless in this frame, while as we move further out, we see the galaxies appear to be organized concentrically around this new 'center', with low velocities near the observer galaxies and higher velocities further away.
This analogy also works in 3-dimensions.
This is why we can say the Hubble expansion cannot define a center. To any observer, located anywhere, they will perceive themselves the center of the expansion.
We're still in the Center of the Circle!
One of the popular responses to the fact demonstrated above is how we appear to be in the center of galaxy catalogs such as the SDSS survey. Note that I could have expanded the assumption of uniformity above by expanding the plot of points (galaxies) such that the edge was not visible - and the conclusions would not be altered.
Delusions of Geocentric Quantization...
However, as I demonstrated in some posts in my quantized redshifts series, a uniform distribution in space, which is magnitude limited (i.e. there are galaxies fainter than can be detected by our instruments) will exhibit a similar distribution with the galaxies which we can detect. For details, see:
- Quantized Redshifts. IX. Testing the Null Hypothesis
- Quantized Redshifts. X. Testing Our "Designer Universe"
- Quantized Redshifts XI. My Designer Universe Meets Some Data and What's Next...
That this works provides a consistency check against our other assumptions, such as the redshifts being an actual measure of distance and not intrinsic properties of the galaxies.