# Why is it that the further a galaxy is, the greater is its recessional velocity?

The exam question is:

Explain how red-shift provides evidence for the Big Bang theory.

One of the points in the answer is:

• the further away the galaxy is, the greater is their recessional speed

Using terminology that a 16-year-old will understand, why is this the case?

• Check the Hubble's law on Wikipedia, there is a good explanation. Oct 5 at 0:25
• It's a feature of uniform scaling (uniform resizing/stretching/expansion in all directions). If you multiply all distances by some scaling factor $s$ (so that the whole object is $s$ times larger), then a distance of $1$ will become $s$, but a distance of $10$ will become $10s$. See this video (and note how the "dot trails" are longer the further away from the local "center" you look): youtube.com/watch?v=W4c-gX9MT1Q Oct 5 at 15:59

Here is a simple way to visualize this.

We simplify the problem to a rubber sheet being stretched into a 2-D plane. The rubber sheet has little dots painted on it all over its surface. As it stretches, the distance between any two next-door-neighbor dots will increase in the same proportion, no matter where they are on the sheet.

Now if you look at three dots in a row on the sheet you'll see that relative to the center dot, the two outer dots are moving away from the center dot, which means the two outer dots are moving away from each other at twice the rate at which they are moving away from the dot in their midst.

So if you look at dot pairs that are further and further apart on that sheet, their recessional velocity becomes greater and greater.

And so it is in 3-D space, where the dots are galaxies, and the universe is expanding.

One major assumption of the Big Bang theory is the "cosmological principle" that the universe is the same everywhere. If the universe is the same everywhere, then every observer should measure it expanding in the same way.

Now suppose two galaxies lie along a sight line, with one galaxy being twice as far as the other. Since the universe is expanding, they are both receding.

But consider in particular that the separation between the two galaxies is the same as the nearer galaxy's distance from us. If the universe is the same everywhere, then the farther galaxy's recession speed relative to the nearer galaxy must be the same as the nearer galaxy's recession speed relative to us.

Thus, the farther galaxy must recede from us twice as fast as the nearer galaxy does.

The more distance there is between two galaxies, the bigger the effect will be from multiplying that distance by a fixed number greater than 1.

The cosmological solutions of general relativity which are consistent with a very small early universe (Big Bang) work by having a scale factor $$a(t)$$ which depends on time. Roughly speaking, we can set $$a(\text{now}) = 1$$ to phrase everything in terms of the current size of the universe. The equations break down when the universe was very small but this formally leads to the value $$a(\text{now} - 14\cdot 10^9 \text{yr}) = 0$$. Since $$a(t)$$ has been getting bigger this whole time, one should expect it to keep getting bigger. And therefore galaxies should recede by an amount proportional to their distance from us.

Probably the best way is to look at the raisin bread animation on Wikipedia. As the raisin bread expands, the raisings that are further away also show a greater recessional velocity.

If a star is moving away from the earth, we see its light red-shifted, that is the wavelength of the light is lengthened, so that a white star would appear more red. When we look at galaxies we see their light is red-shifted, so we know they are moving away from us. Light from more distant galaxies is more strongly red-shifted, so they are moving away from us more quickly.

We don't believe the earth is special, so people in other galaxies would also see other galaxies moving away from them in the same way we do.

If we imagine time running backwards we would see all the galaxies coming together. Furthermore, galaxies twice as far away are moving twice as fast, so all the galaxies would come together in the same place at the same time. This is consistent with the Big Bang theory, that the universe began with all matter and energy in one place, and expanded from there.

There is much more to the Big Bang and expansion than that, for example the galaxies are not so much moving into empty space as being separated by space itself expanding. However the idea above was fundamental in the origin of the Big Bang theory.