Distance of universe's most distant objects in relation to expansion of the universe

The most distant objects here are over 13 billion light years away, and we see them when they were only 500 million years old.

But if these galaxies were closer to us 500 million years after the big bang (and wouldn't they have to be -- since we've been expanding ever since?) then why did it take so long for the light from where they were 500 million years after the big bang to reach us? Shouldn't that light have hit earth a lot sooner since it was a lot closer back then? What principle am I missing here?

Due to the expansion of the universe there are several notions of distance in cosmology:

• Light-travel time distance. This is the 13 billion lyr number you mention. It is based on the time it took for light to travel from the distant object. This distance is "measured" between points at two different times: emission and reception of a light signal.

• Proper distance. This is the distance you would measure if you could lay out ruler sticks between the two objects. This distance is measured between two points at the same (cosmic) time and for distant objects is vastly greater than the light travel distance due to the expansion.

• Angular diameter distance. This distance is based on the apparent size of objects on the sky: the farther away the smaller they appear.

• Luminosity distance. This distance is based on how bright an object appears on the sky. The farther away the dimmer.

In a flat space that is not expanding all of these notions of distance agree. Not so in an expanding universe. Also, they agree for nearby objects but gradually become increasingly different from each other as you go farther away. The differences really only become apparent on a cosmological scale.

The only principle missing is that if you are moving away from a source of light, then you are running away from the light, and so it has to travel more to catch up to you.

Think of two runners: A is at position 0, and B is 500 meters ahead. B is not as fast as A, so eventually A will catch up to B. However, by the time this happens, B will be further along, so A will have traveled more than 500 meters. This will take more than 50 seconds if A is only going 10 meters per second.

The only way this analogy is deceptive is that galaxies are moving apart mostly because the space in between is expanding. So A, the light from a galaxy, is going at a fixed speed; B, which represents us the observers, is standing still; and the track itself is stretching everywhere.

In fact, putting in some rough numbers with my smart phone app of all things, galaxies at a redshift of 10 emitted the light we see at 490 million years post-Big Bang, some 13.4 billion years ago. The universe was 9.1% its current linear size back then, so that gives you some idea about how fast the track has expanded.