How do we define distance in an expanding universe?
If we say that light from a distant galaxy took 10 billion years to reach earth (10 billion light years from Earth), why isn't it true that the galaxy is actually much further away since during that 10 billion year time since the light started its journey toward us, the galaxy has continued to move still farther away?
If this has any truth, then galaxies which are farthest away and traveling close to the speed of light away from us are almost twice the distance now (almost 28 billion light years away if the universe is 14 billion years old) as they were when the light started its journey!
It is completely legitimate to say that the galaxy is farther than 10 billion light years away from Earth now -- if you're using a particular definition of the "distance" to the galaxy. Unfortunately, distance is one of those things that has an intuitive meaning in everyday life but is not so intuitive in our expanding universe! Astronomers (and other people) are not always very clear about what they mean when they talk about an object's "distance", leading to a lot of confusion about this topic. Read on for a further explanation.
First of all, the expansion of the universe doesn't consist of galaxies moving through some static space, but rather the "stretching" of the space itself. The light is moving through this expanding space and has to travel the initial distance plus whatever distance is added due to the universe's expansion during the course of the journey. It's like running on a racetrack that is being stretched -- if the racetrack started off 100 meters long but got stretched to a final length of 400 meters as you were running from start to finish, then the total distance you've run is more than 100 meters.
In fact, when you talk about the "distance" between the start and finish lines in this racetrack, you might mean several different things:
(1) You could mean 100 meters, since that's the distance when you start running; it's also what the markings on the track say the distance is.
(2) You could mean 400 meters, since that's the distance between start and finish at the moment you reach the finish line.
(3) You could mean the actual distance you've run, which is more than 100 meters (since the track stretches while you're running on it), but less than 400 meters (since some of the stretching happens on parts of the track you've already passed through).
[Thanks to a reader for pointing out the difference between (2) and (3) -- in my first attempt at answering this question I did not make any distinction between them!]
You can see from the above example that when astronomers talk about the "distance" to a faraway galaxy, there are several things they might mean! Ned Wright's Cosmology Tutorial has a comprehensive technical discussion of the different types of distances that astronomers use (though it may be a bit hard to understand if you jump into it without reading the earlier parts of his tutorial first) -- some of these distances are similar to those discussed above for the racetrack, while others are completely different. He also has some answers posted to questions that are similar to the one you are asking.
If we somehow know that "light from a distant galaxy took 10 billion years to reach Earth", as your question posits, then clearly, if we are using definition #3 we would say that the distance to the galaxy is 10 billion light years. However, if we are using definition #1 we would say that the distance to the galaxy is less than 10 billion light years (i.e. it was closer than 10 billion light years when the light was emitted), and if we are using definition #2 we would say that the distance to the galaxy is greater than 10 billion light years (i.e. it is greater than 10 billion light years right now, when the light is received by us).
As you can see, saying that a galaxy is "10 billion light years away" is an ambiguous statement! It doesn't really mean much unless you also specify what definition of distance you are using. And while definition #2 is probably the one that corresponds most closely to your intuitive feeling for what "distance" is, in astronomy that is not always the best definition! After all, the light that travels from a faraway galaxy to us is our only source of information about that galaxy, so we might care a little more about the physical distance that the light has traveled (definition #3) than how far away the galaxy is now (definition #2), since how far away the galaxy is now has no bearing on what we see when we look at the galaxy.
All the definitions of distance discussed here suffer from a bit of a practical problem, though. In order to use astronomical measurements to actually calculate any of these distances to a particular galaxy, we need to know something about the history of the universe's expansion (in other words, how did the racetrack stretch as a function of time?). Different models of the universe's expansion give different numbers for the distance to the galaxy, and although recent measurements (in particular those of the WMAP satellite) are helping us learn more about how the universe expands, we still don't know all the details.
Therefore, the most common measurement of distance that astronomers use for faraway galaxies is a lot simpler and less informative than the definitions of distance discussed above, but it is much easier to measure! This distance measurement is known as the redshift of a faraway galaxy. Astronomers take advantage of the fact that as light travels through the expanding universe, the light itself gets stretched by the same factor that the universe does, causing its wavelength to increase and its color to change and become more towards the red end of the spectrum. The redshift of the light refers to the amount by which it has been stretched and is basically a measurement of how much the universe has expanded during the light's trip from the faraway galaxy to us.
Astronomers can measure the wavelength of light that we receive on Earth, and they can also usually figure out what wavelength the light had when it was emitted, based on a knowledge of the chemical processes involved in the light's production (for some information on how this is done, see our answer to a previous question). Therefore, they can easily calculate the redshift for almost any faraway object.
In the example of the racetrack discussed above, the track has expanded by a factor of 4 (from 100 meters to 400 meters). Astronomers would say that the redshift in this case is 3 (the redshift is defined as "one less than the factor by which the universe has expanded", just so it works out that if there is no expansion at all, the redshift will be zero). If you were running on the racetrack and your body behaved like light did, you would reach the finish line and find that your body was 4 times bigger than it was when you started out!
Redshift is not a "traditional" measure of distance in the sense that we are used to. Standing at the finish line and saying that the starting line is at a redshift of 3 doesn't tell us anything about how big the track is or how far you just ran. (That's probably the reason that science journalists almost never use redshift to describe distances even though it's what astronomers use all the time -- it's not something that readers can intuitively connect with.) However, there is some meaning to the concept -- in an expanding universe, objects with larger redshifts are farther away. So if we measure one galaxy to have a redshift of 3 and another to have a redshift of 3.5, we might not have any idea how long it would take us to get to either of them in a spaceship, but at least we can say which one we could get to faster!
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