In another post called the Life of a Well Traveled Photon, I explain the expansion of the universe by using one photon as an example. Since then, I keep thinking about a version where the Earth is expanding. In the other post, the photon travels for 12 billion years. Considering the universe is 13.7 billion years old, that photon was travelling for most of that time. It came from a galaxy that was just 3 1/2 billion light years away. But by the time that photon reached us, the galaxy got 29 billion light years away. It just so happens that the second longest interstate in the US, I-80, is 2900 miles long. It runs from New York to San Francisco. I need to replace the photon with something real slow, like an ant. Some big ants go 1 MPH. So, with a little math, you might think the journey takes 4 months. But remember that at the beginning of the journey, I-80 is much shorter. It's only 360 miles long. That's 1/8 the real distance, which matches the redshift Z=7 used in the other post. Not to give the ending away, but the trip takes 1200 hours.
Click the image to see a full sized map. Or maybe you prefer a Google map.
So let's say the Earth is born right on new years. At that time, the Earth is just a dot. But it's expanding quickly. In the first hour, New York and San Francisco go from the same place to 2 miles apart. And the Earth becomes 18 miles around (radius=3mi). Our ant doesn't begin it's journey until 2am January 8, when the Earth is 1 week and 2 hours old. At that time, New York is 360 miles away. In half a day, the ant passes through Sacramento, which is pretty good considering it would normally take 3 or 4 days at 1 MPH. The problem is New York has gotten an additional 25 miles away. At this rate, our ant will never get there. Still, it reached Reno in another day and Salt Lake City by January 13th. By now, our ant has been on the road for 120 hours. And by now, we notice something promising. I-80 has grown in length a steady rate of just over 2 MPH. But our ant has crossed a 1/4 of that road. So the road in front of it is expanding a little slower. Over the past hour, it grew by only 1 1/2 miles. Still, New York has reached 450 miles away and it's still getting farther.
Nine days later, it's January 22nd and the ant has reached Lincoln, Nebraska. That's more than half way. As depressing as it my be to hear that New York is now 507 miles away, our little ant has crossed an important mile stone. New York is finally getting closer. Even though I-80 continues to expand at over 2 MPH, the part in front of it is expanding at less than 1 MPH. So now we know the ant will get to New York. The only problem is the second half will take a lot longer, 5 more weeks. That's more than twice as long as the first half.