"If someone forces you to go one mile, go with him two miles." The note to Matthew 5:41 in my NIV Study Bible says that the word for "forces" means "pressed into service" and that the same verb is used in Matthew 27:32, where the Roman soldiers press Simon into service to carry Jesus' cross. This idea reminds me of what Jesus says in Matthew 11:28-30:
"Come to me all you who are weary and burdened, and I will give you rest. Take my yoke upon you and learn from me, for I am gentle and humble in heart, and you will find rest upon your souls. For my yoke is easy and my burden is light."
Is there some connection between not simply doing what you have to do, but doing that obligatory thing beyond its obligation, and finding rest? Going the second mile sounds a lot like extra work to me, not rest. But not so, according to the physics of the Kingdom of Heaven.
I read where, during Jesus time, people would mark with stakes from their doorstep or the village limit the beginning and the end of a mile. That way, if one is "pressed into service", he and the soldier know just how far the civilian need carry the soldier's burden. The Roman soldier's own command structure forbade his forcing anyone further. So there would be no second mile mile-marker. The second mile was endless, infinite. Or, at least, the second mile was whatever the relationship seemed to require. It was a complete journey. Like forgiving "seven times seventy". The injunction to forgive doesn't mean we stop forgiving at 490 times, it means we stop forgiving when there is nothing further to forgive.
We put our burden down, then, when we reach with that "other", whose burden we assumed under obligation but then carried further out of a sort of love, the end of the journey. Perhaps by then it is no longer a burden. The rest Jesus promises does not begin at the end of the journey, but some time during the journey. Probably it begins to develop at the beginning of the second mile, when the relationship with the one to whom we are obligated starts to be transformed by our service.