I wish I had a good way of illustrating this, but anyway, doesn't the following travel strategy allow you to get anywhere in arbitrarily little time?
You're at rest at the origin of space-time, and you'd like to meet up with an object some distance away in the positive x-axis - in other words, you'd like to reach the vertical world line of that object. You begin traveling towards the object at t = 0 in the following 3 phase trip:
I. You accelerate nearly to the speed of light relative to the stationary frame of the origin and the object you're trying to reach.
II. You level out your speed and coast.
III. You decelerate right before you reach the object.
As long as the observer travels on a space-time path that's always nearly lightlike, the proper time of his journey will be almost zero. Not that he sees a finite amount of distance go by him in zero time - the proper distance between the endpoints of his nearly lightlike journey is also nearly zero, so the speed of light is never exceeded by objects passed by the traveller.
I realize that the travel time in the stationary reference frame is finite - it's the same as that of a light ray, and that during the acceleration phase (I.), the traveller will see all of this finite time go by at his destination, but he'll never experience that amount of time going by, right?
If the above reasoning is correct, then there isn't really a limit on how far you can travel in a human lifetime, right? People seem to think that the finite maximum speed c is a death knell for, say, trips to Andromeda (and beyond). But if the above is correct, then the only limit on short travel times is the energy required to accelerate (a limit which exists in a Newtonian world as well).
In short, isn't it possible (in principle) to go arbitrarily far in a human lifetime?
