I believe we are on the right track now, so let us take a few more steps.
Hurkyl said:
One thing I wanted to point out is that there is no natural way to map worldlines in 3+1-space to worldlines in 4-space. What you need to work with are "pointed" worldlines; that is, you have to choose a point on the worldline acts as the origin. (e.g. for a worldline in 3+1-space, you have to choose what point corresponds to \tau = 0 before you can map it to 4-space)
Quite right. I would like to rephrase that to make sure we understand exactly where we are: Although geodesics can be mapped from 3+1- to 4-space and back, the same thing cannot be done with points, that is, if three geodesics cross at one point in 3+1-space they will normally not have a common crossing point in 4-space; this has important consequences.
What is known as an event in special relativity, a set of particular values for \langle t, x, y, z \rangle, cannot usually be univocally translated into a set of particular values for \langle x, y, z, \tau \rangle. This is, I think, why Carl proposes that the \tau be curled up in a tight helix, but he will have to explain that himself; I am sticking to flat spaces. Furthermore, it must also be noted that only the spacelike part of 3+1-space is mapped to the full 4-space. We could, if needed, make a separate mapping from the timelike part to another full 4-space.
I said above that points in 3+1-space don't
usually map to points in 4-space but now remember how I modified Bondi's approach to define both t and \tau:
t = (t_0 + t_2)/2
\tau = \sqrt{t_0 \ast t_2}
As we move the distant object closer to the observer t_0 \rightarrow t_2 and so \tau \rightarrow t. So, for the observer, there is no problem in translating from one space to the other. This means, for instance, that I can map a collision event from one space to the other, because it happens at one single point in spacetime and I can place myself, as observer, at that point. Please see ArXiv: physics/0201002 for a collision discussion.
Now let us think a little about how we translate distant or moving objects. We must restrict the discussion to objects moving on geodesics (straight lines); I will say a little bit about dynamics below. The observer's worldline is the t axis in 3+1-space and the \tau axis in 4-space; as we have seen the two measurements coincide for the observer. If an object's worldline crosses the observer's at any point, this point can be mapped from one space to the other; since the worldlines are also mapped, we just need to translate time intervals measured in 3+1-space into distances measured on the worldline in 4-space.
The problem is a bit more tricky if the object's worldline never crossed the observer's. The two definitions above should always solve the problem but there is one philosophical argument I would like to advance: If the Universe is expanding from a big bang it can be argued that all worldlines must have crossed at some time in the past, so justifying the synchronism of all clocks.
Dynamics is a problem and I would say it does not map from one space to the other. I've given some attention to the different dynamics in the two spaces and you can read about in the paper I mentioned above. However I don't think we should care too much about those differences because special relativity, as we know, is not the final answer to dynamics. When we come to discussing equivalence between general relativity and a generalization of ESR, I will deal with dynamics problems.
Jose