> what most people would consider to be fundamental

Most people should then have it explained to them that frame-dependent quantities are not fundamental.

This is something that arises even in their first exposure to Newtonian physics: kinetic energy is just such a frame-dependent quantity.

Generally covariant and invariant quantities are candidates for fundamental quantities.

The stress-energy tensor is, for example, generally covariant. However, one tends to want to slice it up into energy, pressure, stress, and so forth, and the fluxes of those in various directions. Those "sliced" quantities are frame-dependent.

Let's make the ship extremely tiny and have it represented as a contributor to the stress-energy tensor T at some point on the manifold, p \in M. The total of T at p is invariant. We're interested in the content of T at p that represents the ship, practically all of which will be in T_{00} when we write down exactly what axis 0 (the timelike one, unlike the spacelike 1, 2, and 3 axes) is.

Now let's move the ship. We want to shift the relevant covariant content of T at p \in M to p' \in M using parallel transport. With the mild assumption that locally at every point \in M there is a tiny patch of 4d Lorentzian spacetime, we can then talk about any path we like between p and p' as being everywhere timelike, everywhere lightlike, or neither. We still don't need coordinates; this is just a feature of a Lorentzian (sub-)manifold[1].

In perfectly flat spacetime, we have the advantage that there is exactly one everywhere-lightlike path from p to p'. However, that is often not the case when there is real spacetime curvature; and the curvature in the walls of the Alcubierre warp bubble cannot be ignored in this regard. We also have to consider curvature if p and p' are at cosmological distances, even in the non-Alcubierre/always-sublight case.

Coordinates now become useful, and here we will typically want to take a 4d Lorentzian spacetime and slice it into 3d spacelike hypersurfaces arranged by some time coordinate t_{past} < t_{0} < t_{future}. We sprinkle matter[+] on such a surface and use the initial values formulation[2] and covariant laws for the matter content, and predict how it evolves from one time coordinate to the next. In the case of a space ship, we are hoping to evolve to a final values surface, with matter sprinkled differently at some t', t < t'. It turns out that we can do this for matter that is not constrained to an always-timelike (i.e., subluminal) path -- with some mild assumptions[3] one can in principle make concrete predictions of the behaviour of e.g. a sometimes-FTL spaceship. However, what happens is generally very far from intuitive, and so should be determined by actually grinding out each infinitesimally-short-duration hypersurface from initial to final, solving each surface numerically.

Alcubierre, as it happens, has written a textbook about numerical relativity. [4]

So,

> you can change what most people would consider to be fundamental things about the journey

arises here in the choice of a slicing and a set of initial values. In effect, one is choosing an axis, calling it time, labelling it with tiny markings, and choosing a way to reflect "all space everywhere" at that particular time. One has enormous freedom with each of these choices, but some choices would be ridiculously useless to make, and others are seemingly quite sensible. However, note that our choices of initial and final values have ship @ p and ship at p' with p' at a later time coordinate. There is no ambiguity about which came first, and so no opportunity for either case in : "where it disappears from A and arrives at B instantly, and one where it goes from B to A.", where A is p and B is p'.

More rigorously, we would not specify the the slice in which ship is at B at all. We would instead specify all the content of a slice in which ship is at A, and then evolve slice by slice seeing what happens ("case A"). Or alternatively, we could specify all the content of a slice in which ship is at B, and then evolve slice by slice seeing what happens ("case B"). The time-symmetry of the physical laws of matter let us march in either direction from one slice to the next. But for a one-way journey, in "case A" we have only one slice in which we are closer to "case B", and vice-versa. If we evolve backwards from "case A", ship never reaches B. If we evolve forwards from "case B", ship never reaches A. It is only forwards from "case A" or backwards from "case B" that we recover the journey.

However, with realistically specified values surfaces, macroscopic thermodynamics come into play. One direction will have an overall increase in entropy (everywhere, from one slice to the next), the other direction will have an overall decrease in entropy. This difference sharpens as one makes the values surfaces more realistic, and as one applies finer and finer scale laws to the matter so specified. Observing B->A also means seeing eggs unscramble in frying pans at A and martinis being stirred apart into gin and vermouth at B, and many similar things everywhere one looks.

Is this fundamental? No, it's the result of having made a set of choices. However, the arrow of time is important to our experiences, and imagining aliens that experience a backwards arrow of time and interact with us and the things we see marching forward, leads to the latter being washed out by the difference in degrees of freedom (there are a lot more of them when you scramble than when you unscramble) -- by interaction with "our" matter, their arrow of time would be "corrected".

Sean Carroll had a series of blog entries on this topic some years ago [5].

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[+] Here this means the contributions to the stress-energy tensor; we may also need to supply values the left-hand-side of the Einstein Field Equations too, in cases where there is non-negligible gravitational radiation

[1] https://en.wikipedia.org/wiki/Causal_structure#Curves

[2] https://en.wikipedia.org/wiki/Initial_value_formulation_(gen...

[3] Geroch, R. AMS/IP Stud.Adv.Math. 49 (2011) 59-70 ("New Developments in Lorentzian Geometry", held in November 2009 in Berlin, DE), https://arxiv.org/abs/1005.1614 top of p.8 in preprint ("Initial-Value Formulation" subsection).

[4] https://global.oup.com/academic/product/introduction-to-31-n... (and cf. https://arxiv.org/abs/gr-qc/0412019)

[5] http://www.preposterousuniverse.com/blog/2004/10/27/the-arro... and later a series starting at http://www.preposterousuniverse.com/blog/2008/03/31/incompat...