Whenever a discussion thread turns to time travel, somebody will always raise the challenging question: If you jump forward in time, say by a week, shouldn't you reappear in outer space because the Earth has continued to move and is now seven days further along in its orbit?
And I always feel obliged to answer: Time travel does not work like that.
This conception of time travel typically imagines a time machine as a box of some kind that jumps in time but stays in the same place. And written into that conception are some assumptions that mislead. One is the assumption that you can "jump" discontinuously from one point in time to another. A second is the assumption that movement in time can be considered separately from movement in space. And a third is the implication that the expression "in the same place" is well-defined: as Relativity taught us, we have to ask "the same place relative to what? The Earth? The sun? The center of the galaxy?" (and once we answer that question, we realize that we have pretty much answered our own conundrum.) To properly understand what should happen in time travel, we need to overturn each of those assumptions.
So as far as we know* there is no possibility of
"a box that travels through time but doesn't travel through space".
It's not just hard, it's an oxymoron -- although in order to understand why,
you have to teach yourself to think about spacetime properly, instead of
thinking of space and time as separable.
So we have to take a few steps back, discard the presumption that
such a box exists, and ask what an actual time machine could be like. And it
turns out that (realistic) time travel is a characteristic of a region of
spacetime, not a characteristic of a particular machine.
The first thing to understand is that nothing ever moves through space alone
or through time alone**. Everything is moving through spacetime, all the time
(so to speak). And that movement is always continuous. It simply doesn't make
sense to say that a particle X moved from time t1 to time t2 without specifying
space coordinates too. The only thing that makes sense is to say that X moved
from spacetime position (x1, y1, z1, t1) to (x2, y2, z2, t2). This is so basic that, Relativity shows, even basic concepts like "before", "after", and "simultaneous" are not well defined if you don't specific both time and space coordinates.
The second thing to understand is that physicists have dreamed up lots of
ways that time travel might be possible, and all of them involve a continuous
path through spacetime, not a mysterious jump. The trick is, the spacetime curves in such a way that
when you return to the same coordinates in space, you arrive at a different
time from, say, an identical twin who stayed home. However, at no point do you
perceive anything "weird" or discontinuous happening. Think of it as
being like an extreme version of the famous Twins Paradox from Relativity. In
that "paradox" you leave home with a clock that reads 2pm, travel in
a big loop at almost the speed of light, come home, and your clock now reads
3pm while an identical clock that stayed home reads 4pm. (This happens all the
time, on a smaller scale, right here on Earth: the GPS satellites you rely on
for satnav have to correct for the fact that their atomic clocks drift slightly
from identical clocks on the surface of the Earth). If the spacetime path you
follow is sufficiently warped, when you get home your clock might read 3pm while the one on Earth reads 1pm -- congratulations, you've traveled an hour
into your own past!
Of course, we should also note that when you return from
your big loop you might be a little surprised to discover that the Earth isn't
where you expected it to be, it's actually a little earlier in its orbit from
where you would calculate according to the clock that traveled with you,
so you have to adjust your path in spacetime a little to navigate back to it.
But notice there was still no "jump" involved: you traveled away from
Earth, you traveled back to Earth, and found something unexpected, because the
paths you and Earth followed through spacetime put your clocks out of sync.
Note also that I pulled a little sleight of hand in that paragraph. When I
said "where you expected the Earth to be", I should really specify relative
to what for the sentence to have any meaning. And what I really mean
is "relative to where you would expect it to be if you had traveled at low
speed through flat spacetime and your clock had kept time with Earth
clocks". This is kind of a subtle point, but it is at the heart of what's
going on here. Your clock tells you where Earth
"should" be, and relative to your clock Earth has
moved "out from under you" -- but that is only because you assume
that you and your clock traveled in flat spacetime. If you could look back with
a telescope and watch the Earth the whole time you were on your trip you would
see nothing odd happen, other than that it's position drifts slowly more and
more backwards from where your clock says it should be. But at no point does
Earth blink out of view and reappear further back in its orbit, the way a
"time jump" is typically depicted in sci-fi.
So really, when I say "return to the same coordinates", I need to
specify relative to what. If I "return to the same coordinates"
relative to the Earth, it's right where I expect. If I choose some other point
of reference relative to which the Earth is moving, it's not.
By the way, one of the first people to demonstrate rigorously that this kind
of thing could happen in General Relativity was Einstein's good friend Kurt
Godel. He showed that if the universe is rotating and sufficiently large, you
could follow a very long loop around the universe and return to your starting
point at an earlier time. Reportedly, Einstein was quite upset by this.
Now, when we see time travel in TV or movies, one way to think about it is
to assume that the device is creating its own region of curved spacetime that
is extremely small and very severely curved. I like to pretend that the
time travel in 7 Days works this way: the device creates a highly
distorted region of spacetime around the capsule. The capsule travels in that
highly distorted region which takes it both into space (relative to the Earth)
and back in time (again, relative to the Earth). It then emerges from that
distorted region and navigates back to Earth traveling in our (relatively) flat
spacetime. But rather than "jumping 7 days into the past", I think of
it as "traveling 7 days into the past and elsewhere in space".
Another way to think about it is this: the most consistently realistic depiction of time travel in fiction is wormholes: a wormhole connects two different points in both space and time. If you had a pair of wormholes -- or a single wormhole that connects two points close to each other in space but distant in time -- you would effectively have the classic time machine. And in fact, the wormhole is simply an extreme*** example of curved spacetime.
I have an even longer answer than this that also explains why we haven't met time travelers from the future yet, but it runs to several pages, and I'm
saving it for the book I'm never going to write, "Physics for Smarties: an
essential math-free guide for curious arts and humanities students".
Anyway, I hope that helps a little bit, and if you take away nothing else,
remember "continuous path through spacetime" and "doesn't travel
in space -- relative to what?"
*And yes, I readily concede that "as far as we know" is not very
far, but the fundamental character of General Relativity is very, very
suggestive on this point.
**OK, you could imagine such a thing if you wanted to,
but we have no idea what physics would describe it, so basically you could
decide how it behaves completely arbitrarily. It does whatever you want it to
do. You just have to bear in mind that there are no absolute coordinates in
spacetime, so if you define that your box "doesn't travel through
space", it's up to you define what you mean by that: doesn't travel
relative to what frame of reference? In short, your time box does whatever you
choose it to do, because you are making it up.
***Insert your own Wormhole Extreme! reference here.