The Utter Failure of Fictional Time Travel
blogs.scientificamerican.com
blogs.scientificamerican.com
The machine no more needs to calculate its motion through space as it travels through time than a rock or an old building does.
Seriously, the point is that the Time Machine does not leave our space-time and appear in another. It simply exists through time, like a building or a rock or a pyramid. It is a four dimensional object where the Traveller can move along one of its dimensions.
Anyway, enough arguing about a fictional (and probably impossible) machine. Read the introduction to The Time Machine if you are interested in how H. G. Wells explained it: http://www.gutenberg.org/ebooks/35
Big deal that fiction doesn't actually detail this, it adds nothing to the narrative.
It's only a communications device, not a transportation device but it does consider the movement. The signal is sent to the same point in space where the machine currently is, the range limit is how far away from that point you can be and receive it.
This might be why time travel in Terminator movies creates the sphere of energy right before the traveler appears: it clears out the space. But then why doesn't the person experience a moment of disruption being inside a perfect vacuum?
One could easily argue that big whiteboards laden to the brim with integrals and capital sigma (summation) rich equations indicate a data rich problem rather than a theory heavy one, and that the inevitable squashed, disappeared, or exploded melon fits very well into this framework.
Shooting a water melon at several million miles per hour so it hits a small time travel device correctly in all N>3 dimensions is obviously going to make a royal mess quite a few times, as evidenced in many movies!
Frame dragging is a good answer. You are locked to the inertial reference of the nearest big body (add yo mom jokes here if you like).
But I don't think frame dragging solves the problem, at least not without a complete rearrangement of relativity, in which case you don't really need frame dragging.
The problem is that frame dragging occurs in the local environment - if you're being dragged along with a frame, you're not traveling at any unusual vector through time.
Time travel necessarily deviates from the spacetime path of the frame, otherwise you'd just be traveling with the frame as normal. And if you deviate from the path of the frame, then ordinary frame dragging isn't going to help you. Of course you could perhaps invent some sort of more spooky frame dragging to handle this case, but it wouldn't have the same physical basis.