> Most time travel theories ignore the fact that the earth is not fixed in space

This is a misconception that bugs me. The problem isn't that the Earth isn't fixed in space, it's that there's no such thing as a fixed point in space. Position is only defined relative to other objects. If you're going to use time travel in a story or something then it has to use something like an anchor object to determine destination. I.e. the relative location of the traveler and the anchor is replicated from the future to the past.

We would assume that the time-space traveler would have to tell the machine both the time and space directions from their current position in time and space. Assuming the time-space traveler cannot stop to observe his or her “location” in time-space coordinate along the way in small increments, he would have to calculate the entire travel trajectory beforehand.

I am saying this trajectory calculation relative to current coordinates is impossible. Even modern satellites with super precise instruments still need regular ongoing “adjustments.” Time travel requires many order of magnitude more precision than satellite orbital maintenance.

Go backwards in time 15 minutes at a time. At this short distance your calculation error will be small, and you can land your hovercraft back on earth to correct for any drift. Then go backwards another 15 minutes, repeat ad infinitum. Even present day aircraft have autopilot, so surely this can be automated too.

I think you have a good premise for a science fiction story right here: Say some "magic" (i.e. invented) physics quirk allows you to travel both into the future and the past, but all you can do is essentially accelerate and rewind time drastically. You don't "jump" to a time, there's still a physical presence, and colliding would have catastrophic results.

The logistical impacts from that would yield plenty of storytelling material: If you want to travel back in time, you need some ancient cellar that has been undisturbed since the target timeframe. If you want to skip forward, you need to establish that cellar, and round trips are limited by the space available.

That is - essentially - how 2002's _The Time Machine_ showed travel: Alexander's machine was 'stationary' on the earth, but time passed around him in a massively accelerated manner

to take this one step further - go back one infitesmal back in time and adjust position one infitensmal, thusly, a fixed time machine

Relativity just says "nothing about space seems to require a preferred reference frame", not "such a thing as a preferred reference frame can't possibly exist". If we're allowing for the discovery of time travel in the story, I'm willing to allow for such a discovery as well.

In reality I'd bet neither are realistic, but that's what makes the stories interesting.

Even if you could magically arrive at the right point, how would you get the right momentum? If the Earth were standing completely still, it would still be spinning at a horrendous speed.

I'll give a half-baked counter to this: we know gravity impacts the flow of time through relativity. There is currently no evidence that time travel wouldn't be impacted by gravity in some way. Maybe the way time in time travel interacts with gravity protects you from this problem? Probably not, but it has just as much evidence to support it as your claim of time travel will dump you in empty space.

You’re positing some unknown influence will cause everything to work out well in the ends without any evidentiary basis. Occam’s Razor suggests that you’re more likely to be wrong than parent.

Of course the idea that your point of origin must be fixed from time A to time Z if you’re willing to allow for time travel is itself flawed. If you could somehow move an object to an arbitrary time you could move them to an arbitrary point in space, and your ability to calculate may be significantly greater on the grounds that you’d have more advanced technology than us. It’s all scifi woo though until someone actually time travels.

I disagree with this interpretation of what I said. We HAVE evidence that time and gravity interact. It's actually more of a violation of Occam's Razor to suggest that time travel is somehow exempt from that interaction than to claim that yes, time travel should in someway be subject to the influence of gravity.

It’s even crazier if you imagine that whole universe might be countless universe lengths away from its starting point every microsecond for all we know. Acceleration is the only thing we feel.

You’re very likely to travel into an undefined void even if you map out and calibrate the whole system.

That why it is important not to mess up coordinate system. With wrong calculations they fall to ground. Or they are buried under ground. And space is full of frozen bodies.