I have a question. If I traveled through time wouldn’t I just show up in empty space considering not just the movement of the earth but also the solar system as a whole?
Yes you would. This is why "time machines" also have to have the capability to travel through space. Time travel puts you back (or forward) in time in the same position relative to the universe. If you travel far forward in time, the expansion of the universe puts you quite far from where you started. Going back to the exact second you left puts you back physically where you started. Even a jump of just a couple years can cause a lot physical displacement.
This is overcome by doing multiple time jumps, very quickly in succession, while simultaneously moving in physical space, effectively keeping up with the expanding/contracting universe. The math requires a quantum computer.
“wouldn’t I just show up in empty space considering not just the movement of the earth but also the solar system as a whole?”
I concur!
Indeed. That is a generally under-appreciated aspect of time travel. I've actually thought about the subject quite a bit. If you build a time machine, you must have it housed in a spaceship, or you'll die the first time you attempt to use it. It's also important to take a lot of motion into account. Wouldn't it suck to try to take a trip to last week and end up 10 miles under the surface of the sun?
You'e got big issues if you want to travel any serious distance in time. Let's say you want to see the dinosaurs, and decide to go back a nice round 100,000,000 years. Now, do you have any idea how far the solar system is from you. Given apparent motions of the galaxy, it could be on the other side of the galactic disk, or even further.
Though I've given a lot of consideration to the concept of time travel, unlike the previous poster, I think the possibility of FTL travel is more likely.
In theory the time travel (back or forward) is to the local frame of reference of the target scene, not to a particular point in space. The time machine is following the local reference forward or backward in time which is the natural way to interpret the process. Traveling to a particular point in space, OTOH, really doesn't make any sense except with respect to actual physical things.