I cannot find any data on the following subject which I have been thinking about:
Time travel may require the ability to calculate where the surface of the earth will be in relation to the earths rotation, orbit around the sun, the solar systems orbit around galactic central point and the galaxy's movement through space as the universe itself expands, otherwise, if you travel forward or backward just 1 minute you will end up in the vacuum of space (or trapped inside a space object like a sun planet or meteor) because from second to second every object in space is moving...That's a lot of math
Actually the math itself isn't a problem. You can estimate the displacement in timespace using the principles of plane geometry in 4 dimensions (x,y,z,t). Permutations of the Pythagorean Theorum show up everywhere in physics math. You remember it from high school plane (2D) geometry yes? a^2 + b^2 = c^2 - the solution for the hypotenuse of a right triangle. Carpenters use it every day to "square" frames and corners.
In 4D Special Relativity, for example, it shows up as
s^2 = (c^2t^2) - x^2 - y^2 - z^2
indicating a spacetime interval (x,y & z being space and t time). Using that equation in differential calculus form will yield the interval between spacetime events, including the seperation (interval) in spacetime between "when and where you were then" and "when and where you are now". The term "interval" is used rather than "distance" because the equation solves the seperation "spacetime", not just the distance in space between events. In Special Relativity space and time are not treated seperately but as two parts of a whole - spacetime - that cannot be seperated; the spacetime continuum.
The real problem is that the accuracy of the numbers used in the calculation, the instruments used to make it happen and precise knowledge of the relative motion between the origin and target have to be damned accurate. Being "close" but off by only 1/1,000,000 of a second per second, considering that the speed of light is 300,000 km/sec, could result in your missing your target by 300 meters for a 1 second time trip. Pretty close for even a 1 week spacetime interval but you don't want to be embedded in rock almost a thousand feet below the surface or fall a 1000 feet to the ground.
There's also another not-so-evident problem. Again, as you point out, you are in motion if for no other reason because the Earth is orbiting the Sun. You also have mass. One definition of mass in motion is momentum, and it is a vector quantity: it has magnitude and
direction. Momentum is also a
conserved quantity (meaning it doesn't go "poof" and alter itself simply because one time traveled) so there is no obvious reason to believe that when you arrive at the target that you won't be traveling in the same direction and speed as was true at the origin.
The Earth is orbiting the Sun at about 30 km/sec. You time travel forward or back 6 months. The target is your original location on Earth. The Earth is about 180 degrees (6 months) displaced along its orbit from its position when you started the trip. Assuming the physics logic is correct, if you actually end up precisely on target you will be traveling along the -x direction at 30 km/sec and the Earth will be traveling in the opposite direction along the +x direction at 30 km/sec. In that case you should end up as a net 60 km/sec meteroid skimming along the surface. That's about Mach 176 and almost 6 times Earth's escape velocity. Ouch! Not to worry however. The "big pieces" if any probably won't hold the velocity due to the air friction and won't leave the Earth forever behind. So there's an upside...sort of...maybe. Jus' sayin'