When the Borg Sphere left the Cube and flew towards Earth and entered the temporal vortex, why did the Earth stay still? It was my understanding that the planet/solar system/galaxy rotate around the universe. So, as the Earth is moving, it would have been in a different position in the mid 21st century than the 23rd century, so they should have come out of the temporal vortex and not found the Earth there. I was watching Red Dwarf when they tried to time travel and a similar problem was encountered. A time travel device is not the same as instant movement to anywhere.
Frame of reference
All motion is relative. Whenever we say the "Earth is moving", we mean that it is moving relative to something. Most often when we speak of the Earth moving, we are referring to motion relative to the Sun. In physics, everything depends on the frame of reference that an observer resides in.
If you knew nothing about how our solar system works, you might conclude that the Earth is fixed and the the Sun is moving, as it rises above the horizon and then falls below it. There is no conflict here: if your frame of reference is Earth, then the Sun is indeed moving (relative to us).
The point is that there is no absolute frame of reference in which the Earth is definitely moving. It is only moving relative to other objects.
Since the Earth is not moving relative to the sphere and Enterprise-E, they must all be in the same frame of reference. In other words, the temporal vortex opened by the sphere keeps anything travelling inside the vortex locked with the Earth's rotation and spin.
As to whether the Borg can achieve this kind of lock while tunnelling through time, I believe @Thaddeus said it best in the comments above, and so I will quote him here:
Are we unable to accept a series of cybernetically enhanced, hive-minded beings with access to vastly powerful computer technology couldn't do a calculation to account for galactic movement for Earth's Sun during a time-travel event? This is just math, maybe some higher order math but certainly within the capacity of a group of beings who established Warp Conduits allowing them to travel across the galaxy.
Note: I also want to acknowledge @WadCheber and @JMFB for discussing some similar points in the chat room — in particular @JMFB, who also brought in frames of reference. I wasn't influenced by those discussions, as I had joined the chat late and had already posted my answer by the time @JMFB and I had discussed it, but I think it is good to point out their parallel discussion!
According to Scientific American:
our solar system--Earth and all--whirls around the center of our galaxy at some 220 kilometers per second
Let's do the math:
- The Enterprise-E and Borg sphere are coming from 2373
- They arrive in 2063
That's 310 years = 113,227.5 days (assuming a year remains a constant 365.25 days) = 2,717,460 hours = 978,285,600 seconds
Therefore, the solar system as a whole has moved 215,222,832,000 km
Now, the Borg Sphere and Enterprise-E would have arrived at the same position they were assuming they hadn't moved, so they would be 215,222,832,000 km from where Earth is in 2063. But they aren't that far away - they're very close to Earth when they arrive from the temporal vortex. Therefore, they must have been moving through space during that time
So, how did the two vessels travel that distance? Let's put this in perspective: the speed of light is 299,792 km/s, so it would take light 717,907 seconds to travel that distance = 83 days.
Remember, though, that the warp factor is faster than the speed of light. Now this fabulous answer outlines the warp factors in relation to the speed of light, which tells us that warp 9 is 1516.4 times the speed of light! Doing the math again, that makes it at about 20 seconds at warp 9 to travel that distance. Now, the Enterprise-E and Borg sphere probably weren't traveling at warp 9, but just to make the point; this is a tiny distance relative to the speeds a Starfleet vessel can reach.
Now, it is true that the Enterprise-E probably wasn't traveling at warp when it entered the vortex, but we don't know whether the temporal vortex required warp speeds to enter it and maintain it. Also bear in mind that they don't simply enter and exit the vortex instantaneously; they are in it for a brief period of time. If they were traveling at warp (and I can find no evidence to suggest that in the vortex they weren't at warp speeds, whether under their own devices or not) they could easily have traversed this distance in the relatively short period of time, making it seem like they haven't moved when they actually have!
When we consider time travel in TOS, note that the Slingshot Effect was used; this involved accelerating to above Warp 10 on the old scale; it is quite reasonable to expect that the temporal vortex also involved speed. Considering how we see the crew jolt, almost as if they're suddenly accelerating; it seems to me that the Vortex probably involved massive acceleration, which allowed them to traverse such a distance in such a short period of time.
To resolve this problem, other than seeing it as a plot hole, one must accept the conclusion that the two vessels, whilst in the temporal vortex, must have been traveling at a speed of about warp 9 or above. Note that I'm not arguing that they were traveling at this speed under their own power; rather that the Vortex accelerated them to that speed.
As pointed out in Harry Johnston's comment below, the vortex could be created such that at the end, the travelers ended up at a pre-defined destination based on calculations made about where the Earth would have been at that point in time.
A physics or hard sci-fi answer.
The Borg matched earths rotation and spin to minimize atmospheric friction upon reentry.
A time travel answer if you mean why we don't see the earth make a massive jump in perspective.
The time travel was only a few hundred years the relative distance involved to a warp culture is negligible like .002 seconds in warp so the detail was never mentioned.
Or do you mean that the Earth doesn't rotate in the shot because I'm pretty sure that every time we see the Earth it is rotating.