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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.

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    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. – Thaddeus Howze Sep 3 '15 at 2:48
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    @Thaddeus: I don't think they'd even need to do the math; the temporal vortex is probably designed to follow gravitational fields, automatically remaining in the same spatial position relative to the nearest planet or sun. – Harry Johnston Sep 3 '15 at 3:22
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    With regards to Red Dwarf, if I remember rightly that was a fundamentally different time-travel mechanism; a sudden jump, not a tunnel. And it didn't really make sense anyway, because there's no such thing as absolute position; "travelling in time but staying in the same place" is a meaningless phrase unless you specify "the same place" relative to what. – Harry Johnston Sep 3 '15 at 3:24
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    @Thaddeus : I agree 100% with your comment. So much so that I have quoted it in my answer (with due credit to you). I hope that's okay. – Praxis Sep 3 '15 at 3:36
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    @HarryJohnston : I agree with you too. But for the record, I bet designing something that locks on to gravitational fields while tunnelling through time requires some complicated mathematics. ;-) – Praxis Sep 3 '15 at 3:37
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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!

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    Not all movement is relative - linear one, yes, but with circulation it is more complicated. Centrifugal force is something pretty independent on reference frame, isn't it? If Moon would stop circulating the Earth, it would fall. Changing frame of reference does not make it start falling, right? Same with solar system orbiting center of galaxy. So your argument about motion relativity does not fit the kind of motion OP was talking about. – Mołot Sep 3 '15 at 8:56
  • @Mołot Yes all movement is relative. From Earth, the moon is moving in an orbital pattern around the planet. From the moon, Earth is moving in an orbital pattern around the moon, and the moon is stationary. – mike32 Sep 3 '15 at 9:34
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    @mike32 You are mistaken - please read this detailed answer if you want to understand why. – Mołot Sep 3 '15 at 9:37
  • @Mołot I stand corrected - though there are parts of the linked answer that don't follow for me, I shall chalk this up to my lack of understanding for now. – mike32 Sep 3 '15 at 9:45
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    as thaddeus says, the fact that can create warp conduits, to traverse the galaxy close to instantly, its not unrealistic for the time travel vortex to also move them through space. – Himarm Sep 3 '15 at 13:18
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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

(Source)

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.

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    Nice try but I don't see how Warp speed factors [sic] into this whatsoever. – Lightness Races in Orbit Sep 3 '15 at 1:41
  • @LightnessRacesinOrbit the question makes it pretty clear that we're trying to figure out how the sphere and Enterprise landed right in front of Earth when it's moving through time. I'm offering a solution as to how they could travel that distance in time as well – Often Right Sep 3 '15 at 2:50
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    @LightnessRacesinOrbit: it seems relevant to me. It stands to reason that any useful time-travel mechanism must compensate for the movement of the target; the real question is whether it is reasonable for it to be able to do so - or whether, as the OP put it, whether doing so would require "instant movement to anywhere". This answer demonstrates that the necessary movement isn't beyond the already known capabilities of the Borg. – Harry Johnston Sep 3 '15 at 3:17
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    IOW, the real question is "if the Borg can travel that fast, why don't they do it all the time" and the answer is "they usually travel faster!" The only thing I'd add is that the temporal vortex is probably guided to the correct spatial location by the gravitational field of the Earth, and to be honest that's pure speculation. :-) – Harry Johnston Sep 3 '15 at 3:20
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    This answer is terrible. There are far simpler and better explanations, in the comments at the top in some of the other answers. "If this were the case, it means that the time travelers would still be impacted by the forces of the universe, meaning that they were influenced by the various forces, thereby moving them with the Earth and Solar system" makes no sense at all, physical, mathematical, or otherwise. – user51095 Sep 3 '15 at 4:30
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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.

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    Downvoting because you are handwaving instead of doing the math. See N_Soong's answer. – Lexible Sep 3 '15 at 4:20

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