PART THE FOURTH. In Which We Finally Get to Go Home:
Let’s look at the overall effect en toto. Though revolution and rotation are of course simultaneous, we can parse the mathematical effects of both out into separate steps. Hence each displacement becomes the leg of a triangle–one for rotation, one for revolution. Check out figure four–which puts it all together:
We already have D1 and D2. From the calculations in figure 3, we compute the smaller of the two interior angles of the central parallelogram to be 180o – (89.99966o + 20o) = 70.00034o, as is shown in Fig. 4A. From Figure 2, we know that the small isosceles triangle on the right of 4A has angles of 89.875o. As in figure 4B, adding these two gives a total angle of 159.87534o. Applying the law of cosines one last time yields:
D3= The total distance Einstein would have to travel in order to arrive back in the Twin Pines Mall parking lot one minute into the future…
That distance is ~4/5 the length of the eastern sea board of the United States! Or–to put it more juvenile terms–it’s approximately the distance from Bangor, Maine to Morehead, Kentucky. But less sexy. And that’s how far the Earth would travel in just one minute! I don’t even want to go near the calculation for the 1985-1955 jump…
Now, the machine is traveling at 88mph. While that’s hardly fast enough to cover this distance, how fast would it have to go to make the jump in space as well as time?
Speed is just the distance traveled over an interval of time. Doc’s stopwatches would have you believe that Einstein’s trip took no time at all, at least as far as the dog is concerned:
But then, his watches aren’t perfect, and this allows us to try and estimate something we’re unable to measure. Let’s be generous and assume that the stopwatches have an error of ~1 millisecond. That is, Einstein’s watch could be .001s slower than we think it is, without a perceptible effect: we’d still see the two watches changing time in pretty much perfect sync with one another. Hence an upper-limit to the time it took Einstein to jump 1807.56km is 0.001s. This yields an average speed of:
(1807560m)/(0.001s) = 1.808 x 109 m/s
…roughly six times the speed of light in a vacuum. Of course, an object moving faster than the speed of light would theoretically travel backward in time, so that kinda’ bones us right there.
* * *
So what can we take from this? Well first off, were the DeLorean to really just pop-out and pop-in to time in the same absolute place, then even this first tiny future-leap would drop poor Einstein nearly two-thousand kilometers into the vacuum of space. Longer leaps of time would be even more potentially disastrous: you might reappear inside solid rock, or within the fusion furnace of a sun, or near Steven Segal. One shivers at the thought.
But that kind of speculation is moot. We know that the DeLorean does pop back in pretty much the same relative place it popped out. Maybe it’s a conscious choice–maybe it’s a nimble space ship that Doc could have engineered it to go anywhere, but kept it within frame for safety’s sake. After-all, while I’m unable (and certainly unwilling) to do more elaborate calculations, that’s not to say that —given more sophisticated calculation equipment–a computer on board the DeLorean wouldn’t be able to handle them with aplomb. I mean, check out all the buttons and toggle-switches inside that thing–maybe some of them control the distance-computation circuits, in addition to the flux capacitor et. al. This gets less and less plausible when one imagines how Doc Brown could have constructed a second time-ship out of a 19th century train–as is revealed in the closing scene of BTTF 3. After-all, the retroengineering required in fashioning a flux capacitor from pre-period goods would be hard enough without having to fabricate silicon microprocessors as well.
No, my guess is that we’re supposed to assume the DeLorean is somehow coupled to the local reference frame as it travels freely through time–much in the same way that you and I are coupled to it as we travel through time in a constrained fashion. While the easiest way to achieve this coupling is through gravity, the big “G” tends to muck up time in whatever way it sees fit, regardless of your fancy-shmancy time travel plans.
Now that I think of it, one could always synthesize a worm-hole that encloses a timeless space, with its exterior coupled to the Earth’s timeline. This is of course absurd: any object passing through it would be exposed temperatures approaching absolute zero and would hence instantly become incomparably cold.