Okay, I was just thinking about this and am curious to see what sort of responses I get -- I haven't read anything which addresses this argument in the FAQ.
Suppose we have two cities on either end of an axis passing through the equator and the center of the RE model, as shown below.
Now, the equivalent positions of those cities on the FE model are located on a diameter of the 'equator' (a ring of a radius equal to half that of the earth's total radius) as illustrated below. Obviously the position of the white line is meant to be the equator, though I acknowledge that it is placed incorrectly. This will not matter later in this thought experiment, as we will use real numbers to make our point. This only serves as a visual example to clarify the situation.
We must first introduce a reference frame for each model. In both cases, the reference frame will be defined such that the earth is stationary.
Next, we introduce an airplane. This airplane will act in the manner we are all familiar with. It will obey all laws of physics, motion, and time as we experience in our world today (flat or not). The Boeing 747 commercial jet has a cruising speed of approximately
250 m/s (we will use this velocity in our calculations). In order to simplify the experiment, we assume this speed is constant and unaltered by the effects of wind.
Finally we come to the nitty-gritty. In the RE model, the airplane will travel along the Earth's circumference. We will calculate this circumference by using the RE's mean radius, 6371 km (6,371,000 m). We can ignore the altitude of the plane because it is very small compared to the Earth's radius. Using simple arithmetic, we come up with a circumference of: 2*Pi*r = 40,030,173 m.
Presumably, if we travel along the equator or straight over the North Pole, the distance from city A to city B will be equidistant as we see below:
This distance precisely one half the circumference, thus (40,030,173 m)/2 =
20,015,086.5 m.
The equivalent paths for the FE model (one traveling along the Earth's Equator, and one straight over the North Pole are as such:
I was unable to find dimensions of the FE model, so we will assume that the distance of the path around the equator of the FE model is the same as that of the RE model. We can make our point without the aid of this assumption, but for the purposes of coming up with hard numbers, it is safe to assume so.
No matter what path we take in the RE model, since the distance is the same, and the speed of the airplane is the same (also remember that the Earth is stationary in our reference frame), we can easily verify that the time to travel from city A to city B is equal. Time = Distance/speed, thus (20,015,086.5 m)/(250 m/s) =
80,060 s.
In the FE model, the time it will take to travel along Earth's "Equator" is the same as calculated above, since the distance and speed is the same in both models.
The distance of the path which travels straight over the North Pole of the FE model can be defined as the diameter of the circle making up the Equator, so: Radius = Circumference/2*Pi = (40,030,173 m)/2*Pi = 6,371,000 m. Diameter = 2*Radius, thus 2*(6,371,000 m) =
12,742,000 m.
Therefore, the theoretical time it would take the airplane to travel this distance is: Time = Distance/Speed = (12,742,000 m)/(250 m/s) =
50,968 s. Remember, the airplane acts the way we expect it to in our world today, so no arguments can be made that it will travel slower moving in one path or the other.
Obviously 80,060 s is not equivalent to 50,968 s. Here is the point when the frog jumps into the pond. If a plane were to travel along the proposed paths at this very moment, the time elapsed from city A to city B will be roughly equivalent, allowing for some error due to wind and other minor factors. I ask you this: how can a plane, which travels at a constant speed, travel two different distances in the same amount of time?