Celebrating Einstein
How High Up Is That Place? How Far In
The Future Is That Event?
(continued)
B. How high are those balloons?
To begin our analogy, let's say you and your family have stopped at a restaurant. While you're there, you notice a couple of hot air balloons, one hovering nearby, the other a a few kilometers away. From where you are, both balloons are at the same altitude.
From other locations, though, the two balloons are at different altitudes. Because the earth is curved, how high something is depends on the location of the observer. That's because which way is up depends on where one is, as can be seen from the diagram.
The diagram makes a few points clear:

For each location on earth, the vertical and horizontal directions are different.

The angle by which the vertical axes differ at different locations directly corresponds to the angle by which the horizontal planes differ at those locations. (In fact, the angles between the vertical axes and the horizontal planes are equal, but for now we just note that the angles are related.)
Another noteworthy point is that it's easy to understand how the height difference between balloons depends on your location if our diagram takes in a large portion of the earth's circumference. If we compare balloon heights as seen from two places much closer together, we see that the results are more similar. Two people standing next to each other have so little difference in perspective that the balloons' vertical separation would be practically the same to both.
These conclusions all depend on the fact that the earth is a roughly spherical object, which makes vertical distances between objects different which respect to different points on the earth's surface. Einstein found that, in a similar way, time durations between events are different with respect to different velocities through space. A few years after Einstein discovered this, one of his former professors, Hermann Minkowski, found a way to illustrate the relationship with a diagram. (.....continued)