An astronaut in a spaceship flying toward Earth's Equator at half the speed of light observes Earth to be an oblong solid, wider and taller than it appears deep, rotating around its long axis. A second astronaut flying toward Earth's North Pole at half the speed of light observes Earth to be a similar shape but rotating about its short axis. Why does this not present a contradiction?

First, it is essential to grasp that when two objects are moving relative to one another, the observations made by an observer in one reference frame will not necessarily coincide with observations made by an observer in another reference frame. In this case, we have two astronauts viewing Earth from different directions, while moving at half the speed of light (c/2) relative to the Earth. These observations exhibit the principles of the special theory of relativity.

To understand why there is no contradiction in this scenario, we must recall the Lorentz transformation, which describes how coordinates of an observation in one frame of reference (space and time) will transform into coordinates in another reference frame. When an observer is moving with velocity v relative to another observer, the length, time, and mass all are transformed due to the special theory of relativity. The Lorentz transformation equations include: x' = \gamma(x-vt) t' = \gamma(t-\frac{vx}{c^2}) where x and t are the spatial and temporal coordinates in one frame, x' and t' are the coordinates in the other frame, v is the relative velocity between two frames, and \gamma is the Lorentz factor given by \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}.

Due to length contraction, astronauts observe Earth's shape as oblong and more extensive along the direction of motion; this is because when an object moves relative to an observer at a large speed compared to the speed of light, its length in the direction of motion contracts (apparently gets shorter) as observed by that observer. The contraction factor is given by the Lorentz factor \gamma. Furthermore, the relativity of simultaneity affects the observation of Earth's rotation; events simultaneous in one frame might not be simultaneous in another frame.

Although both astronauts observe Earth as oblong, their observations differ in how Earth rotates around its axes. Due to the relativity of simultaneity and the Lorentz transformation, the events observed by the two astronauts do not present a contradiction. The astronauts' different frames of reference cause them to perceive the events taking place on Earth differently. Their observations are valid in their respective reference frames and are transformed consistently between frames, maintaining the coherence of the physical laws.

In conclusion, the scenario does not present a contradiction as the observations made by the two astronauts are influenced by the special theory of relativity, and, in particular, length contraction and the relativity of simultaneity. These relativistic effects change the perception of Earth's shape and rotations for each astronaut, depending on their direction and relative velocity. This exercise exemplifies the idea that physical laws should appear the same regardless of the reference frame and that the events observed might transform accordingly.