The Equivalence Principle
One thing that comes up over and over in special relativity is the role of inertial frames. We talked about accelerated frames as well, but those followed different rules. The most notable of these is the effect that the acceleration of a frame has on the measurement of the proper time. If an observer carries a clock in their non-inertial frame and measures the time between two events that occur at the same place in that frame with it, the time measured is less than when an observer in an inertial frame carries a clock and does the same with those two events.
This phenomenon is reflected in the shape of the world line of the origin of the non-inertial frame in a spacetime diagram in the rest frame of an inertial observer. By contrast, the shape of the world line of the origin of an inertial frame is always straight when drawn in an inertial observer's space time diagram. The intervals defined by all the straight world lines come out the same.
Einstein must have worried if perhaps it is possible for the world line of an inertial frame to be curved in the spacetime diagram of another inertial observer (it is unlikely he worried about it in exactly these terms, but we'll run with this). This would cause problems, because the Lorentz transformation that links inertial frames only rotates world lines, it doesn't cause them to curve. It turns out that Einstein came up with an example that causes this exact problem!
To see how this problem can arise, let's first return to something we studied in Physics 9HA. When we talked about free-fall, we noted that because every object in an enclosed space experiences the same acceleration when that space is in free-fall, an observer in this frame will assume that nothing in the space is experiencing a force. Einstein elevated this to a fundamental principle that sounds very much like the relativity principle, and called this one the equivalence principle:
No experiment can be performed that will distinguish a frame in gravitational free-fall from an inertial frame.
If this principle is correct, then the following thought experiment raises a problem...
Suppose we have two free-falling frames on opposite sides of a planet. They are moving toward each other in a symmetric fashion, with their common \(x\)-axes passing through the center of the planet: