Inertial Frames in General Relativity

I’d like to shift our focus from coding to one of my other passions: physics. I want to begin with a fundamental concept: the inertial frame of reference. This concept, often underappreciated in introductory physics courses, is crucial not only in Newtonian mechanics but even more so in Einsteinian relativity. In fact, it has significantly influenced Einstein and Minkowski in developing the theory of relativity. In classical mechanics, an inertial frame of reference is defined as a frame where no net force acts on a freely moving object. Essentially, it’s a frame that’s either at rest or moving with constant velocity in a straight line. Acceleration disqualifies a frame from being inertial. Why are inertial frames so important? They serve as the foundation for measuring the accelerations of other objects. Importantly, all inertial frames are inherently non-accelerating relative to each other. If one frame observes another accelerating, at least one of them must not be inertial. Newton’s second law, F=ma, holds true only within inertial frames. A simple test for an inertial frame involves observing a freely falling object. In an inertial frame, the only force acting on the object should be gravity. If the frame is non-inertial (like a car turning a corner), the object will appear to accelerate without any apparent external force, violating Newton’s second law.

Now, consider these four frames:

1. A box on Earth’s surface.
2. A box in deep space, far from any significant gravitational influence.
3. A freely falling box (e.g., dropped from an airplane).
4. A box accelerating uniformly in space with an acceleration equal to Earth’s gravity (e.g., inside a constantly accelerating rocket).

According to Newtonian mechanics, frames 1 and 2 are inertial, while frames 3 and 4 are not due to their acceleration. Einstein, however, made a profound observation: from the perspective of an observer within the box, frames 2 and 3 are indistinguishable. Similarly, frames 1 and 4 are indistinguishable. He questioned the conventional assumption that only frames 2 and 1 are truly inertial. This led him to the Equivalence Principle, a cornerstone of relativity: the effects of gravity are indistinguishable from the effects of acceleration. To accommodate this principle, Einstein had to address some apparent inconsistencies. For example, if frame 3 (the freely falling box) is truly inertial, then two freely falling objects within it should not experience any relative acceleration. However, on Earth, two falling objects converge towards the Earth’s center, suggesting a force between them. Furthermore, orbiting objects, which are in a state of free fall, should also be considered inertial. Yet, they accelerate relative to frames on Earth’s surface. Einstein’s resolution to these apparent contradictions lay in the concept of curved spacetime. In a flat geometry, inertial frames cannot accelerate relative to each other. However, in curved spacetime, this restriction no longer applies. In essence, the curvature of spacetime allows for the acceleration of inertial frames. This insight paved the way for Einstein’s theory of General Relativity, where gravity is not a force but a manifestation of the curvature of spacetime. Regarding the “reality” of gravity, in frame 4 (the accelerating rocket), the observer experiences an apparent force due to the rocket’s acceleration. The Equivalence Principle suggests that the gravitational force experienced on Earth’s surface is also an apparent force arising from the acceleration of the Earth’s frame. From a relativistic perspective, the Earth’s surface is effectively accelerating outward, creating the sensation of gravity. To fully understand this concept, we need to dive deeper into the mathematics of curved spacetime and the concept of geodesics, which describe the paths of freely moving objects in curved spacetime.