Note: This post is an English adaptation of my original Chinese article (URL). Some parts have been modified for clarity, cultural relevance, or to better fit the English-speaking audience.
Update:
After reading another article, I realized that I had initially misinterpreted the question. I had understood the term “derive” in its strict mathematical sense, implying a proof without any additional assumptions. Consequently, when I attempted to derive the properties of Special Relativity (SR) directly from the definitional properties (or axioms) of General Relativity (GR) without setting assumptions such as a flat manifold, I concluded that GR cannot derive SR. From a physics perspective, this conclusion is somewhat flawed, and upon reflection, I found it unsatisfactory. I apologize for the initial oversight and will now correct my original answer.
In retrospect, framing the question as “Can GR be reduced to SR?” is more appropriate than “Can GR derive SR?” The term “reduce” better emphasizes the process of derivation under a specific set of conditions and constraints.
The specific conditions required for GR to reduce to SR are the assumptions of a “Lorentzian manifold” and “flatness (as the Riemann curvature tensor is zero everywhere, also known as local flatness)“. With these two conditions, it can be shown that a neighborhood of any point on the manifold is isometric to Minkowski space. This is how a local version of SR is derived.
Original Answer:
The framework of Special Relativity (SR) incorporates the Minkowski metric as a fundamental axiom to guarantee the “constancy of the speed of light in all inertial reference frames.”
The framework of General Relativity (GR), however, does not stipulate the Minkowski metric as an axiom. GR only requires that its metric satisfy the axioms internal to GR; therefore, the metric in GR is not necessarily the Minkowski metric. Therefore, GR cannot be used to derive SR.
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