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A particle following a geodesic moves by inertia.

Yet typically, we think of movement by inertia as moving at some constant velocity. But a particle in free fall does 'accelerate' i.e. its velocity is changing.

What am I misunderstanding?

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In general relativity a falling apple and an orbiting moon aren't accelerating they have geodesics as worldlines. It is the ground and everything on the ground which are accelerating and their worldlines aren't geodesics.

So when observing an apple falling or a moon orbiting from the ground. You mistakenly assume that they are accelerating because you assume that you are inertial, even though in general relativity you aren't.

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