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In General Relativity, these are the curves that a free particle (that is, a particle upon which no force acts, where ‘force’ in this case excludes gravity, since the effects of gravity are felt entirely through the curvature of space-time) will follow in a curved space-time .

A geodesic could be equivalently defined as:

 

The Einstein's Derivation of the Geodetic Equation from a variationnal approach (Extract from the Manuscript "The Foundation of the General Relativity of Relativity §9 1916)

 

 

[1] Refer to our article Geodesics as proper time maximization to see how this applies also in Special Relativity.