Classical Metrics do not Provide an Unambiguous Inner Product Between Timelike and Spacelike Vectors

The unified theory of mass-ENERGY-Matter in motion

Similarly, in Newtonian gravitation, the acceleration of a timelike curve must always be spacelike, and so the total force on a particle at a point must be spacelike as well. A vector ξa at a point in a classical spacetime is timelike if ξata ≠ 0; otherwise it is spacelike. The required result thus follows by observing that given a curve with unit tangent vector ξa, tannξa) = ξnnata) = 0, again because ξa has constant (temporal) length along the curve. Note that one cannot say simply “orthogonal” (as in the relativistic case) because in general, the classical metrics do not provide an unambiguous inner product between timelike and spacelike vectors.

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