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A signature invariant geometric algebra framework for spacetime physics isformulated. By following the original idea of David Hestenes in the spacetimealgebra of signature $(+,-,-,-)$, the techniques related to relative vector andspacetime split are built up in the spacetime algebra of signature $(-,+,+,+)$.The even subalgebras of the spacetime algebras of signatures$(\pm,\mp,\mp,\mp)$ share the same operation rules, so that they could betreated as one algebraic formalism, in which spacetime physics is described ina signature invariant form. Based on the two spacetime algebras and their"common" even subalgebra, rotor techniques on Lorentz transformation andrelativistic dynamics of a massive particle in curved spacetime areconstructed. A signature invariant treatment of the general Lorentz boost withvelocity in an arbitrary direction and the general spatial rotation in anarbitrary plane is presented. For a massive particle, the spacetime splits ofthe velocity, acceleration, momentum, and force four-vectors with thenormalized four-velocity of the fiducial observer, at rest in the coordinatesystem of the spacetime metric, are given, where the proper time of thefiducial observer is identified, and the contribution of the bivectorconnection is considered, and with these results, a three-dimensional analogueof Newton's second law for this particle in curved spacetime is achieved.Finally, as a comprehensive application of the techniques constructed in thispaper, a geometric algebra approach to gyroscopic precession is provided, wherefor a gyroscope moving in the Lense-Thirring spacetime, the precessionalangular velocity of its spin is derived in a signature invariant manner.

A signature invariant geometric algebra framework for spacetime physics  and its applications in relativistic dynamics of a massive particle and  gyroscopic precession