  A signature invariant geometric algebra framework for spacetime physics is
formulated. By following the original idea of David Hestenes in the spacetime
algebra of signature $(+,-,-,-)$, the techniques related to relative vector and
spacetime 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 be
treated as one algebraic formalism, in which spacetime physics is described in
a signature invariant form. Based on the two spacetime algebras and their
"common" even subalgebra, rotor techniques on Lorentz transformation and
relativistic dynamics of a massive particle in curved spacetime are
constructed. A signature invariant treatment of the general Lorentz boost with
velocity in an arbitrary direction and the general spatial rotation in an
arbitrary plane is presented. For a massive particle, the spacetime splits of
the velocity, acceleration, momentum, and force four-vectors with the
normalized four-velocity of the fiducial observer, at rest in the coordinate
system of the spacetime metric, are given, where the proper time of the
fiducial observer is identified, and the contribution of the bivector
connection is considered, and with these results, a three-dimensional analogue
of Newton's second law for this particle in curved spacetime is achieved.
Finally, as a comprehensive application of the techniques constructed in this
paper, a geometric algebra approach to gyroscopic precession is provided, where
for a gyroscope moving in the Lense-Thirring spacetime, the precessional
angular velocity of its spin is derived in a signature invariant manner.
