Einsteinian Relativity in the Tangent Bundle of Spacetime

The tangent bundle of spacetime consists of spacetime in the base manifold and four-velocity space in the fiber [1]. The coordinates of a point in the spacetime tangent bundle are the spacetime and four-velocity coordinates of the observer. Einsteinian relativity plays a central role in the formulation of possible differential geometric structures and embedded fields in the spacetime tangent bundle. The covariant four-acceleration of Einstein's theory of general relativity plays a particularly important role. The quantum mechanics of the vacuum suggests the existence of a limiting proper acceleration, thereby placing restrictions on the differential geometric structure of the spacetime tangent bundle, and also on the structure of embedded classical and quantum fields [2-4]. In the present work, examples are addressed emphasizing the roles of both special-relativistic Lorentz invariance and general relativistic covariance in the theory of the spacetime tangent bundle. \\[4pt] [1] H. E. Brandt, Contemp. Math. \textbf{196}, 273 (1996). \\[0pt] [2] H. E. Brandt, Rep. Math. Phys. \textbf{45}, 389 (2000). \\[0pt] [3] H. E. Brandt, J. Mod. Optics \textbf{50}, 2455 (2003) \\[0pt] [4] H. E. Brandt, Internat. J. Math. and Math. Sci. \textbf{2003}, 1529 (2003). \\[0pt] [4] H. E. Brandt, Nonlinear Analysis \textbf{63}, 119 (2005).