Class. Quantum Grav. 12 (1995) 2549–2563. Printed in the UK Relative observer kinematics in general relativity Donato Bini†, Paolo Carini‡† and Robert T Jantzen§† † Physics Department and International Center for Relativistic Astrophysics, University of Rome, I–00185 Rome, Italy ‡ GP-B, Hansen Labs, Stanford University, Stanford, CA 94305, USA § Department of Mathematical Sciences, Villanova University, Villanova, PA 19085, USA Received 9 January 1995, in final form 23 June 1995 Abstract. The straightforward reformulation of special relativistic concepts about relative observer kinematics in the context of the flat affine geometry of Minkowski spacetime, so that they respect the manifold structure of that spacetime, allows one to derive the general relativistic ‘addition of acceleration law’. This transformation law describes the relationship between the relative accelerations of a single test particle as seen by two different families of test observers. PACS number: 0420C 1. Introduction The 1960s brought modern differential geometry to general relativity. With this new and powerful tool, complicated coordinate/tensor calculus discussions of physical ideas could often be reduced to a simple mathematical idea. A good example of this is the detailed discussion by Landau and Lifshitz [1] of local simultaneity using a local coordinate system based on a family of test observers (whose world lines are the time coordinate lines) and the idea of synchronizing nearby observers using light signals. A rather detailed coordinate calculation involving solving a quadratic equation in coordinate differentials produces a result which from the point of view of spacetime is trivial: the local simultaneity events for a given test observer correspond to points in the local tangent space in the orthogonal complement of the 4-velocity of the observer. Spacetime geometry is simply more powerful than the space-plus-time split geometry. (Of course the light-signal idea is important in connecting the mathematical orthogonality to the physics underlying the mathematical model.) This does not mean that spacetime splitting is just a necessary evil for interpreting spacetime geometry given our space-plus-time mode of experiencing spacetime. On the contrary, it is often quite convenient in analysing many physical and/or theoretical relativistic problems, as reflected in its widespread use in relativity. However, the power of spacetime geometry should not simply be overlooked when spacetime splitting techniques are employed. Indeed the 4-geometry perspective can be crucial to understanding certain splitting questions. For some reason the natural marriage of special relativity and curved spacetime has never been fully appreciated by the relativity community. While the field seems to have survived quite well without it, a straightforward extension of the ideas of special relativity to the 0264-9381/95/102549+15$19.50 c  1995 IOP Publishing Ltd 2549