Some Issues in Relativistic Spacetime Theories

Abstract

The chapter has as main objective to clarify some important concepts appearing in relativistic spacetime theories and which are necessary of a clear understanding of our view concerning the formulation and understanding of Maxwell, Dirac and Einstein theories. Using the definition of a Lorentzian spacetime structure \((M,\boldsymbol{g},D,\tau _{\boldsymbol{g}},\uparrow )\) presented in Chap. 4 we introduce the concept of a reference frame in that structure which is an object represented by a given unit timelike vector field \(\boldsymbol{Z} \in \sec TU\) (\(U \subseteq M\)). We give two classification schemes for these objects, one according to the decomposition of \(D\boldsymbol{Z}\) and other according to the concept of synchronizability of ideal clocks (at rest in \(\boldsymbol{Z}\)). The concept of a coordinate chart covering U and naturally adapted to the reference frame \(\boldsymbol{Z}\) is also introduced. We emphasize that the concept of a reference frame is different (but related) from the concept of a frame which is a section of the frame bundle. The concept of Fermi derivative is introduced and the physical meaning of Fermi transport is elucidated, in particular we show the relation between the Darboux biform \(\Omega \) of the theory of Frenet frames and its decomposition as an invariant sum of a Frenet biform \(\Omega _{F}\) (describing Fermi transport) and a rotation biform \(\Omega _{\mathbf{S}}\) such that the contraction of \(\star \Omega _{\mathbf{S}}\) with the velocity field v of the spinning particle is directly associated with the so-called Pauli-Lubanski spin 1-form. We scrutinize the concept of diffeomorphism invariance of general spacetime theories and of General Relativity in particular, discuss what meaning can be given to the concept of physically equivalent reference frames and what one can understand by a principle of relativity. Examples are given and in particular, it is proved that in a general Lorentzian spacetime (modelling a gravitational field according to General Relativity) there is in general no reference frame with the properties (according to the scheme classifications) of the inertial referenced frames of special relativity theories. However there are in such a case reference frames called pseudo inertial reference frames (PIRFs) that have most of the properties of the inertial references frames of special relativity theories. We also discuss a formulation (that one can find in the literature) of a so-called principle of local Lorentz invariance and show that if it is interpreted as physical equivalence of PIRFs then it is not valid. The Chapter ends with a brief discussion of diffeormorphism invariance applied to Schwarzschild original solution and the Droste-Hilbert solution of Einstein equation which are shown to be not equivalent (the underlying manifolds have different topologies) and what these solutions have to do with the existence of blackholes in the “orthodox”interpretation of General Relativity.