Violations of Lorentz Invariance
All fundamental interactions, except gravity, are based on theories which incorporate the global space–time symmetries of Special Relativity (SR), encoded in the inhomogeneous Lorentz group (also called the Poincaré group). With the inclusion of gravity (curved space–time) the homogeneous Lorentz group still survives as a local symmetry. Our current understanding of relativistic Quantum Field Theory is fundamentally based on the strict validity of the space–time symmetries of Special Relativity. This may, e.g., be seen from the fact that we classify elementary particles according to their mass and spin, which are nothing but the eigenvalues for the Casimir operators for the Poincaré group. Hence we classify elementary particles according to the unitary irreducible representations of that group. Also, the CPT theorem, the spin-statistic theorem, and also the existence of antiparticles are deeply rooted in the Poincaré symmetry. On the other hand, we expect the space–time structure on the smallest scales to violate the symmetries of Special Relativity, even locally, due to effects of Quantum Gravity. This could e.g. be in terms of a fundamental length (Planck length). One could then ask for a discrete version of the Poincaré group, like Z4 x SL (2,Z+iZ), and its unitary irreducible representations. But one may also ask whether present observational evidence for Poincaré symmetry is not, in fact, also compatible with a smaller symmetry group, which is a subgroup of the Poincaré group. That this possibility indeed exists has recently (2006) been argued for by Cohen and Glashow. They considered a class of subgroups of the Poincaré group, which they termed Very Special Relativity (VSR), whose defining feature is that, upon adding the generators of P, or CP, or T, one gets full Poincaré symmetry. This implies that observations that are not sensitive to violations of these discrete symmetries will not see the difference between SR and VSR. The whole scheme is, however, CPT invariant.
The large-scale structure and overall expansion of the universe influences local systems in dynamical and kinematical aspects. In order to be able to reliably estimate upper bounds for such influences in principle, i.e. without limitation to currently achievable observational accuracies and beyond uncontrolled linear perturbation theory, one tries to construct exact solutions for various simple distributions of local over–densities. For example, it has been asked repeatedly whether the measured anomalous inwardly directed acceleration of magnitude 10-9 m/s² of the Pioneer 10 and Pioneer 11 satellites can have anything to do with cosmological expansion; and even though we now know that this this is almost certainly not the case, it remains a strange coincidence why this magnitude is almost the same as Hc, where H is the current value of the Hubble constant and c the velocity of light. In fact, the quantity Hc, build from fundamental constants with the dimension of an acceleration, enters many expressions that do, in fact, influence the dynamics and kinematics of local systems. Ideally, hard statements should be backed up by exact solutions. In the simplest case, such a solution should describe a single spherically symmetric mass that, intuitively speaking, is embedded in the background of a Friedmann-Lemaître (FL) universe. Several suggestions for such solutions exist, but so far none of these can be said to fully capture the full range of physical situations that one may envisage. The proper requirement for a mathematical representation of the envisaged physical situation must, first of all, consist in asymptotic conditions which ensure that the sought for solution approximates the given (e.g. Schwarzschild) one for small distances, and at the same time approximates a FL universe for large distances. Second, it must specify somehow the physics in the intermediate region. Usually this will include a specification of the matter components and their dynamical laws together with certain initial and boundary conditions. Needless to say that this will generally result in a complex system of partial differential equations. Most analytic approaches therefore impose further simplifying assumptions that automatically guarantee the right asymptotic behaviour and at the same time reduce the free functions to a manageable number.