Energy and Geometry

The following sections are included:

• Preface

• Contents

The following sections are included:

• Foundations of Special Relativity

• The Principle of Solidarity

• Locality and Interaction
  • Locality and Nonlocality
  • Potential and Nonpotential Interactions

As is well known, the Minkowski metric

The following sections are included:

• Energy and the Finzi Principle

• Energy as Dynamical Variable

The following sections are included:

• Deformed Special Relativity

• Deriving the Deformed Lorentz Transformations

• Maximal Causal Speed Revisited

• Boosts in DSR
  • Boost in a Generic Direction
  • Symmetrization of Deformed Boosts
  • Velocity Composition Law in and the Invariant Maximal Speed
  • Choosing the Boost Direction in DSR
  • Appendix — Another Derivation of a Generic Boost

From the knowledge of the generalized Lorentz transformations it is easy to derive the main kinematical and dynamical laws valid in DSR.^3,4 In this Chapter, we shall merely list those which are useful to phenomenological purposes…

The following sections are included:

• Deformed Helmholtz Equation

• Propagation in an Undersized Waveguide

• Total Internal Reflection

• Deformation Tensor

The following sections are included:

• Non-Minkowskian Metric for the E.M. Interaction
  • Photon Tunneling and Superluminal Velocities
  • Deformed E.M. Metric from the Cologne Experiment

• An Application: Unified View to Cologne and Florence Experiments
  • Superluminal Propagation and the Friis Law
  • Two-Antenna System as a Barrier

The following sections are included:

• Analysis of Experimental Data on Clock Rates

• Local Metric of Gravitation

• How Fast Do Travel Gravitational Effects?

• Discussion of the Einstein versus Bohr Gedankenexperiment

• Einstein and the Energy-Dependence of the Light Speed

Let us now discuss the leptonic interaction. In order to derive the phenomenological weak metric, a purely leptonic decay has been analyzed, i.e. that of the meson whose meanlife is experimentally known in a wide energy range (30 ÷ 350 GeV).^61,62 This is an almost unique case, motivated by the fact that the decays violate charge conjugation-parity (CP) invariance, and therefore (by virtue of the celebrated CPT theorem) do provide indirect evidence for violation of time reversal…

The following sections are included:

• The Bose–Einstein Correlation

• DSR Treatment of BE Correlation

• Deformed BE Correlation Function and the Strong Metric

• Hadronic Time Deformation

• A Geometric View to Confinement and Asymptotic Freedom

The following sections are included:

• Review of Phenomenological Metrics
  • Energy-Dependent Phenomenological Metrics for the Four Interactions
  • Threshold Energies and Recursive Metrics

• Asymptotic Metrics

Let us now comment on the results obtained by the analysis of the experimental data for all the interactions considered. It is worth stressing that, in all four cases, one obtains evidence for a departure of the space-time metric from the minkowskian one. However, although a quite cautious attitude must be exercised, it would be probably wrong to understate the relevance of such findings. Let us therefore attempt to interpret them. The most straightforward meaning is that (neglecting the gravitational case, where of course a departure from the Minkowskian metric is expected and natural) the three phenomena considered (i.e. the superluminal propagation of evanescent e.m. waves in waveguide, the decay and the BE correlation) do show a breakdown of the usual Lorentz symmetry for the electromagnetic, weak and strong interaction, respectively. But — what's the basic point — one may wonder whether such a breakdown should be considered as an actual or an effective one. Namely, two interpretations can be given to the need for a deformed Minkowski metric in order to describe the above phenomena. The first one is to state that such results correspond to an actual local deformation of the space-time geometry, induced by the interaction considered. This would constitute an evidence in favor of a real analogy of the e.m., weak and strong interactions with the gravitational one, and therefore of the real validity of the solidarity principle for all the four interactions. The second possible interpretation is inspired by the analysis of the e.m. case. Indeed, in the e.m. wave propagation in waveguides, it is perhaps more sound to understand the non-minkowskian metric obtained in Sec. 3.1 as describing, in an effective way, the nonlocal e.m. effects which occur inside the waveguide and give rise to the superluminal propagation. In the same spirit, one can regard the arising of non-minkowskian metrics in the description of decay and BE correlation as due to the presence of nonlocal forces responsible for such phenomena. Otherwise speaking, the non-minkowskian metrics involved in such cases would play an effective role, in the sense that they would actually be a signature of the presence and effectiveness of nonpotential effects in the phenomena considered?…

The following sections are included:

• Theoretical Foundations

• Experimental Setup and Measurement Procedure

• Experimental Results
  • First Experiment
  • Second Experiment
  • An Interpretation by DSR

• LLI Violation Parameter

• Appendix — Einstein and the Aether

The following sections are included:

• LLI Breaking Factor and Relativistic Energy in DSR

• The Electron as a Fundamental Particle and its “Geometrical” Mass

The following sections are included:

• From LLI Breakdown to Energy as Fifth Dimension

• Embedding of Deformed Minkowski Space in a Five-Dimensional Riemannian Space

• Einstein's Equations in Vacuum

The following sections are included:

• References

• Index