Fins nsler r geom ometry ry and nd deSitter mo momentum space Loret Niccolò arXiv:1407.8143 With: Giovanni Amelino-Camelia, Leonardo Barcaroli, Giulia Gubitosi and Stefano Liberati
Gali lilean Relativity Gali lilean Relativity Invariant (Casimir) Transformation (Boost)
Special Relat ativity Special Relat ativity Invariant (Casimir) Transformation (Boost)
??? Relat ativity ??? Relat ativity Invariant (Casimir) Transformation (Boost)
??? Relat ativity ??? Relat ativity Invariant (Casimir) Transformation (Boost) ???
Deformed Spe pecial l Relat ativity Deformed Spe pecial l Relat ativity κ -Poincaré algebra in bicrossproduct basis
Deformed Spe pecial l Relat ativity Deformed Spe pecial l Relat ativity κ -Poincaré algebra in bicrossproduct basis Casimir Symmetry genera rators representation
Curved momentum-space Curved momentum-space Non-trivial properties of Modified symmetries momentum-space geometry
Curved momentum-space Curved momentum-space Non-trivial properties of Modified symmetries momentum-space geometry (Modified) Dispersion Relation obtained through In the κ -Poincaré (bicro rosspro roduct basis) case
Finsl sler Geometry Finsl sler Geometry Finsler Norm ● Positive function in the tangent bundle ● Homogeneus of degree one in ẋ Velocity-dependent genera ralization of Riemannian metric Rund 1959
Finsler Geometry of a a par article with MDR Finsler Geometry of a a par article with MDR Action of a fr free relativistic particle Girelli, Liberati, Sindoni, PRD 2007
Finsler Geometry of a a par article with MDR Finsler Geometry of a a par article with MDR Action of a fr free relativistic particle By using Hamilton's equation We find Girelli, Liberati, Sindoni, PRD 2007
Finsler Geometry of a a par article with MDR Finsler Geometry of a a par article with MDR Action of a fr free relativistic particle By using Hamilton's equation We find Girelli, Liberati, Sindoni, PRD 2007
Finsl sler Geometry an and κ κ -Poincar -Poincar aré sy symmetries Finsl sler Geometry an and aré sy symmetries We apply this procedure re to the case
Finsl sler Geometry an and κ κ -Poincar -Poincar aré sy symmetries Finsl sler Geometry an and aré sy symmetries We apply this procedure re to the case
Finsl sler Geometry an and κ κ -Poincar -Poincar aré sy symmetries Finsl sler Geometry an and aré sy symmetries We apply this procedure re to the case
Finsler spacetime metric Finsler spacetime metric In terms of momenta
Finsler spacetime metric Finsler spacetime metric In terms of momenta Despite its horri rible aspect this metri ric defines some simple re relati tions: ● Its invers rse is related to the part rticle's MDR ● In terms of g momenta are simply related to velocities
On the invar ariance of the Lag agran angian On the invar ariance of the Lag agran angian Edge terms
On the invar ariance of the Lag agran angian On the invar ariance of the Lag agran angian Edge terms = 0
On the invar ariance of the Lag agran angian On the invar ariance of the Lag agran angian Edge terms = 0 Invari riant Lagrangian
Relat ative Locality and Rainbow metrics Relat ative Locality and Rainbow metrics This invariant Lagrangian suggest us to check whether Invariant t “rainbow” line element This would allow us to satisfy one of the key requirments of rainbow metri rics which is Loret, arXiv:1404.5093
Geod odesic equations Geod odesic equations In terms of ζ( ẋ ):
Geod odesic equations Geod odesic equations In terms of ζ( ẋ ): From the homogeneity of F( ẋ ) one can show th that the metric g ( ẋ ) satisfies
Geod odesic equations Geod odesic equations In terms of ζ( ẋ ): From the homogeneity of F( ẋ ) one can show th that the metric g ( ẋ ) satisfies In terms of g ( ẋ ): Where
Wordli lines and symmetries Wordli lines and symmetries In the k-Poincaré ré case We choose a parametrization
Finsler Killing equation Finsler Killing equation We look for solutions and charges
Finsler Killing equation Finsler Killing equation We look for solutions and charges
Finsler Killing equation Finsler Killing equation We look for solutions and charges
Summar ary and outlook Summar ary and outlook ● Finsler r generalization of Riemannian geometr try can be used to describe spacetime geometry seen by a particle with given (modified) dispersion re relation ● This provides a consistent framework to derive physical properties of the particle: propagation, symmetries ● Equivalent to a ‘ra rainbow’ metri ric associated to classical particles with κ -Poincaré inspired symmetries ● Can it be used to tre reat t more complicated cases, when gravity is introduced? ● How to intro roduce interactions?
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