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The Extended Born’s Reciprocal Relativity Theory : Division, Jordan, N-ary Algebras, and Higher Order Finsler spaces

The Extended Born’s Reciprocal Relativity Theory : Division, Jordan, N-ary Algebras, and Higher Order Finsler spaces
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  The Extended Born’s ReciprocalRelativity Theory : Division, Jordan,N-ary Algebras, and Higher OrderFinsler spaces Carlos CastroCenter for Theoretical Studies of Physical SystemsClark Atlanta University, Atlanta, Georgia. 30314, perelmanc@hotmail.comApril 2013 Abstract We extend the construction of Born’s Reciprocal Relativity theoryin ordinary phase spaces to an extended phase space based on Quater-nions. The invariance symmetry group is the (pseudo) unitary quater-nionic group U  ( N  + ,N  − , H ) which is isomorphic to the unitary symplec-tic group USp (2 N  + , 2 N  − , C ). It is explicitly shown that the quaternionicgroup U  ( N  + ,N  − , H ) leaves invariant both the quadratic norm (corre-sponding to the generalized Born-Green interval in the extended phasespace) and the tri-symplectic 2-form. The study of Octonionic, Jordanand ternary algebraic structures associated with generalized spacetimes(and their phase spaces) described by Gunaydin and collaborators is re-viewed. A brief discussion on n -plectic manifolds whose Lie n -algebrainvolves multi-brackets and n -ary algebraic structures follows. We con-clude with an analysis on the role of higher-order Finsler geometry inthe construction of extended relativity theories with an upper and lowerbound to the higher order accelerations (associated with the higher ordertangent and cotangent spaces). 1 Introduction : Born’s Reciprocal Relativityin Phase Spaces Born’s Reciprocal Relativity [1] was an extension of Einstein’s special relativitywhere in addition to a maximal light speed (derivative of the position coordi-nates), by reciprocity (”duality”) , there was a maximal bound to the derivativesof the momentum ( maximal force). Born’s Reciprocal Relativity incorporatesthe principle of maximal proper force (related also to acceleration [2]) from the1  perspective of Phase Spaces. In the case of four spacetime dimensions one has an8 D phase space and the invariance U  (1 , 3) Group. The U  (1 , 3) = SU  (1 , 3) × U  (1)group transformations leave invariant the phase-space intervals under rotations,velocity and acceleration boosts as shown by Low [3]. These transformationscan be simplified drastically when the velocity/acceleration boosts are takento lie in the z -direction, leaving the transverse directions x,y,p x ,p y intact ;i.e., the U  (1 , 1) = SU  (1 , 1) × U  (1) subgroup transformations leave invariant thephase-space interval given by (in units of ¯ h = c = 1)( dω ) 2 = ( dT  ) 2 − ( dX  ) 2 +( dE  ) 2 − ( dP  ) 2 b 2 =( dτ  ) 2 [1 +( dE/dτ  ) 2 − ( dP/dτ  ) 2 b 2 ] = ( dτ  ) 2 [1 − m 2 g 2 ( τ  ) m 2 P  A 2 max ] . (1 . 1)where we have factored out the proper time infinitesimal ( dτ  ) 2 = dT  2 − dX  2 in eq-(1.1) and the maximal proper-force is set to be b ≡ m P  A max . m P  is thePlanck mass 1 /L P  so that b = (1 /L P  ) 2 , may also be interpreted as the maximalstring tension when L P  is the Planck scale.The quantity g ( τ  ) is the proper four-acceleration of a particle of mass m inthe z -direction which we take to be defined by the X  coordinate. The interval( dω ) 2 described by Low [3] is U  (1 , 3)-invariant for the most general transfor-mations in the 8 D phase-space. The analog of the Lorentz relativistic factorin eq-(1.1) involves the ratios of two proper forces  . One variable force is givenby mg ( τ  ) and the maximal proper force sustained by an elementary particle of mass m P  is assumed to be F  max = m Planck c 2 /L P  .The transformations laws of the coordinates in that leave invariant the in-terval (1.1) were given by [3]: T   = Tcoshξ  + ( ξ  v X c 2 + ξ  a P b 2 ) sinhξ ξ . (1 . 2 a ) E   = Ecoshξ  + ( − ξ  a X  + ξ  v P  ) sinhξ ξ . (1 . 2 b ) X   = Xcoshξ  + ( ξ  v T  − ξ  a E b 2 ) sinhξ ξ . (1 . 2 c ) P   = Pcoshξ  + ( ξ  v E c 2 + ξ  a T  ) sinhξ ξ . (1 . 2 d )The ξ  v is velocity-boost rapidity parameter and the ξ  a is the force/acceleration-boost rapidity parameter of the primed-reference frame. They are defined re-spectively : tanh ( ξ  v c ) = vc. tanh ( ξ  a b ) = mam P  A max . (1 . 3)The effective  boost parameter ξ  of the U  (1 , 1) subgroup transformations ap-pearing in eqs-(2-2a, 2-2d) is defined in terms of the velocity and acceleration2  boosts parameters ξ  v ,ξ  a respectively as: ξ  ≡   ξ  2 v c 2 + ξ  2 a b 2 . (1 . 4)Straightforward algebra allows us to verify that these transformations leavethe interval of eq- (1.1) in classical phase space invariant. They are are fully con-sistent with Born’s duality Relativity symmetry principle [1] ( X,P  ) → ( P, − X  ).By inspection we can see that under Born reciprocity, the transformations ineqs-(1.2a-1.2d) are rotated into each other, up to numerical b factors in order tomatch units. When on sets ξ  a = 0 in (1.2a-1.2d) one recovers automatically thestandard Lorentz transformations for the X,T  and E,P  variables separately ,leaving invariant the intervals dT  2 − dX  2 = ( dτ  ) 2 and ( dE  2 − dP  2 ) /b 2 separately.Also the transformations leave invariant the symplectic two-form dT   ∧ dE   − dX   ∧ dP   = dT  ∧ dE  − dX  ∧ dP  (1 . 5)For simplicity, unless otherwise indicated, we shall choose the natural units¯ h = c = G = 1 so that b = m P  = L P  = 1.The most general U  ( D − 1 , 1) transformations leaving invariant the quadraticinterval in phase space and the symplectic 2-form were given by [3], in units¯ h = G = c = b = 1 T   = T coshξ  + ( ξ  iv X  i + ξ  ia P  i ) sinhξ ξ . (1 . 6 a ) E   = E coshξ  + ( ξ  iv P  i − ξ  ia X  i ) sinhξ ξ . (1 . 6 b ) X   i = X  i + X  j ( ξ  iv ξ  jv + ξ  ia ξ  ja ) coshξ  − 1 ξ  2 + ( ξ  iv T  − ξ  ia E  ) sinhξ ξ . (1 . 6 c ) P   i = P  i + P  j ( ξ  iv ξ  jv + ξ  ia ξ  ja ) coshξ  − 1 ξ  2 + ( ξ  iv E  + ξ  ia T  ) sinhξ ξ . (1 . 6 d )where the effective  boost parameter ξ  is defined in terms of the velocity andacceleration boosts parameters ξ  iv ,ξ  ia , respectively, as ξ  ≡   ( ξ  iv ) 2 + ( ξ  ia ) 2 , i = 1 , 2 , 3 ,.....,D − 1 (1 . 7)The Eddington-Dirac large numbers coincidence ( and an ultraviolet/infraredentanglement ) can be easily implemented if one equates the upper bound on theproper-four force sustained by a fundamental particle , ( mg ) bound = m P  ( c 2 /L P  ),with the proper-four force associated with the mass of the (observed ) universe M  U  , and whose minimal acceleration c 2 /R is given in terms of an infrared-cutoff  R ( the Hubble horizon radius ). Equating these proper-four forces gives m P  c 2 L P  = M  U  c 2 R ⇒ M  U  m P  = RL P  ∼ 10 61 . (1 . 8)3  from this equality of proper-four forces associated with a maximal/minimalacceleration one infers M  U  ∼ 10 61 m Planck ∼ 10 61 10 19 m  proton = 10 80 m  proton which agrees with observations and with the Eddington-Dirac number 10 80 [4] N  = 10 80 = (10 40 ) 2 ∼ ( F  e F  G ) 2 ∼ ( Rr e ) 2 . (1 . 9)where F  e = e 2 /r 2 is the electrostatic force between an electron and a proton ; F  G = Gm e m  proton /r 2 is the corresponding gravitational force and r e = e 2 /m e ∼ 10 − 13 cm is the classical electron radius ( in units ¯ h = c = 1 ).One may notice that the above equation (1.8) is also consistent with theMachian postulate [4] that the rest mass of a particle is determined via thegravitational potential energy due to the other masses in the universe. In par-ticular, by equating m i c 2 = Gm i  j m j | r i − r j | = Gm i M  U  R ⇒ c 2 G = M  U  R. (1 . 10)Due to the negative binding energy, the composite mass m 12 of a system of two objects of mass m 1 ,m 2 is not equal to the sum m 1 + m 2 > m 12 . We cannow arrive at the conclusion that the minimal acceleration c 2 /R is also thesame acceleration induced on a test particle of mass m by a spherical massdistribution M  U  inside a radius R . The acceleration felt by a test particle of mass m sitting at the edge of the observable Universe ( at the Hubble horizonradius R ) is | a | = GM  U  R 2 (1 . 11)From the last two equations one gets the same expression for the minimal ac-celeration a = a minimal = c 2 R and which is of the same order of magnitude as theanomalous acceleration of the Pioneer and Galileo spacecrafts a ∼ 10 − 8 cm/s 2 .Let us examine closer the equality between the proper-four forces m P  c 2 L P  = M  U  c 2 R ⇒ m P  L P  = M  U  R = c 2 G. (1 . 12)The last term in eq-(1.12) is directly obtained after implementing the Machianprinciple. Thus, one concludes from eq-(1.12 ) that as the universe evolves intime one must have the conserved ratio of the quantities M  U  /R = c 2 /G = m P  /L P  . This interesting possibility, advocated by Dirac long ago, for thefundamental constants ¯ h,c,G,..... to vary over cosmological time is a plausibleidea with the provision that the above ratios satisfy the relations in eq-(1.12)at any given moment of cosmological time. If the fundamental constants do notvary over time then the ratio M  U  /R = c 2 /G must refer then to the asymptotic values of the Hubble horizon radius R = R asymptotic .We provided in [5] six specific results stemming from Born’s reciprocal Rela-tivity and which are not present in Special Relativity. These were : momentum-dependent time delay in the emission and detection of photons; energy-dependent4  notion of locality; superluminal behavior; relative rotation of photon trajecto-ries due to the aberration of light; invariance of areas-cells in phase-space andmodified dispersion relations. One of the most interesting conclusions was thatthere are null hypersurfaces in a flat phase-spaces where points can have su-perluminal v > c behavior in ordinary spacetime, despite corresponding to anull hypersurface in a flat phase-space. Superluminal behavior in spacetime canoccur without having superluminal behavior in C  -spaces [6].In [8] we extend the construction of Born’s Reciprocal Phase Space Rela-tivity to the case of Clifford Spaces and which involve the use of  polyvectors and a lower/upper length scale. A Clifford Phase-Space Gravitational The-ory based in gauging the generalization of the Quaplectic group and invokingBorn’s reciprocity principle between coordinates and momenta (maximal speedof light velocity and maximal force) was provided. The purpose of this work isto continue this line of research and explore further generalizations. 2 Quaternions algebras and Extended Born’sReciprocal Relativity Let us begin with the quaternionic-valued variable Z  µ = Z  (0) µ e 0 + Z  (1) µ e 1 + Z  (2) µ e 2 + Z  (3) µ e 3 , e i e j = − δ  ij e o +  ijk e k , i,j,k = 1 , 2 , 3(2 . 1)Upon using the units ¯ h = G = c = b = 1, the spacetime coordinates X  µ can beregrouped with a triplet of momenta given by the triad of variables P  µ ,U  µ ,V  µ (the imaginary quaternionic components) as follows Z  (0) µ = X  µ , Z  (1) µ = P  µ , Z  (2) µ = U  µ , Z  (3) µ = V  µ (2 . 2)the indices µ,ν  span the values 1 , 2 , 3 ,.....,D . If one has a different choice of units one needs to introduce physical constants (length/mass scales) in order toensure that all quantities in eq-(2.1) have the same physical units.The quaternionic conjugate is¯ Z  µ = Z  (0) µ e 0 − Z  (1) µ e 1 − Z  (2) µ e 2 − Z  (3) µ e 3 (2 . 3)and the norm squared is | Z  | 2 ≡ ¯ Z  µ Z  µ = Z  µ ¯ Z  µ = X  µ X  µ + P  µ P  µ + U  µ U  µ + V  µ V  µ (2 . 4)for Minkowski signature one has X  µ X  µ = ( X  0 ) 2 − ( X  1 ) 2 − ( X  3 ) 2 − ...... − ( X  D − 1 ) 2 , etc .... (2 . 5)5
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