hadronic mathematics, mechanics and chemistry - Institute for Basic ...

828 RUGGERO MARIA SANTILLIISOAXIOM III. The projection in our spacetime of the isorelativistic laws ofdilation of time t ◦ , contraction of length l ◦ and variation of mass m ◦ with speedare given respectively by:t = ˆγ × t ◦ ,(6.1.13a)l = ˆγ −1 × l ◦ ,m = ˆγ × m ◦ .(6.1.13b)(6.1.13c)ISOAXIOM IV. The projection in our spacetime of the Doppler-Santilli isolawis given by the law (here formulated for simplicity for 90 ◦ angle of aberration):ω = ω o × 1 − ˆβ × cosˆθ ˆ√ , (6.1.14)1 − hatbeta 2ISOAXIOM V. The projection in our spacetime of the isorelativistic law ofequivalence of mass and energy is given by: 7E = m × V 2 max = m × c 2 o × b2 4b 2 3In the above isoaxioms we have= m × c 2 o × n2 3n 2 . (6.1.15)4ˆβ = v × b 3 /c o × b 4 = v × n 4 /c o × n 3 = v/V max , cosˆθ ˆ = cos(θ × b s ) (6.1.16)Since v is always smaller than or equal to the maximal causal speed V max , ˆβ isalways smaller than or equal to one and ˆγ = (1 − ˆβ 2 ) −1/2 cannot take imaginaryvalues as it is the case for special relativity. For isotrigonometric functions, werefer the reader for brevity to EHM Vol. I. For detailed studies of the iso-Dopplerlaw, one may consult EHM Vol./ II, Section 8.5.F.Note that the isoaxioms are not isotopies of the corresponding axioms of specialrelativity, because they characterize major structural departures, such as; the themaximal local speed is not, in general, the speed of light; the energy equivalence isnot given by the familiar expression E = m × c 2 ; etc. These structural deviationsemerge only within physical media and have major implications we shall studylater on, such as the elimination of any need for dark matter..7 As indicated in Section 3.5, the initial formulation of Isoaxiom V wasE = m × c 2 = m × c 2 o × b2 4 = m × c2 on 2 .4However, experimental verifications of isorelativity proved this formulation to be wrong, and had to bereplaced with isolaw (6.1.15). The occurrence reinforced the view that, contrary to popular beliefs inthe 20-th century, the speed of light is not, in general, the maxima. causal speed because physical mediaare generally opaque lo light, in which case the use of the speed of light has no mathematical or physicalmeaning. It happens that in vacuum b 3 = b 4 = 1 and in water b 3 = b 4 , in which case V max = c o, butthis is a mere particular case without universal validity.