Extended Linear and Nonlinear Lorentz Transformations and Superluminality

Dara Faroughy

doi:10.1155/2013/760916

Abstract

Two broad scenarios for extended linear Lorentz transformations (ELTs) are modeled in Section 2 for mixing subluminal and superluminal sectors resulting in standard or deformed energy-momentum dispersions. The first scenario is elucidated in the context of four diverse realizations of a continuous function <svg style="vertical-align:-3.56265pt;width:30.1625px;" id="M1" height="16.6625" version="1.1" viewBox="0 0 30.1625 16.6625" width="30.1625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x1D453" d="M619 670q0 -13 -9 -26t-18 -19q-13 -10 -25 2q-36 38 -66 38q-31 0 -54.5 -50t-45.5 -185h120l-20 -31l-107 -12q-23 -138 -57 -293q-27 -122 -55 -184.5t-75 -109.5q-60 -61 -114 -61q-25 0 -47.5 15t-22.5 31q0 17 31 44q11 8 20 -1q10 -11 31 -19t35 -8q26 0 47 19
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q0 43 -12 57q-11 12 -11 24q0 19 15 35.5t34 16.5q18 0 30.5 -34.5t12.5 -81.5z" /></g><g transform="matrix(.017,-0,0,-.017,24.218,12.162)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg>, with <svg style="vertical-align:-3.56265pt;width:85.324997px;" id="M2" height="16.6625" version="1.1" viewBox="0 0 85.324997 16.6625" width="85.324997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g><g transform="matrix(.017,-0,0,-.017,12.931,12.162)"><path id="x2264" d="M531 71l-474 214v50l474 215v-56l-416 -184l416 -183v-56zM531 -40h-474v50h474v-50z" /></g><g transform="matrix(.017,-0,0,-.017,27.634,12.162)"><use xlink:href="#x1D453"/></g><g transform="matrix(.017,-0,0,-.017,38.548,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,44.429,12.162)"><use xlink:href="#x1D463"/></g><g transform="matrix(.017,-0,0,-.017,51.79,12.162)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,62.397,12.162)"><use xlink:href="#x2264"/></g><g transform="matrix(.017,-0,0,-.017,77.101,12.162)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g> </svg> and <svg style="vertical-align:-3.56265pt;width:107.5375px;" id="M3" height="16.6625" version="1.1" viewBox="0 0 107.5375 16.6625" width="107.5375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D453"/></g><g transform="matrix(.017,-0,0,-.017,10.976,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,16.857,12.162)"><use xlink:href="#x30"/></g><g transform="matrix(.017,-0,0,-.017,25.017,12.162)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,35.607,12.162)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,50.311,12.162)"><use xlink:href="#x1D453"/></g><g transform="matrix(.017,-0,0,-.017,61.224,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,67.106,12.162)"><path id="x1D450" d="M383 397q0 -32 -35 -49q-12 -6 -23 8q-37 45 -84 45t-90 -71q-40 -65 -40 -167q0 -57 22 -86t59 -29q38 0 81.5 24.5t69.5 51.5l16 -21q-44 -53 -104 -84t-109 -31q-56 0 -89.5 41t-33.5 117q0 61 30 124t79 105q33 28 81 50.5t86 22.5q34 0 59 -15.5t25 -35.5z" /></g><g transform="matrix(.017,-0,0,-.017,74.007,12.162)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,84.614,12.162)"><use xlink:href="#x3D"/></g><g transform="matrix(.017,-0,0,-.017,99.318,12.162)"><use xlink:href="#x31"/></g> </svg>, which is fitted in the ELT. What goes in the making of the ELT in this scenario is not the boost speed <svg style="vertical-align:-0.1638pt;width:7.4875002px;" id="M4" height="7.9124999" version="1.1" viewBox="0 0 7.4875002 7.9124999" width="7.4875002" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.638)"><use xlink:href="#x1D463"/></g> </svg>, as ascertained by two inertial observers in uniform relative motion (URM), but <svg style="vertical-align:-3.56265pt;width:55.049999px;" id="M5" height="16.6625" version="1.1" viewBox="0 0 55.049999 16.6625" width="55.049999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D463"/></g><g transform="matrix(.017,-0,0,-.017,11.197,12.162)"><path id="xD7" d="M528 54l-36 -38l-198 201l-198 -201l-36 38l197 200l-197 201l36 38l198 -202l198 202l36 -38l-197 -201z" /></g><g transform="matrix(.017,-0,0,-.017,24.949,12.162)"><use xlink:href="#x1D453"/></g><g transform="matrix(.017,-0,0,-.017,35.862,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,41.743,12.162)"><use xlink:href="#x1D463"/></g><g transform="matrix(.017,-0,0,-.017,49.104,12.162)"><use xlink:href="#x29"/></g> </svg>. The second scenario infers the preexistence of two rest-mass-dependent superluminal speeds whereby the ELTs are finite at the light speed <svg style="vertical-align:-0.1638pt;width:7.0250001px;" id="M6" height="7.9499998" version="1.1" viewBox="0 0 7.0250001 7.9499998" width="7.0250001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D450"/></g> </svg>. Particle energies are evaluated in this scenario at <svg style="vertical-align:-0.1638pt;width:7.0250001px;" id="M7" height="7.9499998" version="1.1" viewBox="0 0 7.0250001 7.9499998" width="7.0250001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D450"/></g> </svg> for several particles, including the neutrinos, and are auspiciously found to be below the GKZ energy cutoff and in compliance with a host of worldwide ultrahigh energy cosmic ray data. Section 3 presents two broad scenarios involving a number of novel nonlinear LTs (NLTs) featuring small Lorentz invariance violations (LIVs), as well as resurrecting the notion of simultaneity for limited spacetime events as perceived by two observers in URM. These inquiries corroborate that NLTs could be potent tools for investigating LIVs past the customary LTs.

Highlights

The standard theory of special relativity (SR) is a simple framework, relying on a set of well establish postulates, for mixing and studying the space and time coordinate transformations between systems of inertial frames (IFs) in relative uniform motion
In the case of nonlinear Lorentz transformations (LT) (NLTs), especially the ones we find most promising and will cover in Section 3, finding the inverse transformations algebraically could be daunting, if not impossible
All available experimental data on the neutrino SL are consistent with the neutrinos moving at the light speed; yet, the quality of the data is such that it can neither confirm nor refute any prospective tiny SL, and even much more so at the level of numbers we provided earlier