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Superluminal speeds , an experiment and some theoretical considerations

Superluminal speeds , an experiment and some theoretical considerations. B. Allés INFN Pisa. Málaga, June 2012. [ Phys . Rev. D85 047501 (2012) or arXiv:1111.0805]. CNGS Experiment. CNGS Experiment. Gran Sasso setup. Cern production. p g raphite π, K μ , ν. CNGS Experiment.

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Superluminal speeds , an experiment and some theoretical considerations

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  1. Superluminalspeeds, an experiment and some theoreticalconsiderations B. Allés INFN Pisa Málaga, June 2012 [Phys. Rev. D85 047501 (2012) or arXiv:1111.0805]

  2. CNGS Experiment

  3. CNGS Experiment GranSassosetup Cernproduction p graphite π, K μ, ν

  4. CNGS Experiment • m(π)=140 MeV • m(K)=494 MeV • m(μ)=106 MeV • m(e)=0.5 MeV • m(ν)≈0 small transversemomentum in π→μν maximizes the number of collimatedν. π, K μ,e ν

  5. CNGS Experiment • From initialprotonsatCern, neutrinoseventswererecordedat Gran Sasso and onlytrue candidates. • Since an analogousexperiment (MINOS atFermilab in 2007) foundan indication of possiblesuperluminal neutrino speed, itwasinteresting to determinethisspeedalsoat CNGS. • Dividing the totaldistanceCern-SPS to Gran Sasso facility by the time of flight of neutrinos, itcame the astonishingresult: (v-c)/c=2.37 ±0.32

  6. Thisresultclashes with the basicprinciples of special relativity. If a faster-than-light messageissent by an observeratresttowardsanotherobserver in motion and the latterreplies back the message, the answermightarrivebefore the messagewassent!!

  7. CNGS Experiment (end) Ereditato makingstatementsbeforediscovering a misconnection among GPS cables. By March 2012 ICARUS collaborationfound a resultcompatible with

  8. …butnotallhasbeen a waste of time… • The surprisingfindings of OPERA team spurred a great deal of research in order to understandwhatwasgoing on aboutneutrinos and relativity.

  9. We are going to seehow the inclusion of gravitychangesdrasticallyallconclusionsaboutsuperluminalspeeds. • The aboveargumentdemonstratingthatsuperluminalsignalswould generate logicalcontradictionsappliesrigorouslyonly in special relativity: the theory of Einstein withoutgravity. • Moreover, whatreally can attainsuperluminalvalues are the meanvelocities, not the instantaneousones… • …butmost of the velocitiesusuallymeasured are preciselymean (in particular the onedescribed by the OPERA team). • An exception are the velocities of celestialobjectswhenthey are determined by studying the Doppler effect on the spectrum of light emitted by the object.

  10. A tour aroundrelativity Don’t be afraid, just smile!!

  11. A tour aroundrelativity The pivotalparadigm in relativityis the metric Greekindices (μ,ν,α,β,…) stand indistinctly for time or spatialcoordinates. Noughtindex (0) stands for the time coordinate. Latin indices (,…) indicate spatialcoordinates. run from 1 to 3. μναβ,… run from 0 to 3.

  12. A tour aroundrelativity =issymmetric, aseverymetric! Given infinitesimalincrements the quantity iscalledsquaredproperlength.

  13. A tour aroundrelativity It isclearthat, under coordinate transformations, goes over to Since, mathematically, a metricis a tensor, ittransforms in such a way as to leave invariant. THEREFORE, UNDER ANY COORDINATE TRANSFORMATION REMAINS UNVARIED.

  14. A tour aroundrelativity Given an energy-momentumtensordescribing the matter contents, the followingexpression provides a non-linear partialdifferentialequation of secondorder for the metric. is the Ricci tensorand the scalar of curvature. is the mattertensor. For instance, for a certain density of matterρ, and all the otherzero. Weshallnot solve thisequation, but show twophysically interestingsolutions…

  15. A tour aroundrelativity NO MATTER: Minkowskimetric SPHERICAL MASS DISTRIBUTION: Schwarzschildmetric

  16. A tour aroundrelativity A VERY IMPORTANT ADVICE Coordinates are mere labels, likestreetnumbers. THUS: Distances or time intervalscannot be calculated by just coordinate differences (like in usualEuclideanspacewith Cartesiancoordinates).

  17. A tour aroundrelativity What is(or )? Itdepends! Much as the spatialEuclideanmetricserves for calculatingdistancesin , (or ) allows to determineboth time intervals and spatialdistances.

  18. A tour aroundrelativity TIME INTERVALS The time displayed by a clock at restat the spatial position (coordinates) is/. It ispresumedthat the 0-coordinate (whichis) changesbut the spatialcoordinatesremainfixed and equal to .

  19. A tour aroundrelativity The case of the Minkowskimetricisverysimple and alsowell-known: the ticking of the clock time equals the successive valuestaken by the time coordinate . However, the case of the Schwarzschildmetricsuggeststhat clocks atdifferentheightstickdifferently. Indeed, given a unique coordinate time interval, , clocks atradialcoordinates and indicate CLOCK TIME AT CLOCK TIME AT Therefore the ratio of the twotimesisnot 1…

  20. A tour aroundrelativity Robert Pound (picture) and Glen Rebkademonstrated experimentally the correctness of the above statement. In 1959 theycomparedatomic clocks separated by a height of 22.5 meters, (resorting to the recentlydiscovered Mösbauereffect). The agreement wasexcellent.

  21. A tour aroundrelativity SPATIAL DISTANCES Spatial distances are the value of atfixed time coordinate.

  22. A tour aroundrelativity Therefore the truedistancebetween and in a Schwarzschildmetric (whatonewouldmeasure by countinghowmanyrulers can be laid in order to exactly cover the separation from to ) isgiven by TRUE DISTANCE = NOTE: The truedistanceisNOT because coordinates are just labels!

  23. A tour aroundrelativity TRUE DISTANCE For the Earth, thiscorrectionisextremely small: mm. For Earth radius and (),

  24. A tour aroundrelativity Ifbothspatial and temporalcoordinates are left to vary, we can follow the trajectory of movingparticles. An interesting case isthat of light. In special relativity Hence, for light rays always.

  25. A tour aroundrelativity Thisistrue for special relativity, thatis, in absence of gravity. WHAT ABOUT GRAVITY? Does light travelat299,792,458meters/secondalso in gravity?

  26. A tour aroundrelativity WHAT TO DO? PRINCIPLE OF EQUIVALENCE

  27. A tour aroundrelativity The PRINCIPLE OF EQUIVALENCE states: Ifyou are small enough and you are falling down, you cannotfeelgravity.

  28. A tour aroundrelativity More mathematically: atanypoint in spacetime a coordinate transformation can be foundsuchthat in a closeneighbourhood of thispoint the metric in the new coordinatesisMinkowski. Since in thesecoordinates(and close to the chosenpoint) everythingbehavesas in special relativity, the speed of light issurely and Ergo, being coordinate invariant, itremainsconstant in the original (notmotivated by the principle of equivalence) coordinates. Hence light rays go atLOCALLY and .

  29. CONGRATULATIONS!! WE HAVE NOW GOT ALL NECESSARY INGREDIENTS.

  30. Let usconsider a light rayfollowing the spatialtrajectory from point to pointwhereparametervaries from at to at Our goal is to calculate the MEAN VELOCITY of the light ray. The totaltravelleddistanceis whereoverdotsdenoteλderivatives.

  31. As clocks rundifferently in differentspatial positions, wehave to clearlyspecify WHERE the observerlies. We choose to placehimat the end of the trajectory, , althoughthisdetailislargelyimmaterial. So, according to whatwehave just learnt, FIRSTLYitshould be obtained the coordinate time interval needed by the light ray to travel from to and SECONDLY the time reallydisplayed by the observer’s clock atwill be determined.

  32. To findweresort to the vanishingproperty of in light. Specifically: Consequently,

  33. The ratio gives the desiredMEAN VELOCITY. The meanvaluetheoremenablesus to simplify the expression. There isindeed an intermediate for which

  34. Finally, The surpriseisthatthis ratio isbarelyequal to .

  35. In summary, meanvelocitiesneednot be But instantaneousvelocities ARE always, (asprescribed by the PRINCIPLE OF EQUIVALENCE). Indeed, ifwemakepoints and collapse, thenalso the intermediate pointlabelled with tends to and so the above ratio of radicandstends to 1.

  36. The Schwarzschildmetricoffers an analyticallycalculableinstance of the aboveeffect:

  37. The Schwarzschildmetricoffers an analyticallycalculableinstance of the aboveeffect:

  38. The terrestrialgravityistooweak. Effectslike the onedescribed in this seminar are completelynegligible. In particular, the excessvelocityclaimed by the OPERA team isalmostthreeorders of magnitudelargerthanourprediction. Thereis, though, an oldexperimentthatnecessarily hasdetectedsuch an effect: the LUNAR LASER RANGING It consists in sending a laser pulse to the Moon and observing the reflectedlight. The (round trip time)/2 multiplied by yields the Earth-Moon distance… BUT THE SPEED OF THE SIGNAL IS NOT

  39. Lunar Laser Ranging Retroreflectors:

  40. Lunar Laser Ranging If the effectdescribedhereistakeninto account, the correctionturns out to be about 53 cm., well beyond the stipulatedprecision of the experiment (about 1-2 cm.). Nevertheless, the LLR collaborationrecorded the coordinates of the Moon and otherplanets and satellites (in a certain solar system coordinate set). In this way, theyimplicitlyincluded the effectsstudiedhere.

  41. Conclusions • Care must be taken in defining a velocity in the context of general relativity. • Local definitionscomply with the basictenets of the theory: theyyieldexactlyfor light rays but… • non-localdefinitions (like the above-analysed MEAN VELOCITY) do notnecessarilyobservethisrule. • Relativitymaystillteachusmanysurprisingaspects of Nature…

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