On the invariance of the speed of light

Transcription

1 On he inariance of he speed of ligh Shan Gao Uni for he Hisory and hilosophy of Science and Cenre for Time, Uniersiy of Sydney, NSW 006, sralia. Insie for he Hisory of Naral Sciences, Chinese cademy of Sciences, Beijing 0090,. R. China. I has been arged ha he eisence of a minimm obserable ineral of space and ime (MOIST) is a model-independen resl of he combinaion of qanm field heory and general relaiiy. In his paper, I promoe his resl o a fndamenal poslae, called he MOIST poslae. I is arged ha he poslae leads o he eisence of a maimm signal speed and is inariance. This new resl may hae wo ineresing implicaions. On he one hand, i sggess ha he MOIST poslae can eplain he inariance of he speed of ligh and hs i migh proide a deeper logical fondaion for special relaiiy. Moreoer, i sggess ha he speed consan c in modern physics is no he acal speed of ligh in acm, b he raio of he minimm obserable lengh o he minimm obserable ime ineral. On he oher hand, he resl also sggess ha he eising eperimens confirming he inariance of he speed of ligh already proide obseraional eidence o sppor he MOIST poslae.. Inrodcion I has been widely arged ha he eisence of a minimm obserable ineral of space and ime (MOIST) is a model-independen resl of he proper combinaion of qanm field heory and general relaiiy (see, e.g. [-]). This sggess ha he eisence of a MOIST may hae a firmer basis beyond he eising heories, and i is probably a more fndamenal feare of nare. On he oher hand, he eising heories are sill based on some neplained poslaes. For eample, special relaiiy, he common basis of qanm field heory and general relaiiy, poslaes he inariance of he speed of ligh in all inerial frames, b he heory does no eplain why. In he long rn, hese poslaes need o be eplained by some more fndamenal ones. Therefore, i may be sefl o eamine he relaionship beween MOIST and he eising heories from he opposie direcion. In his paper, I will inesigae he implicaions of he eisence of a MOIST for special relaiiy. I will arge ha i is a more fndamenal poslae ha can eplain he inariance of he speed of ligh. I is well known ha he key ingrediens for he appearance of minimm obserable space and ime inerals Noe ha he minimm obserable space and ime inerals are ofen loosely called minimm lengh and minimm ime in he lierare. Thogh Einsein originally based special relaiiy on wo poslaes: he principle of relaiiy and he consancy of he speed of ligh he laer hogh ha he niersal principle of he heory is conained only in he poslae: The laws of physics are inarian wih respec o Lorenz ransformaions beween inerial frames [3]. Noe ha he consancy of he speed of ligh denoes ha he speed of ligh in acm is consan independenly of he moion of he sorce, in a leas one inerial frame.

2 a he lanck scale are qanm field heory (h, c) and general relaiiy (G, c), and he formlae of he hg hg lanck ime and lanck lengh narally conain h, c and G, namely and l 5. 3 c c When iewing he connecion from he opposie direcion, we can obain he speed of ligh c, from he minimm obserable space and ime inerals, and l, namely c l /. s we will see, his is no merely a simple mahemaical ransform; raher, i may hae some ineresing implicaions, no only for he meaning of special relaiiy, b also for he eperimenal eidence of he eisence of a MOIST.. The MOIST poslae lhogh qanm field heory and general relaiiy are boh based on he concep of coninos spaceime, i has been arged ha heir proper combinaion leads o he eisence of he lanck scale, a lower bond o he ncerainy of disance and ime measremens [-]. For eample, when we measre a space ineral near he lanck lengh he measremen will ineiably inrodce an ncerainy comparable o he lanck lengh, and as a resl we canno accraely measre a space ineral shorer han he lanck lengh. Moreoer, differen approaches o qanm graiy also lead o he eisence of a minimm lengh, a resolion limi in any eperimen []. In his paper, I will promoe his resl o a fndamenal poslae 3 : The MOIST oslae: There are minimm obserable space and ime inerals, which are he lanck lengh and lanck ime respeciely. The poslae implicily assmes he alidiy of he principle of relaiiy. I means ha he minimm obserable lengh and he minimm obserable ime ineral are he same in all inerial frames. If he minimm obserable space and ime inerals are differen in differen inerial frames, hen here will eis a preferred Lorenz frame, while his conradics he principle of relaiiy. 3. Maimm signal speed and is inariance Now I will analyze he possible conseqences of he MOIST poslae. In pariclar, I will arge ha i imposes a ery sringen resricion on he coninos ransmission of a physical signal, and een a he normal energy scale i also has an ineresing conseqence. Firs of all, he MOIST poslae reqires ha any physical change dring a ime ineral shorer han he lanck ime,, is nobserable, or in oher words, a physically obserable change can only happen dring a ime ineral no shorer han he lanck ime. Oherwise we can measre a ime ineral shorer han he lanck ime by obsering he physical change, which conradics he MOIST poslae. Howeer, he poslae does no reqire ha a nonphysical change (e.g. moemen of a shadow) or an nobserable 3 I is worh noing ha his poslae is no implied b only moiaed by he eising argmens for he eisence of minimm obserable space and ime inerals. One reason is ha hese argmens implicily assme he (approimae) alidiy of boh qanm heory and general relaiiy down o he lanck scale (e.g. []), b his assmpion may be debaable (see also [4]).

3 physical change (e.g. see below) canno happen dring a ime ineral shorer han he lanck ime. Ne he MOIST poslae reqires ha he coninos ransmission of a physical change oer a disance shorer han he lanck lengh, l, is nobserable4, b i does no prohibi he happening of sch ransmissions eiher. Cerainly, a ransmission oer a disance no shorer han he lanck lengh is sill obserable 5. Noe ha he ransmission of an obserable physical change (e.g. change of a ligh plse from being absen o being presen) corresponds o he ransmission of informaion or energy, which is sally called he ransmission of a physical signal 6. For conenience I will se his common parlance in he following discssions. Le s consider he coninos ransmission of a physical signal in an inerial frame 7. If he signal moes wih a speed larger han, hen i will moe more han one c l / l dring one, and hs moing one l, which is physically obserable in principle, will correspond o a ime ineral shorer han one dring he ransmission. This conradics he MOIST poslae, which reqires ha a physically obserable change can only happen dring a ime ineral no shorer han he lanck ime,. By comparison, he coninos ransmission of a physical signal wih a speed smaller han c is permied, as dring he ransmission he signal will moe less han one l dring a ime ineral shorer han one, while he displacemen smaller han one l is physically nobserable according o he MOIST poslae8. This argmen shows ha he MOIST poslae leads o he eisence of a maimm signal speed for he coninos ransmission of a physical signal, which is eqal o he raio of he minimm obserable lengh o he minimm obserable ime ineral, namely l / c ma. Since he minimm obserable ime ineral and he minimm obserable lengh are he same in all inerial frames, he maimm signal speed for he coninos ransmission of a physical signal will be c in 4 This kind of nobserabiliy is no only a he indiidal leel b also a he saisical leel. We may ndersand his resl by hinking ha he signal (e.g. he wae fncion of a microscopic paricle) has a spaial fzziness no smaller han one l. 5 For a signal wih a posiion ncerainy mch larger han is ransmission disance, which is no shorer han he lanck lengh, he ransmission is sill obserable a he saisical leel. 6 For he sake of laer discssions, i is also worh noing ha defining he ransmission speed of a physical signal is no so simple. n acal physical signal wih a finie een e.g. a plse of ligh raels a differen speeds in a media. Roghly speaking, he larges par of he plse raels a he grop elociy, and is earlies par raels a he fron elociy. Under condiions of normal dispersion, he grop elociy can represen he signal speed, namely he acal propagaion speed of informaion or energy. In pariclar, for a microscopic paricle moing in acm as a physical signal, he signal elociy can be defined as he grop elociy of is wae packe. B in an anomalosly dispersie medim where he grop elociy eceeds he speed of ligh in acm [], he grop elociy no longer represens he signal elociy. For hese siaions, he signal elociy is sally defined as he fron elociy, namely he speed of he leading edge of he signal [3]. Howeer, his definiion is no operaional in acal eperimens. n operaional definiion of signal elociy may be based on he signal-o-noise raio, which closely relaes o qanm flcaions [4]. 7 In he discssions of his secion he speed of signal always denoes he wo-way speed, and he speed of ligh is always he wo-way speed of ligh. This wo-way speed can be eperimenally measred independen of any clock synchronizaion scheme. For an arbirary conenion of simlaneiy, he sal Lorenz ransformaions in special relaiiy, which is based on he Einsein synchronizaion, will be replaced by he Edwards-Winnie ransformaions [5-6]. 8 This sggess ha space and ime ms boh hae a minimm obserable ineral; oherwise eiher he coninos ransmission of a physical signal is impossible or he signal speed can be infinie, boh of which conradic eperience. For insance, consider he siaion ha ime has a minimm obserable ineral b space has no. Then a physical signal can moe an arbirarily shor disance ha is physically obserable. B for a signal moing wih any finie speed, moing an obserable disance shorer han will correspond o a ime ineral shorer han one, while his conradics he eisence of a minimm obserable ime ineral.

4 eery inerial frame. Now I will arge ha his maimm speed c is inarian in all inerial frames. Sppose a physical signal moes in he direcion wih speed c in an inerial frame S. Then is speed will be eiher eqal o c or larger han c in anoher inerial frame S wih a elociy in he direcion relaie o S. Since c is he maimm signal speed in eery inerial frame, he speed of he signal in S can only be eqal o c. This resl also means ha when he signal moes in he direcion wih speed c in he inerial frame S, is speed will be also c in he inerial frame S wih a elociy in he direcion relaie o S. Since he inerial frames S and S are arbirary, we can reach he conclsion ha if a signal moes wih he speed c in an inerial frame, i will also moe wih he same speed c in all oher inerial frames. This proes he inariance of speed c. Here is anoher argmen for he inariance of speed c. Sppose a signal moes in he direcion wih speed c in an inerial frame S. Then is speed will be eiher c or smaller han c in anoher inerial frame S wih a elociy in he direcion relaie o S. If is speed is smaller han c in S, say c-, hen here ms eis a speed larger han c- and a speed smaller han c- in S ha correspond o he same speed in S de o he coniniy of elociy ransformaion and he maimm of c. This means ha when he signal moes wih a cerain speed in frame S is speed in frame S will hae wo possible ales, which is impossible. Ths he signal moing wih speed c in S also moes wih speed c in S, which has a elociy in he direcion relaie o S. This resl also means ha when a signal moes in he direcion wih speed c in S, is speed is also c in S wih elociy in he direcion relaie o S. Since he inerial frames S and S are arbirary, his also proes ha he maimm signal speed c is inarian in all inerial frames 9. To sm p, I hae arged ha he MOIST poslae (i.e. assming he eisence of a minimm obserable ineral of space and ime a he lanck scale) leads o he eisence of a maimm signal speed c, and his speed is inarian in eery inerial frame. I is well known ha he inariance of he speed of ligh has been confirmed by eperimens wih ery high precision [8], and no iolaion of Lorenz inariance has been fond eiher [9]. Therefore, we may say ha he MOIST poslae already has eperimenal sppor. On he oher hand, he poslae can eplain he inariance of he speed of ligh and hs migh proide a deeper logical fondaion for special relaiiy, he common basis of qanm field heory and general relaiiy. Le me gie a more deailed analysis of his implicaion. 4. Relaiiy wiho ligh Special relaiiy is originally based on wo poslaes: he principle of relaiiy and he consancy of he speed of ligh. B as Einsein laer admied o some een [0], i is an incoheren mire []; he 9 Here one may objec ha I shold firs sae clearly he spaceime ransformaions before he analysis of speed ransformaion. Howeer, on he one hand, I are rying o derie he fndamenal poslae ha deermines he spaceime ransformaions, and on he oher hand, as I hae arged aboe, he spaceime ransformaions are no needed o derie he relaion beween he maimm speeds in wo inerial frames, and he laer can be obained only from some basic reqiremens sch as he coniniy of speed ransformaion ec. Besides, i is worh noing ha a similar argmen for he inariance of a maimm speed was also gien by Rindler in [7].

5 firs principle is niersal in scope, while he second is only a pariclar propery of ligh which has obios elecrodynamical origins in Mawell s heory. In iew of his problem, here has been a lasing aemp ha ries o drop he ligh poslae from special relaiiy, which can be raced back o Ignaowski [] (see also [3-4]). I has been fond ha based only on homogeneiy of space and ime, isoropy of space and he principle of relaiiy, one can dedce Lorenz-like ransformaions wih an ndeermined inarian speed / K : K K Unlike special relaiiy ha needs o assme he inariance of he speed of ligh an inarian speed narally appears in he heory, which is sally called relaiiy wiho ligh. This is a srprise indeed. Howeer, since he ale of he inarian speed can be infinie or finie, he heory of relaiiy wiho ligh acally allows wo possible ransformaions: Galilean and Lorenzian. This raises serios dobs abo he connecion beween he heory and special relaiiy. Some ahors dobed ha he heory is indeed relaiisic in nare [4], and ohers sill insised ha he ligh poslae in special relaiiy is sill needed o derie he Lorenz ransformaions [5-7]. Indeed, een if eperience can help o deermine he inarian speed and eliminae he Galilean ransformaions, and een if he eperience may no refer o any properies of ligh in an essenial way [8-9], here is sill one mysery neplained. I is why he inarian speed is finie. In oher words, we need o frher eplain he finieness of he inarian speed by some more fndamenal poslaes. If sccessfl, his will esablish a genine heory of relaiiy wiho ligh which may hen lead s o a deeper ndersanding of spaceime and relaiiy. s I hae arged in he las secion, he MOIST poslae leads o a maimm signal speed, ma l / c, which is inarian in all inerial frames. Ths he inarian speed () / K in he eising heory of relaiiy wiho ligh can be deermined by he minimm obserable space and ime inerals 0, and he deermining relaion is K / l /. Besides, his also proides a possible new eplanaion of he speed consan c in special relaiiy (as well as in qanm field heory and general relaiiy); i is no he acal speed of ligh in acm (hogh which may be also eqal o c), b he raio of he minimm obserable lengh o he minimm obserable ime ineral. Once we hae dedced he inariance of speed c in erms of he MOIST poslae, special relaiiy will gain a new formlaion. I can be based on he MOIST poslae, which saes ha he minimm obserable space and ime inerals, l and, are inarian in all inerial frames. The heory can be aken as a more complee heory of relaiiy wiho ligh. 0 Noe ha he MOIST poslae can be compaible wih he homogeneiy of space and ime and he isoropy of space, and hs he deriaion of he Lorenz-like ransformaions in he heory of relaiiy wiho ligh is sill alid nder he poslae.

6 5. Frher discssions In his secion, I will presen some frher discssions on he sggesed relaion beween he MOIST poslae, maimm signal speed and special relaiiy. I will also briefly discss oher possible implicaions of he poslae and is relaion wih discree spaceime. Firs of all, alhogh he MOIST poslae leads o he eisence of a maimm signal speed c for coninos ransmissions, i does no preclde he sperlminal coninos ransmissions ha do no correspond o acal informaion or energy ransmissions. Two well-known eamples are sperlminal ligh plse propagaion and he hypoheical achyons. Eperimens hae shown ha he grop elociy of a ligh plse in an anomalosly dispersie media (e.g. aomic caesim gas) can be mch larger han c []. B he sperlminal ligh plse propagaion does no correspond o he sperlminal ransmission of a physical signal, and i can be shown ha he signal speed is sill eqal o or smaller han c in his case [4]. Similarly, a consisen heory of achyons also reqires ha he achyons canno be sed o send signals wih a speed larger han c from one place o anoher [30]. Besides, he MOIST poslae does no preclde he eisence of sperlminal nonlocal signals eiher. If here is some mechanism o realize nonlocal signal ransmission, hen is signal speed can be larger han c, and he nonlocal process may also iolae he Lorenz inariance [3]. B he signal speed in his case also has an pper limi depending on he disance de o he limiaion of he MOIST poslae, which is eqal o he raio of ransmission disance o he lanck ime. Ne he MOIST poslae may hae more implicaions. The eisence of an inarian speed is only is implicaion for he coninos eolion of he wae fncion or qanm field. There may eis anoher disconinos and nonlinear qanm eolion, he dynamical collapse of he wae fncion, and he MOIST poslae may also impose resricions for his process [35]. Since he effec of a dynamical collapse eolion depends no only on ime draion b also on he wae fncion iself (e.g. is energy densiy disribion), dring an arbirarily shor ime ineral he effec can always be obserable a he saisical leel for some wae fncions. Howeer, he MOIST poslae demands ha all obserable processes shold happen dring a ime ineral no smaller han he lanck ime,, and hs each iny collapse ms happen dring one or more. Moreoer, since here are infiniely many possible posiions where he collapse can happen a any ime, he draion of each iny collapse will be eacly one for mos ime; when he ime ineral becomes larger han one he collapse will happen in oher posiions wih a probabiliy almos eqal o one. This means ha he dynamical collapse of he wae fncion canno be coninos b be essenially discree. I has been shown ha sch a discree model of dynamical collapse is consisen wih he eising eperimens and also has some ineresing predicions [36]. Thirdly, I will briefly discss he relaion beween he MOIST poslae and discree spaceime. On he one hand, he poslae reqires ha a space and ime ineral shorer han he lanck scale is nobserable, and hs i can be regarded as a minimm reqiremen of spaceime discreeness in he This also implies ha he main dynamical collapse models based on coninos spaceime are inconsisen wih he MOIST poslae.

7 obseraional sense. If an arbirarily shor ineral of space and ime is always obserable, hen space and ime will be infiniely diisible and canno be discree. On he oher hand, he MOIST poslae does no imply ha spaceime is discree in he onological sense. I is also possible ha spaceime iself is sill coninos b physical laws do no permi he resolion of spaceime srcres below he lanck scale. By comparison, here is a sronger reqiremen of spaceime discreeness, namely ha spaceime iself is discree. For insance, one may frher impose a limiaion sronger han he MOIST poslae, e.g., ha an nobserable change does no happen or no change happens dring a ime ineral shorer han he lanck ime. Howeer, here are a leas wo worries abo sch eension. Firs i can neer be esed wheher any change happens or no wihin he lanck ime de o he eisence of a minimm obserable ime ineral. Ne i seems ha coninos spaceime may be sill sefl as a descripion framework, een hogh all obserable physical changes saisfy he reqiremens of he MOIST poslae 3. Lasly, i is worh noing ha he MOIST poslae has no sal problem of Lorenz conracion faced by discree space. I is well known ha he lengh conracion in special relaiiy apparenly conradics he onological discreeness of space. There are some possible approaches o sole his apparen inconsisency (see [3] for a reiew). For eample, one may resor o he eisence of a preferred Lorenz frame or sill insis on he Lorenz inariance b resor o some form of deformed Lorenz ransformaions, as in some models of dobly special relaiiy [9-0] 4. By comparison, he MOIST poslae is compaible wih he Lorenz conracion and does no lead o he eisence of a preferred Lorenz frame. B he obserabiliy of space and ime inerals will become relaie. For insance, in some inerial frames, he moing disance of a signal is longer han he lanck lengh and is obserable, while in oher inerial frames he disance may be shorer han he lanck lengh by Lorenz conracion and hs is nobserable. Wheher his is a poenial problem deseres frher sdy. In conclsion, I hae arged ha he eisence of a minimm obserable ineral of space and ime (MOIST) leads o he eisence of a maimm signal speed and is inariance. This resl may hae wo ineresing implicaions. Firs i sggess ha he MOIST poslae is a fndamenal poslae ha can eplain why he speed of ligh is inarian in all inerial frames, and hs i migh proide a deeper logical fondaion for special relaiiy. Besides, i sggess ha he speed consan c in modern physics is no he acal speed of ligh in acm, b he raio of he minimm obserable lengh o he minimm obserable ime ineral. Ne he resl also sggess ha he eising eperimens confirming he inariance of he speed of ligh already proide eperimenal sppor o he MOIST poslae. There are already many models of discree spaceime in he onological meaning. For eample, in loop qanm graiy, he discreeness of space is represened by he discree eigenales of area and olme operaors [5-6], while in he casal se approach o qanm graiy, one has a direc discreizaion of he casal srcre of coninm Lorenzian manifolds [7-8]. Besides, in dobly special relaiiy [9-0] and riply special relaiiy [] here is also an objecie inarian minimm lengh. In hese heories, he classical Minkowski spaceime is replaced by a qanm spaceime sch as κ-minkowski noncommaie spaceime. 3 In my opinion, here are wo reasons o sppor his iew. Firs i seems ha here is no physical limiaion on he difference of he happening imes of wo casally independen eens, e.g., he difference is no necessarily an inegral mliple of he lanck ime. Ne i seems ha coninos spaceime is sill needed o describe nonphysical sperlminal moion, e.g., sperlminal ligh plse propagaion in an anomalosly dispersie media. Dring sch sperlminal propagaions, moing one l will correspond o a ime ineral smaller han one. 4 There was a recen debae on wheher he model of deformed special relaiiy is consisen [33-34].

8 cknowledgmens I am ery graefl o Dean Rickles, Hw rice, and Sabine Hossenfelder for helpfl discssions. I am also graefl o he paricipans of Fondaions of hysics Seminar a he Uniersiy of Sydney for discssions. This work was sppored by he osgradae Scholarship in Qanm Fondaions proided by he Uni for Hisory and hilosophy of Science and Cenre for Time (SOHI) of he Uniersiy of Sydney. References [] Garay, L. J. (995). Qanm graiy and minimm lengh. In. J. Mod. hys. 0, 45. [] dler, R. J. and Saniago, D. I. (999). On graiy and he ncerainy principle. Mod. hys. Le. 4, 37. [3] Einsein,. (949) obiographical Noes. In lber Einsein: hilosopher-scienis... Schilpp, ed. The Library of Liing hilosophers, ol. 7. Eanson, Illinois: Norhwesern Uniersiy ress. [4] Calme X., Hossenfelder, S., and ercacci, R. (00). Deformed special relaiiy from asympoically safe graiy, hys. Re. D 8, 404. [5] Roelli, C. and Smolin, L. (995). Discreeness of area and olme in qanm graiy, Ncl. hys., B 44, 593. [6] Roelli, C. (004). Qanm Graiy. Cambridge: Cambridge Uniersiy ress. [7] Bombelli, L., Lee, J., Meyer, D., and Sorkin, R. D. (987). Spaceime as a casal se hys. Re. Le. 59, 5. [8] Dowker, F. (006) Casal ses as discree spaceime, Conemporary hysics, 47(),. [9] melino-camelia, G. (000). Relaiiy in spaceimes wih shor-disance srcre goerned by an obserer-independen (lanckian) lengh scale. arxi:gr-qc/0005. In. J. Mod. hys. D, 35. [0] Kowalski-Glikman, J. (005). Inrodcion o dobly special relaiiy. Lec. Noes hys., 669, 3. [] Kowalski-Glikman, J. and Smolin, L. (004). Triply special relaiiy. hys. Re. D 70, [] Wang, L. J., Kzmich,., and Dogari,. (000) Gain-assised sperlminal ligh propagaion, Nare 406, 77. [3] Brilloin, L. (960) Wae ropagaion and Grop Velociy. New York: cademic ress. [4] Kzmich,. e al (00) Signal elociy, casaliy, and qanm noise in sperlminal ligh plse propagaion. hys. Re. Le. 86, 395. [5] Edwards, W. (963). Special relaiiy in anisoropie space. m. J. hys. 3, 48. [6] Winnie, J. (970). Special relaiiy wiho one-way elociy assmpions, ar I; ar II. hilosophy of Science, 37, 8, 3. [7] Rindler, W. (98) Inrodcion o Special Relaiiy. Oford: Oford Uniersiy ress, p.. [8] Zhang, Y. Z. (997). Special Relaiiy and is Eperimenal Fondaions. Singapore: World Scienific. [9] Maingly D. (005). Modern ess of Lorenz inariance, Liing Re. Rel. 8, 5. [0] Einsein,. (935). n elemenary deriaion of he eqialence of mass and energy. Bllein of he

9 merican Mahemaical Sociey. 4, 3. [] Sachel, J. (995). Hisory of relaiiy. In L. M. Brown,. ais, and B. ippard, (eds.) Twenieh Cenry hysics, ol., pp New York: merican Insie of hysics. [] Ignaowski, W. V. (9). Eine Bemerkng z meiner rbei Einige allgemeine Bemerkngen zm Relaiiäsprinzip. hys. Zeis., 779. Ignaowski, W. V. (9). Das Relaiiäsprinzip. rch. Mah. hys. Lpz. 7, -4, and 8, 7-4. [3] Torrei, R. (983). Relaiiy and Geomery. Oford: ergamon ress. [4] Brown, H. (005). hysical Relaiiy: Spaceime srcre from a dynamical perspecie. Oford: Clarendon ress. [5] ali, W. (9). Theory of Relaiiy, Encyclopedia of Mahemaics, V9. Leipzig: Tebner. [6] Resnick, R. (968). Inrodcion o Special Relaiiy. New York: John Wiley & Sons. [7] Miller,. I. (98). lber Einsein s Special Theory of Relaiiy: Emergence (905) and Early Inerpreaion (905 9). New York: ddison-wesley. [8] Léy-Leblond, J. M. (976). One more deriaion of he Lorenz ransformaion. m. J. hys. 44, [9] Mermin, N. D. (984). Relaiiy wiho ligh. m. J. hys. 5, 9-4. [30] Chase, S. I. (993) Tachyons. Rerieed 7 December 00. [3] Gao, S. (004) Qanm collapse, consciosness and sperlminal commnicaion, Fond. hys. Le. 7(), 67. [3] Hagar,. (009). Minimal lengh in qanm graiy and he fae of Lorenz inariance. Sdies in he Hisory and hilosophy of Modern hysics 40, 59. [33] Hossenfelder, S. (00). Bonds on an energy-dependen and obserer-independen speed of ligh from iolaions of localiy. hys. Re. Le. 04, nd arxi: , arxi: , arxi: [34] Smolin, L. (00). Classical paradoes of localiy and heir possible qanm resolions in deformed special relaiiy. arxi: , arxi: melino-camelia, G. e al (00). Taming nonlocaliy in heories wih deformed oincare symmery. arxi: [35] Gao, S. (006). Qanm Moion: Uneiling he Myserios Qanm World. Bry S Edmnds: rima blishing. [36] Gao, S. (006). model of waefncion collapse in discree spaceime. In. J. Theor. hys. 45 (0), 965. [37] al,. B. (003). Nohing b relaiiy. Er. J. hys. 4, ppendi: Relaiiy wiho ligh There are many differen dedcions of he Lorenz-like ransformaions wiho resoring o he ligh poslae. Ye he assmpions hey are based on are basically he same, namely homogeneiy of space and ime, isoropy of space and he principle of relaiiy. Here I will inrodce a ery clear and simple

10 dedcion (see also [37]). Consider wo inerial frames S and S, where S moes wih a speed relaie o S and when 0 he origins of he wo frames coincide. The space-ime ransformaion eqaions in wo-dimensional space-ime can be wrien as follows: X (, ) () T (, ) () where, denoe he space and ime coordinaes in he frame S, and, denoe he space and ime coordinaes in he frame S. Now I will inoke he aboe assmpions o derie he space-ime ransformaions. () Homogeneiy of space and ime The homogeneiy of space reqires ha he lengh of a rod does no depend on is posiion in an inerial frame. Sppose here is a rod in he frame S, which ends are a posiions and ( > ). De o he homogeneiy of space, he lengh of he rod is he same when is ends are a posiions + Δ and + Δ. Correspondingly, he lengh of he rod in he frame S is also he same for hese wo siaions. Then we hae: X + Δ, ) X ( + Δ, ) X (, ) X (, ) (3) or X Diiding boh sides by ( ( + Δ, ) X (, ) X ( + Δ, ) X (, ) (4) Δ and aking he limi Δ 0, we ge: X (, ) X (, ) Since he posiions and are arbirary, he parial deriaie ms be consan. Therefore, he fncion X (, ) will be a linear fncion of. In a similar way, X (, ) is also a linear fncion of de o he homogeneiy of ime, and he same for T (, ). In conclsion, he homogeneiy of space and ime reqires ha he space-ime ransformaions are linear wih respec o boh space and ime 5. Considering ha he origins of he wo frames S and S coincide when 0, we can wrie down he linear space-ime ransformaions in a mari noaion: where C B D, B, C, D are only fncions of he relaie elociy. Frhermore, since he origin of S (6) (5) 5 The idea ha he homogeneiy of space and ime reqires space-ime ransformaions are linear can be raced back o Einsein, and was laer deeloped by more ahors (see, e.g. Terleskii 968; Léy-Leblond 976; Berzi and Gorini 969). Howeer, i can be arged ha he principle of relaiiy, ogeher wih he law of ineria, can also lead o he lineariy of space-ime ransformaions (Fock 969; Torrei 983; Brown 005). Ths he homogeneiy of space and ime may be dropped from he assmpions needed for dedce a heory of relaiiy wiho ligh.

11 moes a a speed relaie o he origin of S, i.e., 0 when, we also hae he following relaion: B (7) () Isoropy of space The isoropy of space demands ha he space-ime ransformaions do no change when he -ais is reersed, i.e., boh and change sign, and so does. pplying his limiaion o Eqaion (6) we hae 6 : B C D B C D (8) (3) rinciple of relaiiy The principle of relaiiy reqires ha he inerse space-ime ransformaions assme he same form as he original ransformaions. This means ha he ransformaions from S o S assme he same fncional forms as he ransformaions from S o S. Moreoer, he combinaion of he principle of relaiiy wih isoropy of space frher implies reciprociy (Berzi and Gorini 969; Bdden 997; Torrei 983), namely ha he speed of S relaie o S is he negaie of he speed of S relaie o S. Ths we hae: B C D Combining he condiions (8) and (9) we can ge: C D D BC B D BC C D BC D B C D (0) () B Then considering Eqaion (7) he space-ime ransformaions can be formlaed in erms of only one (9) 6 Isoropy of space plays an imporan role in he dedcion. Since isoropy of space and is resling condiion of reciprociy hold only for he sandard conenion of simlaneiy, we only dedce a heory of relaiiy wiho ligh consisen wih he sandard conenion. If simlaneiy is really a conenion (for a differen iew see Malamen 977), hen i seems ha in order o hae a heory of relaiiy wiho ligh we shold dedce he general Edwards-Winnie ransformaions for any conenion (Edwards 963; Winnie 970), no only he Lorenz-like ransformaions. B his seems o be an impossible ask, as symmeries sch as isoropy of space and reciprociy play an indispensable role in he dedcion.

12 nknown fncion, namely () or (3) Now consider a hird frame S which moes wih a speed relaie o S, and we hae: ) ( (3) The principle of relaiiy demands ha his ransformaion assmes he same form as he ransformaion from S o S, and hs he wo diagonal elemens of he mari also saisfy Eqaion (0), namely hey are eqal. Ths we hae: + + (4) or (5) Since and are arbirary, his eqaion means ha is boh sides are consans. Denoing his consan by K and considering he condiion when 0, we hae: K (6) Therefore, we dedce he final space-ime ransformaions in erms of he homogeneiy of space and ime, isoropy of space and he principle of relaiiy, namely: K K (7) The elociy addiion law can be frher dedced. Sppose he speed of he frame S relaie o S is w. Then sing Eqaion (6) and Eqaion (3), in which he firs diagonal elemen of he mari is w by definiion, we can direcly dedce he elociy addiion law, namely:

13 + w (8) + K / I can be seen ha K is an inarian speed, independen of any inerial frame. The possible ales of K can be deermined as follows. Eqaion (6) indicaes > 0 for any. Moreoer, he firs diagonal elemen of he mari in Eqaion (3) frher demands, for if < hen for some ales of and (e.g. >> ) we can ge < 0. Therefore, we hae K 0 according o Eqaion (6). When K 0 we obain he Galileo ransformaions, while when K > 0 we obain he Lorenz ransformaions. Ths he heory is he mos general one consisen wih he principle of relaiiy, which can accommodae boh Galilean and Einseinian relaiiy. B in his meaning i is no ye relaiisic in nare, as he ale of K or an inarian speed needs o be frher deermined in order o esablish is connecion wih Einsein s relaiiy. Noe ha his does no mean we need o deermine he concree ale of K sch as K / c. Wha we need o deermine is only K 0, as K and c are qaniies wih dimension and heir ales can assme he ni of nmber in principle. Cerainly we can resor o eperience, also wiho ligh o eliminae he possibiliy of K 0, and we hae more oday indeed. This, howeer, is nsaisfacory in seeral aspecs. Firs of all, we hae no dedced a heory of relaiiy wiho ligh consisen wih Einsein s relaiiy in his way. There is sill one sep lef which may be more imporan. This obiosly depars from he iniial aim of dropping he ligh poslae from special relaiiy. We hope ha by dropping he ligh poslae, we can sill dedce a heory consisen wih special relaiiy. Ne alhogh we can deermine he ale of K by eperience, here is sill one deep mysery neplained. I is why here eiss an inarian and maimm speed, independen of any inerial frame. For Galilean relaiiy here is no sch mysery, b for Einsein s relaiiy here is one. Lasly, he deerminaion of K by heoreical consideraions may lead s o a deeper ndersanding of space-ime and relaiiy, and will probably bring a frher deelopmen of special relaiiy. The eising heory of relaiiy wiho ligh is only a firs sep owards his direcion. To sm p, we hae no had a heory of relaiiy wiho ligh consisen wih Einsein s relaiiy ye. Only afer answering why here is an inarian and maimm speed and hs deermining he finieness of K by a deeper poslae can we claim we hae. I hae proided a possible answer in my paper. w