The Theory of Invariance A Perspective of Absolute Space and Time

Transcription

1 The Thery Invariane A Perspetive Abslute Spae and Time Thanh G Nuyen Massahusetts, USA thanhn@htmailm Abstrat In this artile, by usin undamental nepts in lassial mehanis, we derive equatins desribin ravitatinal red shit and Dppler eet r liht as well as equatins desribin the relatins amn mass, mmentum, and enery inludin mass-enery equivalene Althuh ur equatins are dierent than thse in Newtnian mehanis r speial relativity, they yield results that are apprximate results alulated with Newtn mehanis r speial relativity r values velity, whih are muh less than speed liht Sine the nepts in lassial mehanis are nt separated rm the perspetive abslute spae and time, this thery is named the thery invariane Keywrds: mass-enery equivalene, liht Dppler eet, ravity, red shit, baryenter 1

2 1 Intrdutin In 17th entury, lassial mehanis develped by Sir Isaa Newtn and ther natural philsphers beame an aurate thery desribin the mtin marspi bjets under the atin res Hwever, in 1905, the sientist Albert instein published a thery whih later was alled the thery speial relativity A revlutin in siene started with the perspetive relative spae and time The equatin = m derived rm speial relativity has beme amus thruhut the wrld [1, ] Is it imprtant t investiate i the amus equatin als hlds in the perspetive abslute spae and time? Can we derive the equatin = m rm nepts in lassial mehanis? I yes, what des this equatin tell us abut the nature abslute spae and time? Liht under the eet ravity Let us nsider the llwin neptual experiment: Fiure 1: Liht travel dwn rm A t B and C in unirm ravitatin A phtn with enery is emitted at a pint A in a unirm ravitatinal ield Pints B and C are psitined belw the pint A suh that AB = BC = h The phtn enery measured at pint B is The enery and the heiht h are variable amunts Hene, the rati an be written as a untin the heiht, (h), as llws: h, and ( h) Hene, h h 1 We als have h h h Therere, (h) is an expnential untin: Thus, and kh h e kh e, (1) kh e 1 () Beause the enery a liht ray is prprtinal t its requeny, quatin (1) an als be interpreted as a rati the requenies the phtn: kh e (3)

3 The Thery Invariane In real experiments, i h <<, the rati the requenies is measured as h 1 () Cmparin quatins (3) and (), we btain: k Substitutin k = / int quatins (1), (), and (3), we btain: h exp, (5) and h exp 1, (6) h exp (7) This equatin desribes the eet ravity n liht Hwever, in the Universe, there is n unirm ravitatinal ield, nly ravitatinal ields arund astrnmial bdies Fr a spherial bdy rest mass M, the requeny a liht ray emitted at a distane R rm the enter the bdy will be haned t as the liht ray mes t a distane R rm the enter the bdy: GM R R exp, (8) RR where G the ravitatinal nstant Hene, i the liht ray apprahes ininity, then its requeny will be redued t : GM exp (9) R This equatin implies that the requeny apprahes zer as R apprahes zer This nsequene indiates that i a blak hle exists, then it has n dimensins, and n event hrizn arund it 3 Mass-enery equivalene Let us nsider an bjet m that is drpped reely rm a pint A in a unirm ravitatinal ield Pints B and C are psitined belw pint A suh that AB = BC = h Fiure : Objet m all reely rm A t B and C in unirm ravitatinal ield 3

4 Call m the rest mass and m the ravitatinal mass the bjet m The ravitatinal mass m and the heiht h are variable amunts Hene, we an write the rati the masses as a untin the heiht, (h), as llws: m m h, and h m m Hene, We als have h h 1 h hh Therere, (h) is an expnential untin: Thus, m kh h e h kh m me (10) Call F the ravitatinal re that is exertin a pull n the bjet m in the ravitatinal ield : F m Frm the deinitin wrk, W, we btain: W h h Fdh 0 0 m dh Substitutin quatin (10) int the equatin abve, we btain: W h 0 m e kh dh 1 kh W m e 1 (11) k Nw, let us nsider the bjet m that deays int tw piees liht preisely when the bjet is drpped One piee emits upwards, and the ther piee emits dwnwards The upwards-emittin piee immediately hits a mirrr and relets dwnward with the ther piee Fiure 3: Objet m deay int liht preisely when drpped Cmparin quatins (6) and (11), let us pay attentin at the prtins [exp(h/ )-1] and (e kh -1) in the equatins, respetively, we btain: k

5 The Thery Invariane Cmparin quatins (6) and (11), let us pay attentin at the prtins ( ) and (m /k) in the equatins, respetively, we btain: 1 m k Substitutin k = / int the equatin abve, we btain: m (1) This equatin desribes the mass-enery equivalene r an bjet at rest Interestinly, this equivalene is exatly the same as that in speial relativity Nw, substitutin k = / int quatin (10), we btain: h m m exp (13) Substitutin quatin (1) int quatin (5), we btain: h m exp (1) Substitutin quatin (13) int quatin (1), we nw btain: m (15) quatins (13) and (1) desribe the ravitatinal mass m and the mass-enery equivalene r the bjet m when it is allin thruh pint B, respetively (See iure ) Mass-enery equivalene r an bjet and its ravitatinal mass an als be expressed in its velity These relatins will be derived in the llwin setins Dppler eet Let us nsider the llwin neptual experiment: Imaine a liht sure S and an bserver O The liht sure emits a lash liht twards the bserver O Call the requeny the lash reeived by the bserver when he is at rest with respet t the liht sure Call the requeny the lash reeived by the bserver when he is mvin at a velity v twards the liht sure The requeny and the velity v are variable amunts Hene, we an write the rati the requenies as a untin the velity, (v), as llws: v, and v Hene, v v 1 We als have v v v Therere, (v) is an expnential untin: kv v e Thus, kv e (16) In real experiments, i v <<, then the rati the requenies is measured as llws: v 1 (17) 5

6 Cmparin quatins (16) and (17), we btain: 1 k Substitutin k = ±1/ int quatin (16), we btain: v exp (18) This equatin desribes the Dppler eet r liht And beause the enery a liht ray is prprtinal t its requeny, quatin (18) an als be interpreted as v exp (19) 5 Ttal enery, ravitatinal mass, kineti enery, ptential enery, and linear mmentum 51 Ttal enery Let us nsider the llwin experiment: An bjet m is allwed t deay int tw piees" liht when an bserver is mvin twards the bjet at a velity v Call m and the rest mass and the rest enery the bjet m, respetively Applyin quatin (19) t the experiment, the bserver reeives a ttal quantity enery, 1 v v exp exp Usin quatin (1), we then btain: 1 v v m exp exp v m sh (0) This equatin desribes the ttal enery an bjet m mvin at a velity v 5 Gravitatinal mass Cmparin quatins (15) and (0), we btain: m v m sh (1) In invariane, the rest mass is dierent rm the ravitatinal mass The rest mass an bjet is unhaned The ravitatinal mass is dependent n velity and is therere an indiatr kineti enery as desribed in the llwin setin 53 Kineti enery Kineti enery is the dierene between ttal enery and rest enery: K Substitutin quatins (15) and (1) int the equatin abve, we btain: K m m Usin quatin (1), we then btain: This is the invariant kineti enery equatin It an be expanded as 6 v K m sh 1 ()

7 The Thery Invariane 6 1 v 1 v 1 v K m!! 6! Hene, r lw values velity v, this equatin apprahes the Newtnian mehanis kineti enery equatin 1 K mv r v << 5 Ptential enery Let us return t setin 3 Substitutin k = / int quatin (11), we btain: h W m exp 1 (3) By the law nservatin enery, quatin (3) als desribes the ptential enery an bjet m at rest at heiht h in a ravitatinal ield, h P m exp 1 () This is the invariant ptential enery equatin It an be expanded as 3 h 1 h 1 h P m! 3! Hene, r lw values h, this equatin apprahes the Newtnian mehanis ptential enery equatin P m h r h << In eneral, the ptential enery an bjet m at rest at a distane R 1 with respet t a distane R rm a spherial bjet M is UR1 UR P m exp 1, GM where UR, where G the ravitatinal nstant R 55 Linear mmentum Frm the experiment desribed in subsetin 51, we an als determine the ttal quantity mmentum p the tw piees liht with respet t the bserver: 1 v v p exp exp v p sinh Substitutin quatin (1) int the equatin abve, we btain: v p m sinh (5) In vetr dentatin, this equatin is written as sinhv p = m v (6) v 7

8 This is the invariant linear mmentum equatin It an be expanded as 1 v 1 v p = m v 1 3! 5! Hene, r lw values velity v, this equatin apprahes the Newtnian mehanis mmentum equatin p m v r v << Nw rm quatins (0) and (5), we btain: p m sh v sinh v p m (7) ven thuh the ttal enery and linear mmentum desribed by quatins (0) and (5) are dierent rm thse values in speial relativity, it is interestin that the ttal enery a partile rest mass m desribed by quatin (7) is exatly the same as that in speial relativity Frm quatins (0), (1), and (5), we als renize that the relatins amn the enery, linear mmentum salar, and ravitatinal mass a mvin bjet m an be desribed as llws: d dp p, and m dv dv 6 Free alls 61 Velity Applyin the law nservatin enery t the equatins ptential enery () and kineti enery (), we btain: h v m exp 1 m sh 1 v h sh exp (8) This equatin desribes the velity v an bjet drpped allin reely rm a heiht h in a ravitatinal ield The equatin an be expanded as 1 v 1 v h 1 h 1 1!!! Hene, r lw values h, this equatin apprahes the Newtnian mehanis equatin: v h r h << 6 Aeleratin Dierentiatin bth sides quatin (8) with respet t time, we btain: 1 v dv h dh sinh dt exp dt v h a sinh exp v Substitutin quatin (8) int the equatin abve, we btain: 8

9 The Thery Invariane sinh a v v sh v (9) v v a tanh (30) This equatin desribes the aeleratin an bjet whih is allin at a velity v in a ravitatin ield The equatin an be expanded as 1 v v a Hene, r lw values velity v, this equatin apprahes the Newtnian mehanis viewpint abut the relatin between aeleratin and ravitatin a r v <<, and a = r v = 0 7 Inertial mass Let us nsider an bjet m allin reely dwnward in a ravitatinal ield Call m i and m the inertial mass and the ravitatinal mass the bjet, respetively Frm the deinitin inertial mass, m i = F/a and rm the deinitin ravitatinal mass, m = F/, we btain: mi (31) m a Substitutin quatins (1) and (30) int quatin (31), we btain: sinhv mi m (3) v Hene, the equatin linear mmentum (6) an als be written as p = m v (33) In invariane, the inertial mass is a untin velity In additin, linear mmentum an be deined as the prdut inertial mass and velity 8 Baryenter and ravitatinal re Let us imaine tw astrnmial bjets M 1 and M rbitin arund eah ther Call R 1 and R the distanes rm M 1 and M t their baryenter, respetively In Newtnian mehanis, the psitin the baryenter and the ravitatinal re between M 1 and M are as llws: GM1M M1R1 M R, and F, [3] R where R = R 1 + R, and G the ravitatinal nstant Beause the thery invariane is based n lassial nepts and perspetive, invariant expressins baryenter and ravitatinal re are very similar with thse in Newtnian mehanis Hwever, ravitatinal masses are used in the expressins abve instead masses, beause, by the deinitin ravitatinal mass, ravitatinal mass is rrespndin t ravitatinal re Hene, the psitin the baryenter and the ravitatinal re are desribed by the llwin expressins: M1 R1 M R, (3) and GM1 M F, (35) R i 9

10 where v1 v M1 M1 sh, and M M sh, where v 1 and v the velities M 1 and M n their rbit, respetively (See setin 5, quatin (1)) 9 Summary The dierent masses an bjet m are its Rest mass m, whih is unhaned, sinhv Inertial mass mi m, and v Gravitatinal mass, whih an be expressed as dp v m, and m m sh dv The linear mmentum an bjet m an be expressed as d sinhv p = mi v, p, and p mv dv v The mass-enery equivalene is m The ptential enery an bjet m at rest at heiht h in a ravitatinal ield is h P m exp 1 In eneral, the ptential enery an bjet m at rest at a distane R 1 with respet t a distane R rm a spherial bjet M is UR1 UR GM P m exp 1, where UR R The kineti enery an bjet m mvin at a velity v is v K m sh 1 The ttal enery a partile m is p m The Dppler eet r liht is v exp The ravitatinal red/blue shit eet r liht is h exp A blak hle, i it exists, is a pint with n vlume and n event hrizn 10

11 The Thery Invariane The relatin between inertial mass and ravitatinal mass is mi tanhv m v The relatin between aeleratin and ravitatin is tanhv a v The psitin the baryenter between tw bjets M 1 and M is desribed as M1 R1 M R The ravitatinal re is GM1 M F R 10 Cnlusin We have studied the thery invariane, whih is based n the perspetive abslute spae and time Our analysis prdues invariant equatins whih yield results that are apprximate results alulated with either Newtnian mehanis r speial relativity In additin, we reprdue tw instein s equatins speial relativity: = m and = p + m These utmes indiate that the thery invariane an prvide a distintive view the natural wrld, and the perspetive abslute spae and time is an apprpriate perspetive in prressins understandin reality Reerenes 1 instein A, 1905, Ann Phys, 33, 639 Wrld Year Physis, 005, 3 Arthur Beiser, 1991, Physis, Addisn-Wesley, 10,